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One Variable Advanced Calculus

Kenneth Kuttler klkuttler@gmail.com

January 2, 2021
2
Contents

1 Introduction 7

2 The Real and Complex Numbers 9

2.1 Real and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Set Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Well Ordering and Archimedean Property . . . . . . . . . . . . . . . . . 19
2.8 Arithmetic of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.10 Completeness of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.11 Existence of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.13 The Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.14 Dividing Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.15 The Cauchy Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . . 34
2.16 Integer Multiples of Irrational Numbers . . . . . . . . . . . . . . . . . . 35
2.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Set Theory 39
3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 The Schroder Bernstein Theorem . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Hausdorff Maximal Theorem∗ . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Functions and Sequences 49

4.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 The Limit of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6 The Nested Interval Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.8 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.8.1 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . . . 60
4.8.2 Closed and Open Sets . . . . . . . . . . . . . . . . . . . . . . . . 61
4.8.3 Compactness and Open Coverings . . . . . . . . . . . . . . . . . 64
4.8.4 Complete Separability . . . . . . . . . . . . . . . . . . . . . . . . 65
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.10 Cauchy Sequences and Completeness . . . . . . . . . . . . . . . . . . . . 67

3
4 CONTENTS

4.10.1 Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.10.2 lim sup and lim inf . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.10.3 Shrinking Diameters . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Infinite Series of Numbers 77

5.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Absolute Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 More Tests for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.1 Convergence Because of Cancellation . . . . . . . . . . . . . . . . 84
5.4.2 Ratio And Root Tests . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Double Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Continuous Functions 93
6.1 Equivalent Formulations of Continuity . . . . . . . . . . . . . . . . . . . 98
6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 The Extreme Values Theorem . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 The Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . 103
6.5 Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.7 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.9 Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . 111
6.10 Weierstrass Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.11 Ascoli Arzela Theorem∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.12 Space Filling Continuous Curves . . . . . . . . . . . . . . . . . . . . . . 118
6.13 Tietze Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7 The Derivative 123

7.1 Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 The Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . . . 129
7.4 Continuous and Nowhere Differentiable . . . . . . . . . . . . . . . . . . 133
7.5 Finding the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.6 Local Extreme Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.8 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.10 Derivatives of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . 145
7.11 Derivatives and Limits of Sequences . . . . . . . . . . . . . . . . . . . . 147
7.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 Power Series 151

8.1 Functions Defined in Terms of Series . . . . . . . . . . . . . . . . . . . . 151
8.2 Operations on Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.3 The Special Functions of Elementary Calculus . . . . . . . . . . . . . . . 156
8.3.1 Sines and Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.3.2 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . 159
8.4 ln and logb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.5 The Complex Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.6 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
CONTENTS 5

8.8 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.8.1 Interest Compounded Continuously . . . . . . . . . . . . . . . . 169
8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.10 Multiplication of Power Series . . . . . . . . . . . . . . . . . . . . . . . . 171
8.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.12 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . 175
8.13 Some Other Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9 Integration 181
9.1 The Integral of 1700’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.2 The Riemann Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . 185
9.3 Fundamental Definitions and Properties . . . . . . . . . . . . . . . . . . 185
9.4 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . 199
9.5 Uniform Convergence and the Integral . . . . . . . . . . . . . . . . . . . 200
9.6 A Simple Procedure for Finding Integrals . . . . . . . . . . . . . . . . . 201
9.7 Stirling’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.8 Fubini’s Theorem an Introduction . . . . . . . . . . . . . . . . . . . . . 205
9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10 Improper Integrals 219

10.1 The Dirichlet Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.2 The Riemann Lebesgue Lemma and Convergence . . . . . . . . . . . . . 224
10.3 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
10.4 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

11 Functions of One Complex Variable 237

11.1 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
11.2 Cauchy Goursat, Cauchy Integral Theorem . . . . . . . . . . . . . . . . 243
11.3 The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
11.4 The Method of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11.5 Counting Zeros, Open Mapping Theorem . . . . . . . . . . . . . . . . . 256
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

12 Series and Transforms 265

12.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
12.2 Criteria for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
12.3 Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . 270
12.4 Ways of Approximating Functions . . . . . . . . . . . . . . . . . . . . . 272
12.5 Uniform Approximation with Trig. Polynomials . . . . . . . . . . . . . . 273
12.6 Mean Square Approximation . . . . . . . . . . . . . . . . . . . . . . . . 275
12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12.8 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
12.9 The Inversion of Laplace Transforms . . . . . . . . . . . . . . . . . . . . 282
12.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

13 The Generalized Riemann Integral 287

13.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . 287
13.2 Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . 295
13.3 Computing Generalized Integrals . . . . . . . . . . . . . . . . . . . . . . 297
13.4 Integrals of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
6 CONTENTS

14 The Lebesgue Integral 307

14.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
14.2 Dynkin’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
14.3 The Lebesgue Stieltjes Measures and Borel Sets . . . . . . . . . . . . . . 311
14.4 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
14.5 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
14.6 Riemann Integrals for Decreasing Functions . . . . . . . . . . . . . . . . 319
14.7 Lebesgue Integrals of Nonnegative Functions . . . . . . . . . . . . . . . 319
14.8 Nonnegative Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . 320
14.9 The Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . 322
14.10The Integral’s Righteous Algebraic Desires . . . . . . . . . . . . . . . . . 323
14.11Integrals of Real Valued Functions . . . . . . . . . . . . . . . . . . . . . 323
14.12The Vitali Covering Theorems . . . . . . . . . . . . . . . . . . . . . . . 326
14.13Differentiation of Increasing Functions . . . . . . . . . . . . . . . . . . . 330
14.14Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

15 Integration on Rough Paths∗ 339

15.1 Finite p Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
15.2 Piecewise Linear Approximation . . . . . . . . . . . . . . . . . . . . . . 343
15.3 The Young Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

A Construction of Real Numbers 353

B Classification of Real Numbers 359

B.1 Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
B.2 The Symmetric Polynomial Theorem . . . . . . . . . . . . . . . . . . . . 361
B.3 Transcendental Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Copyright c 2018, You are welcome to use this, including copying it for use in
classes or referring to it on line but not to publish it for money. I do constantly upgrade
it when I find things which could be improved.
Chapter 1

Introduction

The difference between advanced calculus and calculus is that all the theorems are
proved completely and the role of plane geometry is minimized. Instead, the notion
of completeness is of preeminent importance. Formal manipulations are of no signifi-
cance at all unless they aid in showing something significant. Routine skills involving
elementary functions and integration techniques are supposed to be mastered and have
no place in advanced calculus which deals with the fundamental issues related to exis-
tence and meaning. This is a subject which places calculus as part of mathematics and
involves proofs and definitions, not algorithms and busy work. Roughly speaking, it is
nineteenth century calculus rather than eighteenth century calculus.
An orderly development of the elementary functions is included but I assume the
reader is familiar enough with these functions to use them in problems which illustrate
some of the ideas presented. I have placed the construction of the real numbers at the
end in an appendix to conform with the historical development of analysis. Complete-
ness of the real line was used and all the classical major theorems proved long before
Dedekind and Cantor showed how to construct the real numbers from the rational
numbers. However, this could be presented earlier if desired.

7
8 CHAPTER 1. INTRODUCTION
Chapter 2

The Real and Complex

Numbers

2.1 Real and Rational Numbers

To begin with, consider the real numbers, denoted by R, as a line extending infinitely far
in both directions. In this book, the notation, ≡ indicates something is being defined.
Thus the integers are defined as

Z ≡ {· · · − 1, 0, 1, · · · } ,

the natural numbers, N ≡ {1, 2, · · · } and the rational numbers, defined as the numbers
which are the quotient of two integers.
nm o
Q≡ such that m, n ∈ Z, n 6= 0
n
are each subsets of R as indicated in the following picture.
−4 −3 −2 −1 0 1 2 3 4

1/2
As shown in the picture, 21 is half way between the number 0 and the number, 1. By
analogy, you can see where to place all the other rational numbers. It is assumed that
R has the following algebra properties, listed here as a collection of assertions called
axioms. These properties will not be proved which is why they are called axioms rather
than theorems. In general, axioms are statements which are regarded as true. Often
these are things which are “self evident” either from experience or from some sort of
intuition but this does not have to be the case.

Axiom 2.1.1 x + y = y + x, (commutative law for addition)

Axiom 2.1.2 x + 0 = x, (additive identity).

Axiom 2.1.3 For each x ∈ R, there exists −x ∈ R such that x + (−x) = 0, (existence
of additive inverse).

Axiom 2.1.4 (x + y) + z = x + (y + z) , (associative law for addition).

Axiom 2.1.5 xy = yx, (commutative law for multiplication).

Axiom 2.1.6 (xy) z = x (yz) , (associative law for multiplication).

9
10 CHAPTER 2. THE REAL AND COMPLEX NUMBERS

Axiom 2.1.7 1x = x, (multiplicative identity).

Axiom 2.1.8 For each x 6= 0, there exists x−1 such that xx−1 = 1.(existence of multi-
plicative inverse).

Axiom 2.1.9 x (y + z) = xy + xz.(distributive law).

These axioms are known as the field axioms and any set (there are many others
besides R) which has two such operations satisfying the above axioms is called a field.

and subtraction are defined in the usual way by x − y ≡ x + (−y) and x/y ≡
Division
x y −1 . It is assumed that the reader is completely familiar with these axioms in the
sense that he or she can do the usual algebraic manipulations taught in high school and
junior high algebra courses. The axioms listed above are just a careful statement of
exactly what is necessary to make the usual algebraic manipulations valid. A word of
advice regarding division and subtraction is in order here. Whenever you feel a little
confused about an algebraic expression which involves division or subtraction, think of
division as multiplication by the multiplicative inverse as just indicated and think of

subtraction as addition of the additive inverse. Thus, when you see x/y, think x y −1
and when you see x − y, think x + (−y) . In many cases the source of confusion will
disappear almost magically. The reason for this is that subtraction and division do not
satisfy the associative law. This means there is a natural ambiguity in an expression
like 6 − 3 − 4. Do you mean (6 − 3) − 4 = −1 or 6 − (3 − 4) = 6 − (−1) = 7? It
makes a difference doesn’t it? However, the so called binary operations of addition and
multiplication are associative and so no such confusion will occur. It is conventional to
simply do the operations in order of appearance reading from left to right. Thus, if you
see 6 − 3 − 4, you would normally interpret it as the first of the above alternatives. This
is no problem for English speakers, but what if you grew up speaking Hebrew or Arabic
in which you read from right to left?
In the first part of the following theorem, the claim is made that the additive inverse
and the multiplicative inverse are unique. This means that for a given number, only one
number has the property that it is an additive inverse and that, given a nonzero number,
only one number has the property that it is a multiplicative inverse. The significance
of this is that if you are wondering if a given number is the additive inverse of a given
number, all you have to do is to check and see if it acts like one.

Theorem 2.1.10 The above axioms imply the following.

1. The multiplicative inverse and additive inverses are unique.

2. 0x = 0, − (−x) = x,

3. (−1) (−1) = 1, (−1) x = −x

4. If xy = 0 then either x = 0 or y = 0.

Proof : Suppose then that x is a real number and that x + y = 0 = x + z. It is

necessary to verify y = z. From the above axioms, there exists an additive inverse, −x
for x. Therefore,
−x + 0 = (−x) + (x + y) = (−x) + (x + z)
and so by the associative law for addition,

((−x) + x) + y = ((−x) + x) + z

which implies 0 + y = 0 + z. Now by the definition of the additive identity, this implies
y = z. You should prove the multiplicative inverse is unique.
2.1. REAL AND RATIONAL NUMBERS 11

Consider 2. It is desired to verify 0x = 0. From the definition of the additive identity

and the distributive law it follows that

0x = (0 + 0) x = 0x + 0x.

From the existence of the additive inverse and the associative law it follows

0 = (−0x) + 0x = (−0x) + (0x + 0x)

= ((−0x) + 0x) + 0x = 0 + 0x = 0x

To verify the second claim in 2., it suffices to show x acts like the additive inverse of
−x in order to conclude that − (−x) = x. This is because it has just been shown that
additive inverses are unique. By the definition of additive inverse, x + (−x) = 0 and so
x = − (−x) as claimed.
To demonstrate 3.,
(−1) (1 + (−1)) = (−1) 0 = 0
and so using the definition of the multiplicative identity, and the distributive law,

(−1) + (−1) (−1) = 0.

It follows from 1. and 2. that 1 = − (−1) = (−1) (−1) . To verify (−1) x = −x, use 2.
and the distributive law to write

x + (−1) x = x (1 + (−1)) = x0 = 0.

Therefore, by the uniqueness of the additive inverse proved in 1., it follows (−1) x = −x
as claimed.
To verify 4., suppose x 6= 0. Then x−1 exists by the axiom about the existence of
multiplicative inverses. Therefore, by 2. and the associative law for multiplication,

y = x−1 x y = x−1 (xy) = x−1 0 = 0.

This proves 4. 
Recall the notion of something raised to an integer power. Thus y 2 = y × y and
−3
b = b13 etc.
Also, there are a few conventions related to the order in which operations
are per-
formed. Exponents are always done before multiplication. Thus xy 2 = x y 2 and is not
2
equal to (xy) . Division or multiplication is always done before addition or subtraction.
Thus x − y (z + w) = x − [y (z + w)] and is not equal to (x − y) (z + w) . Parentheses
are done before anything else. Be very careful of such things since they are a source of
mistakes. When you have doubts, insert parentheses to resolve the ambiguities.
Also recall summation notation.

Definition 2.1.11 Let x1 , x2 , · · · , xm be numbers. Then

m
X
xj ≡ x1 + x2 + · · · + xm .
j=1
Pm
Thus this symbol, j=1 xj means to take all the numbers, x1 , x2 , · · · , xm and add them
all together. Note the use of the j as a generic variable which takes values from 1 up
to m. This notation will be used whenever there are things which can be added, not just
numbers.

As an example of the use of this notation, you should verify the following.
P6
Example 2.1.12 k=1 (2k + 1) = 48.
12 CHAPTER 2. THE REAL AND COMPLEX NUMBERS

Pm+1 Pm
Be sure you understand why k=1 xk = k=1 xk +xm+1 . As a slight generalization
of this notation,
m
X
xj ≡ xk + · · · + xm .
j=k

It is also possible to change the variable of summation.

m
X
xj = x1 + x2 + · · · + xm
j=1

while if r is an integer, the notation requires

m+r
X
xj−r = x1 + x2 + · · · + xm
j=1+r

Pm Pm+r
and so j=1 xj = j=1+r xj−r .
Summation notation will be used throughout the book whenever it is convenient to
do so.
x y
Example 2.1.13 Add the fractions, x2 +y + x−1 .

You add these just like they were numbers. Write the first expression as (x2x(x−1)
+y)(x−1)
y (x2 +y )
and the second as (x−1)(x2 +y) . Then since these have the same common denominator,
you add them as follows.


x y x (x − 1) y x2 + y x2 − x + yx2 + y 2
2
+ = 2
+ 2
= .
x +y x−1 (x + y) (x − 1) (x − 1) (x + y) (x2 + y) (x − 1)

2.2 Exercises
1. Consider the expression x + y (x + y) − x (y − x) ≡ f (x, y) . Find f (−1, 2) .
2. Show − (ab) = (−a) b.
3. Show on the number line the effect of multiplying a number by −1.
x x−1
4. Add the fractions x2 −1 + x+1 .

2 3 4
5. Find a formula for (x + y) , (x + y) , and (x + y) . Based on what you observe
8
for these, give a formula for (x + y) .
n
6. When is it true that (x + y) = xn + y n ?
7. Find the error in the following argument. Let x = y = 1. Then xy = y 2 and so
xy − x2 = y 2 − x2 . Therefore, x (y − x) = (y − x) (y + x) . Dividing both sides by
(y − x) yields x = x + y. Now substituting in what these variables equal yields
1 = 1 + 1.
√
Find the error in the following argument. x2 + 1 = x + 1 and so letting x = 2,
8. √
5 = 3. Therefore, 5 = 9.
1 1 1 1
9. Find the error in the following. Let x = 1 and y = 2. Then 3 = x+y = x + y =
1 + 12 = 32 . Then cross multiplying, yields 2 = 9.
10. Find the error in the following argument. Let x = 3 and y = 1. Then 1 = 3 − 2 =
3 − (3 − 1) = x − y (x − y) = (x − y) (x − y) = 22 = 4.
2.3. SET NOTATION 13

xy+y
11. Find the error in the following. x = y + y = 2y. Now let x = 2 and y = 2 to
obtain 3 = 4.

12. Show the rational numbers satisfy the field axioms. You may assume the associa-
tive, commutative, and distributive laws hold for the integers.
Pn Pn
13. Show that for n a positive integer, k=0 (a + bk) = k=0 (a + b (n − k)) . Explain
why
Xn n
X
2 (a + bk) = 2a + bn = (n + 1) (2a + bn)
k=0 k=0
Pn a+(a+bn)
and so k=0 (a + bk) = (n + 1) 2 .

2.3 Set Notation

A set is just a collection of things called elements. Often these are also referred to as
points in calculus. For example {1, 2, 3, 8} would be a set consisting of the elements
1,2,3, and 8. To indicate that 3 is an element of {1, 2, 3, 8} , it is customary to write
3 ∈ {1, 2, 3, 8} . 9 ∈
/ {1, 2, 3, 8} means 9 is not an element of {1, 2, 3, 8} . Sometimes a rule
specifies a set. For example you could specify a set as all integers larger than 2. This
would be written as S = {x ∈ Z : x > 2} . This notation says: the set of all integers, x,
such that x > 2.
If A and B are sets with the property that every element of A is an element of
B, then A is a subset of B. For example, {1, 2, 3, 8} is a subset of {1, 2, 3, 4, 5, 8} , in
symbols, {1, 2, 3, 8} ⊆ {1, 2, 3, 4, 5, 8} . The same statement about the two sets may also
be written as {1, 2, 3, 4, 5, 8} ⊇ {1, 2, 3, 8}.
The union of two sets is the set consisting of everything which is contained in at least
one of the sets, A or B. As an example of the union of two sets, {1, 2, 3, 8}∪{3, 4, 7, 8} =
{1, 2, 3, 4, 7, 8} because these numbers are those which are in at least one of the two sets.
In general
A ∪ B ≡ {x : x ∈ A or x ∈ B} .
Be sure you understand that something which is in both A and B is in the union. It is
not an exclusive or.
The intersection of two sets, A and B consists of everything which is in both of the
sets. Thus {1, 2, 3, 8} ∩ {3, 4, 7, 8} = {3, 8} because 3 and 8 are those elements the two
sets have in common. In general,

A ∩ B ≡ {x : x ∈ A and x ∈ B} .

When with real numbers, [a, b] denotes the set of real numbers x, such that a ≤ x ≤ b
and [a, b) denotes the set of real numbers such that a ≤ x < b. (a, b) consists of the set
of real numbers, x such that a < x < b and (a, b] indicates the set of numbers, x such
that a < x ≤ b. [a, ∞) means the set of all numbers, x such that x ≥ a and (−∞, a]
means the set of all real numbers which are less than or equal to a. These sorts of sets
of real numbers are called intervals. The two points, a and b are called endpoints of
the interval. Other intervals such as (−∞, b) are defined by analogy to what was just
explained. In general, the curved parenthesis indicates the end point it sits next to
is not included while the square parenthesis indicates this end point is included. The
reason that there will always be a curved parenthesis next to ∞ or −∞ is that these
are not real numbers. Therefore, they cannot be included in any set of real numbers.
It is assumed that the reader is already familiar with order which is discussed in the
next section more carefully. The emphasis here is on the geometric significance of these
intervals. That is [a, b) consists of all points of the number line which are to the right
14 CHAPTER 2. THE REAL AND COMPLEX NUMBERS

of a possibly equaling a and to the left of b. In the above description, I have used the
usual description of this set in terms of order.
A special set which needs to be given a name is the empty set also called the null set,
denoted by ∅. Thus ∅ is defined as the set which has no elements in it. Mathematicians
like to say the empty set is a subset of every set. The reason they say this is that if it
were not so, there would have to exist a set A, such that ∅ has something in it which is
not in A. However, ∅ has nothing in it and so the least intellectual discomfort is achieved
by saying ∅ ⊆ A.
If A and B are two sets, A \ B denotes the set of things which are in A but not in
B. Thus
A \ B ≡ {x ∈ A : x ∈/ B} .
Set notation is used whenever convenient.

2.4 Order
The real numbers also have an order defined on them. This order may be defined
by reference to the positive real numbers, those to the right of 0 on the number line,
denoted by R+ which is assumed to satisfy the following axioms.

Axiom 2.4.1 The sum of two positive real numbers is positive.

Axiom 2.4.2 The product of two positive real numbers is positive.

Axiom 2.4.3 For a given real number x one and only one of the following alternatives
holds. Either x is positive, x = 0, or −x is positive.

Definition 2.4.4 x < y exactly when y + (−x) ≡ y − x ∈ R+ . In the usual way,

x < y is the same as y > x and x ≤ y means either x < y or x = y. The symbol ≥ is
defined similarly.

Theorem 2.4.5 The following hold for the order defined as above.

1. If x < y and y < z then x < z (Transitive law).

2. If x < y then x + z < y + z (addition to an inequality).

3. If x ≤ 0 and y ≤ 0, then xy ≥ 0.

4. If x > 0 then x−1 > 0.

5. If x < 0 then x−1 < 0.

6. If x < y then xz < yz if z > 0, (multiplication of an inequality).

7. If x < y and z < 0, then xz > zy (multiplication of an inequality).

8. Each of the above holds with > and < replaced by ≥ and ≤ respectively except for
4 and 5 in which we must also stipulate that x 6= 0.

9. For any x and y, exactly one of the following must hold. Either x = y, x < y, or
x > y (trichotomy).

Proof: First consider 1, the transitive law. Suppose x < y and y < z. Why is
x < z? In other words, why is z − x ∈ R+ ? It is because z − x = (z − y) + (y − x) and
both z − y, y − x ∈ R+ . Thus by 2.4.1 above, z − x ∈ R+ and so z > x.
2.4. ORDER 15

Next consider 2, addition to an inequality. If x < y why is x + z < y + z? it is

because

(y + z) + − (x + z) = (y + z) + (−1) (x + z)
= y + (−1) x + z + (−1) z
= y − x ∈ R+ .

Next consider 3. If x ≤ 0 and y ≤ 0, why is xy ≥ 0? First note there is nothing

to show if either x or y equal 0 so assume this is not the case. By 2.4.3 −x > 0 and
−y > 0. Therefore, by 2.4.2 and what was proved about −x = (−1) x,
2
(−x) (−y) = (−1) xy ∈ R+ .
2 2
Is (−1) = 1? If so the claim is proved. But − (−1) = (−1) and − (−1) = 1 because
−1 + 1 = 0.
Next consider 4. If x > 0 why is x−1 > 0? By 2.4.3 either x−1 = 0 or −x−1 ∈ R+ .
It can’t happen that x−1 = 0 because then you would have to have 1 = 0x and as was
shown earlier, 0x = 0. Therefore, consider the possibility that −x−1 ∈ R+ . This can’t
work either because then you would have

(−1) x−1 x = (−1) (1) = −1

and it would follow from 2.4.2 that −1 ∈ R+ . But this is impossible because if x ∈ R+ ,
then if −1 ∈ R, (−1) x = −x ∈ R+ and contradicts 2.4.3 which states that either −x or
x is in R+ but not both.
Next consider 5. If x < 0, why is x−1 < 0? As before, x−1 6= 0. If x−1 > 0, then as
before,
−x x−1 = −1 ∈ R+

which was just shown not to occur.

Next consider 6. If x < y why is xz < yz if z > 0? This follows because

yz − xz = z (y − x) ∈ R+

since both z and y − x ∈ R+ .

Next consider 7. If x < y and z < 0, why is xz > zy? This follows because

zx − zy = z (x − y) ∈ R+

by what was proved in 3.

The last two claims are obvious and left for you. 
Note that trichotomy could be stated by saying x ≤ y or y ≤ x.

x if x ≥ 0,
Definition 2.4.6 |x| ≡ −x if x < 0.

Note that |x| can be thought of as the distance between x and 0.

Theorem 2.4.7 |xy| = |x| |y| .

Proof: You can verify this by checking all available cases. Do so. 

Theorem 2.4.8 The following inequalities hold.

|x + y| ≤ |x| + |y| , ||x| − |y|| ≤ |x − y| .

Either of these inequalities may be called the triangle inequality.

16 CHAPTER 2. THE REAL AND COMPLEX NUMBERS

Proof: First note that if a, b ∈ R+ ∪ {0} then a ≤ b if and only if a2 ≤ b2 . Here

is why. Suppose a ≤ b. Then by the properties of order proved above, a2 ≤ ab ≤ b2
because b2 − ab = b (b − a) ∈ R+ ∪ {0} . Next suppose a2 ≤ b2 . If both a, b = 0 there is
nothing to show. Assume then they are not both 0. Then

b2 − a2 = (b + a) (b − a) ∈ R+ .
−1
By the above theorem on order, (a + b) ∈ R+ and so using the associative law,
−1
(a + b) ((b + a) (b − a)) = (b − a) ∈ R+

Now
2 2
|x + y| = (x + y) = x2 + 2xy + y 2
2 2 2
≤ |x| + |y| + 2 |x| |y| = (|x| + |y|)

and so the first of the inequalities follows. Note I used xy ≤ |xy| = |x| |y| which follows
from the definition.
To verify the other form of the triangle inequality, x = x − y + y so |x| ≤ |x − y| + |y|
and so |x| − |y| ≤ |x − y| = |y − x| . Now repeat the argument replacing the roles of x
and y to conclude |y| − |x| ≤ |y − x| .Therefore, ||y| − |x|| ≤ |y − x| . 

Example 2.4.9 Solve the inequality 2x + 4 ≤ x − 8

Subtract 2x from both sides to yield 4 ≤ −x − 8. Next add 8 to both sides to get
12 ≤ −x. Then multiply both sides by (−1) to obtain x ≤ −12. Alternatively, subtract
x from both sides to get x+4 ≤ −8. Then subtract 4 from both sides to obtain x ≤ −12.

Example 2.4.10 Solve the inequality (x + 1) (2x − 3) ≥ 0.

If this is to hold, either both of the factors, x + 1 and 2x − 3 are nonnegative or they
are both non-positive. The first case yields x + 1 ≥ 0 and 2x − 3 ≥ 0 so x ≥ −1 and
x ≥ 23 yielding x ≥ 23 . The second case yields x + 1 ≤ 0 and 2x − 3 ≤ 0 which implies
x ≤ −1 and x ≤ 23 . Therefore, the solution to this inequality is x ≤ −1 or x ≥ 32 .

Example 2.4.11 Solve the inequality (x) (x + 2) ≥ −4

Here the problem is to find x such that x2 + 2x + 4 ≥ 0. However, x2 + 2x + 4 =

2
(x + 1) + 3 ≥ 0 for all x. Therefore, the solution to this problem is all x ∈ R.

Example 2.4.12 Solve the inequality 2x + 4 ≤ x − 8

This is written as (−∞, −12].

Example 2.4.13 Solve the inequality (x + 1) (2x − 3) ≥ 0.

3
This was worked earlier and x ≤ −1 or x ≥ 2 was the answer. In terms of set
notation this is denoted by (−∞, −1] ∪ [ 23 , ∞).

Example 2.4.14 Solve the equation |x − 1| = 2

This will be true when x − 1 = 2 or when x − 1 = −2. Therefore, there are two
solutions to this problem, x = 3 or x = −1.

Example 2.4.15 Solve the inequality |2x − 1| < 2

From the number line, it is necessary to have 2x − 1 between −2 and 2 because

the inequality says that the distance from 2x − 1 to 0 is less than 2. Therefore, −2 <
2x − 1 < 2 and so −1/2 < x < 3/2. In other words, −1/2 < x and x < 3/2.
2.5. EXERCISES 17

Example 2.4.16 Solve the inequality |2x − 1| > 2.

This happens if 2x − 1 > 2 or if 2x − 1 < −2.
Thus the solution is x > 3/2 or
x < −1/2. Written in terms of intervals this is 23 , ∞ ∪ −∞, − 12 .

Example 2.4.17 Solve |x + 1| = |2x − 2|

There are two ways this can happen. It could be the case that x + 1 = 2x − 2 in
which case x = 3 or alternatively, x + 1 = 2 − 2x in which case x = 1/3.

Example 2.4.18 Solve |x + 1| ≤ |2x − 2|

In order to keep track of what is happening, it is a very good idea to graph the two
relations, y = |x + 1| and y = |2x − 2| on the same set of coordinate axes. This is not a
hard job. |x + 1| = x + 1 when x > −1 and |x + 1| = −1 − x when x ≤ −1. Therefore,
it is not hard to draw its graph. Similar considerations apply to the other relation.
Functions and their graphs are discussed formally later but I assume the reader has
seen these things. The result is

y = |x + 1|

1/3 3

Equality holds exactly when x = 3 or x = 31 as in the preceding example. Consider x

between 13 and 3. You can see these values of x do not solve the inequality. For example
x = 1 does not work. Therefore, 13 , 3 must be excluded. The values of x larger than 3

do not produce equality so either |x + 1| < |2x − 2| for these points or |2x − 2| < |x + 1|
for these points. Checking examples, you see the first of the two cases is the one which
holds. Therefore, [3, ∞) is included. Similar reasoning obtains (−∞, 31 ]. It follows the
solution set to this inequality is (−∞, 31 ] ∪ [3, ∞).

Example 2.4.19 Suppose ε > 0 is a given positive number. Obtain a number, δ > 0,
such that if |x − 1| < δ, then x2 − 1 < ε.

First of all, note x2 − 1 = |x − 1| |x + 1| ≤ (|x| + 1) |x − 1| . Now if |x − 1| <
1,

it follows |x| < 2 and so for |x − 1| < 1, x2 − 1 < 3 |x − 1| .Now let δ = min 1, 3ε .
This notation means to take the minimum of the two numbers, 1 and 3ε . Then if
|x − 1| < δ, x2 − 1 < 3 |x − 1| < 3 3ε = ε.

2.5 Exercises
1. Solve (3x + 2) (x − 3) ≤ 0. 6. Solve (x − 1) (2x + 1) > 2.

2. Solve (3x + 2) (x − 3) > 0. 7. Solve x2 − 2x ≤ 0.

x+2 2
3. Solve 3x−2 < 0. 8. Solve (x + 2) (x − 2) ≤ 0.
3x−4
4. Solve x+1 9. Solve ≥ 0.
x+3 < 1. x2 +2x+2

3x+9
5. Solve (x − 1) (2x + 1) ≤ 2. 10. Solve x2 +2x+1 ≥ 1.
18 CHAPTER 2. THE REAL AND COMPLEX NUMBERS

x2 +2x+1
11. Solve 3x+7 < 1. 23. Describe the set of numbers, a such
that there is no solution to |x + 1| =
12. Solve |x + 1| = |2x − 3| . 4 − |x + a| .
13. Solve |3x + 1| < 8. Give your answer
in terms of intervals on the real line. 24. Suppose 0 < a < b. Show a−1 > b−1 .

14. Sketch on the number line the solu- 25. Show that if |x − 6| < 1, then |x| <
tion to the inequality |x − 3| > 2. 7.

15. Sketch on the number line the solu- 26. Suppose |x − 8| < 2. How large can
tion to the inequality |x − 3| < 2. |x − 5| be?
√
16. Show |x| = x2 . 27. Obtain a number, δ > 0, such that if
|x − 1| < δ, then x2 − 1 < 1/10.
17. Solve |x + 2| < |3x − 3| .
18. Tell when equality holds in the trian- 28. Obtain a number, √ δ > 0, such that if
gle inequality. |x − 4| < δ, then | x − 2| < 1/10.

19. Solve |x + 2| ≤ 8 + |2x − 4| . 29. Suppose ε > 0 is a given positive

number. Obtain a number, δ >
20. Solve (x + 1) (2x − 2) x ≥ 0.
√ such that if |x − 1| < δ, then
0,
21. Solve x+3
> 1. | x − 1| < ε. Hint: This δ will de-
2x+1
pend in some way on ε. You need to
x+2
22. Solve 3x+1 > 2. tell how.

2.6 The Binomial Theorem

Consider the following problem: You have the integers Sn = {1, 2, · · · , n} and k is an
integer no larger than n. How many ways are there to fill k slots with these integers
starting from left to right if whenever an integer from Sn has been used, it cannot be
re used in any succeeding slot?
k of these slots
z }| {
, , , ,··· ,

This number is known as permutations of n things taken k at a time and is denoted

by P (n, k). It is easy to figure it out. There are n choices for the first slot. For each
choice for the fist slot, there remain n − 1 choices for the second slot. Thus there are
n (n − 1) ways to fill the first two slots. Now there remain n − 2 ways to fill the third.
Thus there are n (n − 1) (n − 2) ways to fill the first three slots. Continuing this way,
you see there are
P (n, k) = n (n − 1) (n − 2) · · · (n − k + 1)
ways to do this.
Now define for k a positive integer, k! ≡ k (k − 1) (k − 2) · · · 1, 0! ≡ 1. This is called k
n!
factorial. Thus P (k, k) = k! and you should verify that P (n, k) = (n−k)! . Now consider
the number of ways of selecting a set of k different numbers from Sn . For each set of
k numbers there
 are
 P (k, k) = k! ways of listing these numbers in order. Therefore,
n
denoting by the number of ways of selecting a set of k numbers from Sn , it must
k
be the case that  
n n!
k! = P (n, k) =
k (n − k)!
 
n n!
Therefore, = k!(n−k)! . How many ways are there to select no numbers from Sn ?
k
2.7. WELL ORDERING AND ARCHIMEDEAN PROPERTY 19

Obviously one way. Note the above formula gives the right answer in this case as well
as in all other cases due to the definition which says 0! = 1.
n
Now consider the problem of writing a formula for (x + y) where n is a positive
integer. Imagine writing it like this:
n times
z }| {
(x + y) (x + y) · · · (x + y)
Then you know the result will be sums of terms of the form ak xk y n−k . What is ak ? In
other words, how many ways can you pick x from k of the factors above
 and  y from the
n
other n − k. There are n factors so the number of ways to do it is . Therefore,
k
ak is the above formula and so this proves the following important theorem known as
the binomial theorem.
Theorem 2.6.1 The following formula holds for any n a positive integer.
n  
n
X n
(x + y) = xk y n−k .
k
k=0

2.7 Well Ordering and Archimedean Property

Definition 2.7.1 A set is well ordered if every nonempty subset S, contains a
smallest element z having the property that z ≤ x for all x ∈ S.
Axiom 2.7.2 Any set of integers larger than a given number is well ordered.
In particular, the natural numbers defined as N ≡ {1, 2, · · · } is well ordered.
The above axiom implies the principle of mathematical induction.
Theorem 2.7.3 (Mathematical induction) A set S ⊆ Z, having the property that
a ∈ S and n + 1 ∈ S whenever n ∈ S contains all integers x ∈ Z such that x ≥ a.
Proof : Let T ≡ ([a, ∞) ∩ Z) \ S. Thus T consists of all integers larger than or equal
to a which are not in S. The theorem will be proved if T = ∅. If T 6= ∅ then by the well
ordering principle, there would have to exist a smallest element of T, denoted as b. It
must be the case that b > a since by definition, a ∈ / T. Then the integer, b − 1 ≥ a and
b−1 ∈ / S because if b − 1 ∈ S, then b − 1 + 1 = b ∈ S by the assumed property of S.
Therefore, b − 1 ∈ ([a, ∞) ∩ Z) \ S = T which contradicts the choice of b as the smallest
element of T. (b − 1 is smaller.) Since a contradiction is obtained by assuming T 6= ∅, it
must be the case that T = ∅ and this says that everything in [a, ∞) ∩ Z is also in S. 
Mathematical induction is a very useful device for proving theorems about the inte-
gers.
Pn
Example 2.7.4 Prove by induction that k=1 k 2 = n(n+1)(2n+1) 6 .
By inspection, if n = 1 then the formula is true. The sum yields 1 and so does the
formula on the right. Suppose this formula is valid for some n ≥ 1 where n is an integer.
Then
n+1 n
X X 2 n (n + 1) (2n + 1) 2
k2 = k 2 + (n + 1) = + (n + 1) .
6
k=1 k=1
The step going from the first to the second equality is based on the assumption that
the formula is true for n. This is called the induction hypothesis. Now simplify the
expression in the second line,
n (n + 1) (2n + 1) 2
+ (n + 1) .
6
20 CHAPTER 2. THE REAL AND COMPLEX NUMBERS
 
n(2n+1)
This equals (n + 1) 6 + (n + 1) and

n (2n + 1) 6 (n + 1) + 2n2 + n (n + 2) (2n + 3)

+ (n + 1) = =
6 6 6
Therefore,
n+1
X (n + 1) (n + 2) (2n + 3) (n + 1) ((n + 1) + 1) (2 (n + 1) + 1)
k2 = = ,
6 6
k=1

showing the formula holds for n + 1 whenever it holds for n. This proves the formula by
mathematical induction.
1 3
Example 2.7.5 Show that for all n ∈ N, 2 · 4 · · · 2n−1
2n <
√ 1
2n+1
.

1 √1
If n = 1 this reduces to the statement that 2 < 3
which is obviously true. Suppose
then that the inequality holds for n. Then
√
1 3 2n − 1 2n + 1 1 2n + 1 2n + 1
· ··· · <√ = .
2 4 2n 2n + 2 2n + 1 2n + 2 2n + 2

The theorem will be proved if this last expression is less than √ 1 . This happens if
2n+3
and only if
 2
1 1 2n + 1
√ = > 2
2n + 3 2n + 3 (2n + 2)
2
which occurs if and only if (2n + 2) > (2n + 3) (2n + 1) and this is clearly true which
may be seen from expanding both sides. This proves the inequality.
Lets review the process just used. If S is the set of integers at least as large as 1 for
which the formula holds, the first step was to show 1 ∈ S and then that whenever n ∈ S,
it follows n + 1 ∈ S. Therefore, by the principle of mathematical induction, S contains
[1, ∞) ∩ Z, all positive integers. In doing an inductive proof of this sort, the set, S is
normally not mentioned. One just verifies the steps above. First show the thing is true
for some a ∈ Z and then verify that whenever it is true for m it follows it is also true
for m + 1. When this has been done, the theorem has been proved for all m ≥ a.

Definition 2.7.6 The Archimedean property states that whenever x ∈ R, and

a > 0, there exists n ∈ N such that na > x.

This is not hard to believe. Just look at the number line. Imagine the intervals
[0, a), [a, 2a), [2a, 3a), · · · . If x < 0, you could consider a and it would be larger than x.
If x ≥ 0, surely, it is reasonable to suppose that x would be on one of these intervals, say
[pa, (p + 1) a). This Archimedean property is quite important because it shows every
fixed real number is smaller than some integer. It also can be used to verify a very
important property of the rational numbers.

Axiom 2.7.7 R has the Archimedean property.

Theorem 2.7.8 Suppose x < y and y − x > 1. Then there exists an integer, l ∈
Z, such that x < l < y. If x is an integer, there is no integer y satisfying x < y < x + 1.

Proof: Let x be the smallest positive integer. Not surprisingly, x = 1 but this
can be proved. If x < 1 then x2 < x contradicting the assertion that x is the smallest
natural number. Therefore, 1 is the smallest natural number. This shows there is no
integer y, satisfying x < y < x + 1 since otherwise, you could subtract x and conclude
0 < y − x < 1 for some integer y − x.
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The Project Gutenberg eBook of The war myth in
United States history
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Title: The war myth in United States history

Author: C. H. Hamlin

Author of introduction, etc.: Charles F. Dole

Release date: May 13, 2024 [eBook #73623]

Language: English

Original publication: New York: The Vanguard Press, 1927

Credits: Al Haines

*** START OF THE PROJECT GUTENBERG EBOOK THE WAR MYTH

IN UNITED STATES HISTORY ***
 The War Myth in
United States History

By C. H. HAMLIN
Atlantic Christian College

With Introduction by
CHARLES F. DOLE

THE VANGUARD PRESS

NEW YORK

Copyright, 1927, by
VANGUARD PRESS, INC.

VANGUARD PRINTINGS
Fifth Printing
 PRINTED IN THE UNITED STATES OF AMERICA

To my brother,
ASKEW H. HAMLIN
A life full of promise cut short
in early manhood

CONTENTS

Chapter

Introduction

I. Patriotism and Peace

II. The Revolutionary War

III. The War of 1812

IV. The War With Mexico

V. The Civil War

VI. The War With Spain

VII. The World War

 THE WAR MYTH IN UNITED STATES
HISTORY

INTRODUCTION

Professor Hamlin's book seems to me not merely interesting but

extremely important. No man who cares for the story of his country can
afford to neglect it.

The plan of the work is suggested by the title;—the time has come to
ask soberly regarding every war in which the United States has been
engaged from the beginning, whether it had to be, and if it had to be, why?
We want to know frankly if our wars have brought us glory. It is already
easy to see that the wars of other nations, and especially of those who have
fought against us, have entailed upon them shame, cruel measures,
oppression of the poor, suppression of liberties, violation of law, destruction
of wealth and immeasurable futility. But we were told that our wars had
been different; our wars had been sacred; our sovereignty "could do no
wrong." Had we not solemnly thanked God for his help in winning every
one of our wars?

The great World War has brought mankind to a new and surprising
conclusion such as probably never before prevailed at the end of a war.
Leading people in all nations are at one in the conclusion, that no thoughtful
person in any country which entered the war knew of any adequate reason
why his government should spend the blood of its people. As Mr. Lloyd
George has said: "No one intended the war, but we all 'staggered and
stumbled' into it." It came upon the world like an epidemic of mania. It is
evident also that its coming was directly related to the prevailing fashion of
"preparedness" for war and to the fears and suspiciousness that everywhere
attended this preparedness. It had been the barbarous expectation for ages
that war must come every once in so often, as a plague comes. Was not the
world full of barbarous people, and therefore of barbarous nations?

Professor Hamlin boldly carries over all the wars of our own United
States into the broad generalization which includes the wars of other
nations. They all belong together among the old world evils, like slavery or
witchcraft, which it is our business to clear away from the earth. We
apologize for them no longer. We propose not to expect them nor prepare at
tremendous expense to suffer and die when they come; we propose rather
through simple, humane and rational measures to provide never to endure
them again.

Professor Hamlin accordingly takes up in a rapid survey and analysis

each one of the six major wars through which the larger part of our total
national expenditures have been devoured. He proceeds, like a skilful
surgeon, without passion or partisanship, with a trace of sympathy for all
groups and parties, in so far as all were alike victims of misunderstanding,
ignorance of the facts, and hereditary prejudices and delusions. Save for the
great common human characteristics which gleam out among all peoples
and on both sides in times of calamity—the patience, the heroism, the self-
sacrifice, the exceptional acts of magnanimity—he finds nothing whatever
holy in a single one of our national wars, but rather the manifestation of
every mean, cruel and cowardly trait which has ever debased human nature.

He can discover in the case of no one of these wars any evidence that
the body of the people or an intelligently informed majority in it, or even
the government, had taken pains to assure themselves either of the justice or
the necessity of going to war, or that their leaders were ever able to assign a
just and sufficient reason and purpose compelling them to resort to war.
Thus he brings to light, what every one ought by this time to know, that the
Revolutionary War, far from being undertaken by the will of a free
citizenry, was actually forced upon the American people by a small
minority in the teeth of the earnest opposition of a highly respectable
percentage of thoughtful citizens, while another large part of the colonists
was quite indifferent to the issue. Professor Hamlin also makes clear that in
all our wars, exactly as in those we usually reprobate, our people were
presently found practising the same injustices, indignities, lying
defamations, detestable acts of revenge, outrages on innocent women and
children, upon the fears of which we had hastily assumed excuse for
ourselves in rushing into war.

In all our wars we have boasted of our American ardor for liberty.
Professor Hamlin's book shows how every great war requires the most
terrible form of slavery, namely conscription, in which the individual is
stripped of the normal use of his conscience and judgment. In order to drive
men to submit to this degradation the government itself, even in the hands
of its "best" men, must resort to the employment of unscrupulous lying,
reckless propaganda in abuse of the enemy, and the suppression of truth, of
free speech and open-mindedness—in short, to a debauch of miseducation,
and a general corruption of the whole population. Once in war, it never will
do to let good be known of the enemy! War counts upon the plentiful
outpouring of passion and hatred.

The churches also are pressed in war-time to undertake the defense of

doing evil that good may come, and to strain their arguments over the verge
of hypocrisy in making the worse appear the better reason. So altogether,
"hell is let loose." The worst of it is that the lower passions, once let loose,
do not willingly return under control, but remain to haunt the earth.

Once more, Professor Hamlin shows how in each case after a war the
whole horrible storm flattens out into waste, corruption and futility. The
World War is the most colossal demonstration of this condition. If a people
thought they knew what they were fighting for, they failed to get it; the
victor proves often at last to be the vanquished. It is curious now in looking
back to the Civil War to note that the reason which chiefly persuaded
"good" and chivalrous souls to engage in it was to put an end to slavery.
This at best was dealing in the wrong way with evil, that is, overcoming it
with evil, as was abundantly proved after the war. But Mr. Lincoln would
not admit that we were at war against slavery! We were at war, as the
government held, to put down secession, whereas we had begun the
national union by a war of secession; our government would have liked at
the time of the war of 1812 to get Canada by secession or capture; we
fought with Mexico to secure the results of the secession of Texas; we
refused in 1898 to accept a peaceful method to separate Cuba from Spain
but insisted upon fighting to effect the separation; and we still keep armed
forces in the Philippine Islands against the protest of the inhabitants. Mr.
Roosevelt was quick to postulate the right of secession in the case of
Panama. As to the Great War, our President Wilson's proclamation in favor
of the natural right of small nations to secede has become one of the slogans
of mankind! As has been often remarked: "This is a queer world." Professor
Hamlin's little book is at least an easy reductio ad absurdum for war.

CHARLES F. DOLE.

Southwest Harbor, Me.,

August, 1926.

CHAPTER I

PATRIOTISM AND PEACE

For the first one hundred and thirty-five years of the history of this
republic the total expenditure of the federal treasury was approximately
$66,000,000,000. Of this total expenditure approximately $56,000,000,000
was for warfare. From 1775 to 1923 the United States Army was engaged in
no conflicts comprising about 8,600 battles and a casualty list of
approximately 1,280,000 men. (See Ganoe, History of U. S. Army, page
490.) Of course most of these conflicts were minor. This study will include
only the six major wars in which we have been participants.

A most common fallacy in the study of history is the blind acceptance of

that which has happened as inevitable in the course of events. This is a form
of collective fatalism. It reduces history to a study of the dead past with no
message for today. This view is the very opposite of democracy. Democracy
assumes that the group has control over its actions and that they are not the
result of a blind fatalism. To look upon past events as inevitable makes man
the victim of forces over which he can have no control. It makes man a
slave. This fatalism is incompatible with democracy. The democrat must
study history not to discover the forces of fate but to discover more perfect
rules of human conduct. Primarily, the study of the past should be to throw
light on the present and future, so that we might profit by the wisdom and
the mistakes of the past. But to do this we can not accept collective fatalism
as our attitude toward history.

Until the beginning of the nineteenth century the study of history was a
study of the Greeks and the Romans. It was a study of the ancients only.
Early in the nineteenth century, with the rise of nationalism especially
intensified by the French Revolution, all nations began introducing the
study of their national history in their elementary schools. The object of this
was to teach patriotism. Examine their meaning of patriotism and you find
it meant the support of the king on the throne. All texts and instruction
exalted the nation to show its superiority to others. Patriotism meant
national propaganda. With the rise of democracy patriotism began to shift to
mean the support of the group,—pro-group rather than pro-king. This was
the cause and the result of the national mind set. Patriotism became
international hatred, measured in terms of military service. This attitude
toward history caused the teaching and writing of history to be largely
national propaganda, by interpreting all the wars of a nation as defensive
with the opponent always the offensive nation.

The greatest difference between the present peace movement and

previous ones is that now among many of those who study the problem the
offensive-defensive relationship in warfare is being not only questioned but
rejected. All nations picture their side as defensive. Previous peace
movements accepted this attitude. Accordingly, when a conflict arose, these
opponents of war usually yielded to the pressure because they thought their
nation was being attacked by an aggressor. But a careful study of history
does not warrant such an idea. The effective element of the present peace
movement is based chiefly on the fact that there is no nation of "sole guilt"
in any war once the facts are studied carefully. The following study is an
attempt to show that in our wars there has not been the "sole innocence" of
the United States as opposed to the "sole guilt" of our opponents. That its
wars are defensive against an offensive enemy, is the war myth of every
country. This national bias makes it easy for the military party to
predominate and to precipitate war. Yet warfare is not popular if measured
in terms of voluntary support of the citizenship in time of war. It was hard
for the colonies to induce as many as 250,000 men to join the
Revolutionary forces out of a total population of over 3,000,000, and only a
part of the 250,000 were enlisted at any one time. In the Civil War both
sides were forced to use the draft, or the war would have collapsed. No
major war of modern times could have been fought without the draft. This
would be enough to show that warfare is not popular if judged by actual
voluntary support on the field of battle.

One often hears that warfare is a manifestation of human nature and will
be eliminated only through a long evolutionary process. But the same thing
has been said of slavery, duelling, witchcraft, and many other evils now
eliminated. Warfare is not dependent upon human nature, but upon the
human point of view, and this point of view can be altered by education,—
education which is honest, which can sift the true from the false, which
does not close its eyes to the powerful role played by economic and social
forces in the wars of the nation.

Whether there was another way out in these conflicts, whether the
results aimed at were achieved, whether the ruin and destruction which
went hand in hand with these conflicts could ever be balanced by material
acquisitions,—these are questions the reader must decide for himself. This
book simply lays the facts before him.

CHAPTER II
 THE REVOLUTIONARY WAR

In no sense is an attempt being made here to give a complete history of

the causes of the war for the independence of the United States. This is
simply a brief analysis of the ten outstanding causes and the nature of the
conflict, without defending or opposing either side in the struggle.

The common opinion in the United States regarding the American

Revolution is that it was a war waged against Great Britain in which the
American people as a whole rose up against the mother country in order to
protect themselves against unjustifiable and unbearable oppression. This is
the position taken in the Declaration of Independence, and we have always
looked upon the conflict through the eyes of the Declaration of
Independence. The thirteen colonies declared themselves free and
independent on July 2, 1776, and then on July 4, 1776 adopted the
Declaration of Independence proclaiming to the world their reasons for
declaring themselves free. Thus the Declaration of Independence was not a
declaration of independence, but a publication to the world of the causes
which led the colonies to the point of such a declaration. It was an effort to
put their side before the world and justify it. It was written by Thomas
Jefferson in the heat of a great emotion. Twenty-seven grievances were held
against Great Britain to justify the course taken by the colonies. We shall
not attempt here to study the real nature of "freedom" which is much more
than a question of national boundaries, and is even independent of national
boundaries,—but we shall accept the term in its usual narrow legal sense.

The outstanding causes of the Revolutionary War were the following:

the expulsion of the French from Canada in 1763, the attempt on the part of
Great Britain to enforce the navigation acts, the British western land policy,
the British financial legislation regarding the colonies, the stamp act of
1765, the Townshend act of 1767, the Boston "tea party" of 1773, the five
punitive acts of 1776, the general economic depression during the 70's, and
religious conflicts. Let us examine briefly these ten causes.

(1) After the French were defeated by Great Britain in 1763 and lost
Canada, the colonies did not feel the same need for protection by the
mother country as formerly. The French on the north were defeated. The
Indians gave some trouble but were not a great power to be dreaded. As a
result, the colonies felt themselves to be self supporting. Georgia was an
exception because as the youngest of the thirteen colonies it was dependent
on England for subsidies and protection from the Indians. Thus, because the
people recognized their dependence on Great Britain for protection, the
movement for independence made slower headway in Georgia.

(2) By far the most important cause of the American Revolution was the
effort on the part of George III to enforce the navigation laws of Great
Britain. It was customary then for every mother-country to regard its
colonies as trading posts. The colonies were considered necessary as the
source of raw materials for the home manufacturers and also as a market for
the surplus manufactured goods of the home country. This economic
principle was a phase of mercantilism which was the dominant economic
doctrine of the time. In harmony with this theory, Great Britain as early as
1651 began passing navigation acts requiring her colonies to trade only with
British merchants. All the export trade of the colonies had to be sent to
Great Britain, and all their imported goods had to come from Great Britain.
In addition, the ships transporting these goods had to be owned by British
subjects.

This law, however, was openly violated by the colonial merchants. They
traded with the Dutch or with any other foreigners they could. British
officials in America were bribed and co-operated in this illegal trade. The
leading people of New England at this time were merchants, and it has been
estimated that nine-tenths of these merchants were smugglers. John
Hancock, who was to become president of the First Continental Congress in
1775, was a smuggler on a great scale, and at one time was sued for
$500,000 as a penalty for smuggling. John Adams was his counsel. (See
Simons, "Social Forces in United States History," pages 61-62.) It was these
merchants of New England and especially of Boston, who were among the
leaders in the Revolution. After the close of the French and Indian War in
1763, English merchants and English business in general had to be heavily
taxed in order to pay the enormous national debt. Accordingly, pressure was
brought to bear on the British government to have the navigation laws
enforced, which would give the English the colonial trade, thus enabling
them to meet more easily the financial demands of taxation. Efforts were
then made by Great Britain to enforce these navigation laws which had
been openly violated for more than a century. Their legality had never been
questioned. It was the usual policy of all countries of that age in dealing
with their colonies. These navigation laws were no doubt unwise
interferences with trade but their legality was not questioned, as all modern
tariffs are trade barriers, which does not make their violation legal. Besides,
these laws did not entirely disregard the interests of the colonies. Great
Britain gave them a monopoly of tobacco raising, prohibiting Ireland from
raising it. Bounties or sums of money were often paid by the British
Government to the colonial producers to encourage industry. These bounties
were paid on indigo, tar, pitch, hemp, and many other industries which
Great Britain was attempting to establish in the colonies in order to keep the
empire from finding it necessary to buy them from a foreign nation. These
navigation laws aroused New England rather than the South, for it was the
commercial section of the country.

(3) Another cause of friction between the colonies and the mother
country was the British land policy proclaimed in 1763. This policy ordered
the colonial governors to grant no more land to settlers beyond a certain
western border extending south from the New England States along the
western part of New York, Pennsylvania, Virginia, North Carolina, South
Carolina, and Georgia. (See Hockett, "Political and Social History of the
United States," Vol. 1, page 115.) This line extended down just east of the
mountains and was to leave to the Indians the territory west of it. This
western land was then to be purchased from the Indians for the king. After
that the Indians would go further west and their original territory was to be
opened to settlers as soon as it was purchased. This arrangement was made
by Great Britain to avoid conflict between the Indians and the frontier
settlers. The frontier settlers, however, objected, preferring to drive the
Indians back by more ruthless methods even if it caused trouble. The
western land speculators also did not like it because they could not sell their
land until Great Britain had first pushed the Indians back. The royal
government immediately began making treaties with the Indians for the
purchase of their territory. The policy was wise and humane but the settlers
were too impatient to abide by it. (The Washington family was prominent in
these western land speculations.) A land lobby was kept in London by these
speculators in their efforts to get large grants of western land from the
crown and then to sell it off as the country became more and more settled.

(4) The next principal cause of trouble was the British financial
legislation regarding the colonies. The colonies had issued fiat money or
colonial bills of credit, which were a form of paper money. These could not
be redeemed, and immediately began to depreciate in value. Yet they were
made legal tender by the colonial legislature, so that they had to be accepted
in payment of debt. Often the colonies would buy goods from the English
and pay them with this colonial money. The southern planters were
especially active in using it to pay their debts to their British creditors. The
merchants of London soon complained of this practice. Finally, in 1764,
Great Britain prohibited all the colonies from issuing these bills of credit or
fiat money as such a procedure was considered unfair to their creditors.
This, of course, aroused great opposition from those profiting by this
currency when paying their debts. Yet no one now would defend such a
financial policy on the part of the colonies.

(5) The popular conception today is that the Stamp Act of 1765 was the
principal if not the sole cause of the American Revolution. This fact is
greatly exaggerated but it is the easiest to understand, and for that reason
has been given the chief place among the many causes of the conflict. The
Stamp Act was an act passed by Great Britain requiring the placing on all
legal documents of stamps to be sold to the colonies by Great Britain. The
usual impression is that this revenue was to go to the mother country and
was to be a continual tax upon the colonies for the sole benefit of the
crown. This impression is entirely false, however. The revenue from these
stamps was to be used to pay one-third of the expense of a colonial army of
about 10,000 men to be kept here for the defense of the colonies. Not one
penny was to go to Great Britain. Examine any elementary text on United
States history. They speak of taxing the colonies, but leave the impression
that the money was to go to Great Britain, whereas actually it was all to be
spent for the protection of the colonies against possible trouble with the
Indians and the French. This colonial army had been proposed before by the
colonies. In 1739 colonial leaders under the leadership of the Governor of
Pennsylvania had themselves proposed such an army supported by such a
tax. But at that time they had felt the danger of the French in Canada. After
the defeat of the French in 1763 this danger was no longer so threatening.
When this Stamp Act was passed in 1765 its operation was delayed for one
year in order to give the colonies an opportunity to agree among themselves
upon some other method of raising the money if they objected to the Stamp
Act. The act was repealed in 1766 because of the bitter opposition of the
colonies, who disliked a tax of any sort. "No Taxation Without
Representation" has been greatly over-emphasized. It is only half true, for it
implies that taxation with representation would have been accepted.

(6) When the colonies objected to the Stamp Act, calling it an "internal"
tax, Great Britain repealed it and in 1767 passed the Townshend Act, which
provided for a tariff on imports to the colonies. The imported goods,
however, were boycotted and Great Britain was forced to repeal the tariff on
imports in 1770. The amount of imported goods in the New England
colonies alone dropped from 1,363,000 pounds in 1768 to 504,000 pounds
in 1769. After the repeal in 1770 the imports in 1771 were doubled. Thus
the boycott was a powerful weapon in the hands of the colonies. With it the
colonies were in a position to enforce almost any demand they liked upon
Great Britain.

(7) When the Townshend duties were repealed in 1770 a tax was still
left on tea, in order to assert the right to levy such a tax. In 1773, Great
Britain allowed a tea company known as the East India Company to bring
over a large quantity of tea. This company had been given a monopoly of
the colonial tea market. When this tea arrived in Boston, on December 16,
1773, a group of men entered the ship and threw overboard the cargo valued
at about £15,000. But why was this tea destroyed? Simply because the
leaders in this act were tea merchants in Boston, whose trade would have to
compete with the newly arrived tea had it been permitted to enter the
market. The act was the destruction of private property on the part of the
participants. The more moderate element in Boston wanted the tea paid for
and the action repudiated.

(8) As a punishment for this performance, Great Britain passed the five
punitive or coercive acts of 1774. These five acts were the following: Close
the port of Boston until the tea should be paid for. Revise the charter of
Massachusetts. Send to England for trial colonial agents accused of
violence in the execution of their duties. Station soldiers in Massachusetts
to aid in the execution of law. Annex to Quebec the land between the Ohio
River and the Great Lakes. These acts were all legal. Great Britain had as
much right to demand that Boston pay for the tea destroyed as we have to
demand that a foreign power compensate our subjects for property lost there
through the mob action of its subjects.

(9) Another cause of the Revolution often overlooked was the general
economic depression both in Great Britain and the colonies following the
close of the French and Indian War in 1763. This was felt in all industries.
Depressions of this sort always create political unrest and a desire for
change in government, even though the authorities in power are in no way
responsible for the condition. This is especially true in American political
history. Presidential elections have been determined by economic
conditions having no direct bearing upon the issues involved.

(10) The tenth and last cause we shall give of the American Revolution
was the religious cause. There was a movement on foot to locate an
Episcopal bishop in the colonies. At that time all the clergy of the Episcopal
Church were ordained in England as there was no bishop here.
Consequently, all the Episcopal ministers came from abroad and they were
often mediocre, for the more efficient among them were kept in England. In
1770 there were about two hundred and fifty Episcopal clergy in the
colonies, most of whom were in Virginia. The rumor of locating a bishop
here aroused resentment in the other denominations who unanimously
opposed the plan. But the most effective religious cause of the Revolution
came from still another source. When Great Britain extended Quebec down
between the Ohio River and the Great Lakes, the Catholic Church was
made the established church of these regions, as it was in Quebec. This
greatly incensed all Protestants and "no pope no king" became one of the
slogans of the Revolution. John Adams considered this religious animosity
"as much as any other a cause" of the war for independence. Both these
attitudes on the part of the colonies were unwise. An Episcopal bishop was
badly needed here to elevate the Episcopal clergy and remove the unworthy
ministers. The prejudice against Catholics was simply folly. The Catholic
priests in the colonies unanimously supported the Revolution.
 If we examine the acts of Great Britain which brought on the
Revolution we find that they were all legal. They were all in harmony with
the spirit of the age. There was simply a general breakdown of
mercantilism. Patrick Henry especially talked about "rights as British
subjects," but there were no such rights of which the colonies were being
deprived. Had they remained in England they would have enjoyed no
privileges of which they were deprived by coming to America. Talk of this
sort made effective oratory, but was false when examined. "No Taxation
without Representation" is not a legal matter but commonplace political
philosophy. We have many other examples of taxation without
representation. The great majority of people in England were then
disfranchised yet taxed. Women were taxed before they were given the
ballot. Many people are now taxed even in those states where they are
deprived of the ballot. Phrases, as this regarding taxation, were merely
effective generalities without real meaning. The mistake of Great Britain
was not in the passage of any illegal or unusual laws for governing the
colonies, but it was in trying to rule a group of people against their will.
Such a policy invariably invites trouble.

Instead of thirteen units, as we usually regard the thirteen colonies, there

were three units differing in economic and political ideals. The coastal
plains extending from New Hampshire to Pennsylvania constituted one,
which was dominated by commercial interests. The second was the
tidewater section from Maryland to Georgia, which was primarily
agricultural and was dominated by the planters. The third unit or section
was the frontier with extreme ideas about political democracy. The first unit
was commercial and interested in trade and shipbuilding. Great mercantile
families had grown up there accumulating their wealth largely through
smuggling with the West Indies. To them the navigation laws were
especially offensive. Their chief desire was to restore the commercial
conditions before 1763, yet they bitterly opposed a withdrawal from the
British Empire, for they wanted its protection. They dominated Boston,
Newport, New York, and Philadelphia. They were Whig in opposing trade
restrictions, but Tory in opposing separation. They had no sympathy with
the political radicalism of Jefferson, Henry, and such leaders. The second
region was the tidewater region of the South. It was dominated by the
planters, many of whom were heavily in debt to British creditors. They
secured the passage of lax bankruptcy laws detrimental to non-resident
creditors. These laws, however, were vetoed by the king as were the laws
providing for colonial bills of credit. These planters felt themselves
aristocrats. Although they opposed British financial policy, they likewise
objected to the democracy of Jefferson. The third section was the frontier.
This section had often been discriminated against by the older sections in
matters of representation in the colonial assemblies, administration of
justice, and taxation. Its inhabitants were zealous for popular rights and had
no economic interests to the contrary. In domestic politics they were out of
harmony with the commercial and planter sections. Their zeal for imaginary
"rights of man" gave great impetus to the movement for independence.
Henry and Jefferson were the leaders of this section and their point of view
prevailed when the Declaration of Independence was written, the ideas of
which were shocking to the other sections.

These three sections reacted differently to the various British Acts. In

Georgia, the frontier people were pro-British because they were dependent
upon Great Britain for subsidies and protection from the Indians. The
frontier people of North Carolina were also Tory because they had a sharp
difference with the eastern part of the state. Had the frontier of all the
colonies had a similar sharp difference with the coastal plains they would
no doubt have been Tory and defeated the Revolution. The frontier of
Virginia got possession of the state and furnished such leaders as Henry and
Jefferson.

The Revolution was the American phase of an English civil war. It was
not so much a conflict between England and the colonies as between
different classes of the English people. It was struggle between liberals and
conservatives. The liberals were in control in the colonies while the
conservatives were in control in England. In both countries there was a
large and influential minority group. The thirteen colonies were a part of the
British Empire and simply seceded, as the South did in 1860.

The terms "Whig" and "Tory" are often misleading or vague when
applied to this period. Many Whigs of Great Britain, such as Burke, Fox
and Pitt, were opposed to the British policy of regulating the colonies, but
they were equally opposed to granting them independence. Many of the
American moderates were Whig in opposing the British navigation policy,
but wanted to pay for the tea destroyed in Boston. Many advocated an
imperial union to handle such questions in the future. The radicals were for
complete home rule and got control of the First Continental Congress of
1774. There was never a general uprising of the whole colonial population.
John Adams estimated that about one-third of the population were opposed
to separation. The greatest problem of the Revolutionists was to keep the
spirit of revolt alive. About 25,000 Americans enlisted in the British army.

When the radicals declared the colonies independent in 1776 many men
of property were shocked—Henry Laurens wept when he heard the
Declaration of Independence read—but there was rejoicing among the
radicals. A horse-jockey neighbor said to John Adams: "Oh! Mr. Adams,
what great things you and your colleagues have done for us! There are no
courts of justice now in the Province and I hope there never will be any."

There are many facts regarding our conduct during the Revolution
which are not pleasant to relate. For example, on June 1, 1775, Congress
passed a resolution disclaiming any intention of invading Canada. The
report of this decision was widely circulated in Canada. About four weeks
later Congress secretly made plans for the invasion of Canada that fall. The
invasion took place in September, 1775, but Canada drove the invaders
back. (See Lecky, "The American Revolution," page 215.) Is there any
difference between our invasion of Canada and the German invasion of
Belgium? Many people suspected of being Tories were terribly badly
treated. The New York legislature passed a resolution that Tories should be
"deemed guilty of treason and should suffer death." They were often hunted
by mobs, tarred and feathered, and killed. American troops set fire to the
houses of the people to plunder and rob. In fact in some sections the
colonists looked upon the British army with as much favor as the American
army. New York alone confiscated $3,600,000 worth of property belonging
to Tories, and all the states did likewise. During that entire period the Tories
were the great sufferers. It is obvious that a person had as much legal and
moral right to be a Tory as to be a Whig, provided he committed no act of
violence against society, and the great majority of Tories had committed
none. It was simply a question of difference in opinion. To punish a person
for a difference of opinion cannot of course be harmonized with democracy,
—majority rule does not mean coercion of minorities. Dictatorship of the
majority can be the worst kind of despotism. When Great Britain
recognized the independence of the colonies in 1783, one provision of the
treaty agreed to by both parties was that the Tories should be compensated
by the states for the property confiscated during the conflict. The states,
however, did nothing about it, but treated that provision as a "scrap of
paper."

Was our separation from Great Britain a wise or an unwise step? It is

impossible to answer a question of this sort with certainty. We assume that
it was wise and beneficial. But to determine that, it would be necessary to
roll history back, to let us remain a part of Great Britain, and then compare
the two conditions. It has been argued that if we had remained a part of the
British Empire the democratic spirit of the colonies would have been a great
help to the democratic element in Great Britain, that these elements co-
operating would have democratized and federated all the English-speaking
peoples, which, in turn, would have aided in democratizing the world. Such
an idea cannot be upheld with assurance, but neither can one say
dogmatically that the American Revolution resulting in our separation was
for the best. We use the terms "freedom" and "independence" in too loose a
sense when we say that we then gained our freedom or independence.
Would the South have been free and independent if it had been the winning
faction in the Civil War? Secession or the changing of national boundaries
does not give freedom. Canada is free although a part of the British
Commonwealth; Texas is free although a part of the United States.

BIBLIOGRAPHY

Faulkner, Harold Underwood—American Economic History, pages 137-

139.

Hayes, C. J. H.—Political and Social History of Modern Europe. Vol. I,

chapter 10.

Hockett, H. C.—Political and Social History of United States. Vol. I,

chapters 5, 6, 7, 8.
Lecky, E. H.—The American Revolution.

Muzzey, D. S.—The United States of America Through the Civil War. Vol.
I, chapter 2.

Schlesinger, A. M.—New Viewpoints in American History. Chapter 7.

Simons, A. M.—Social Forces in American History. Chapters 6 and 7.

CHAPTER III

THE WAR OF 1812

There were two different causes of the war with Great Britain in 1812,
and it is necessary to examine each separately. These causes were maritime
rights and land hunger.

The general European upheaval from 1789 to 1815, known as the

French Revolution, soon developed into a war between Great Britain and
Napoleon. All Europe was divided into two camps, with Great Britain and
Napoleon as the leaders on their respective sides. Almost a decade before
1812 Great Britain began issuing decrees known as Orders in Council.
These "Orders in Council," issued in the name of the king, attempted to
prohibit neutral nations from shipping goods to France. In this manner, a
blockade was proclaimed against France, and ships attempting to get
through the lines were subject to capture and confiscation.

Napoleon issued similar decrees, known as the Berlin and Milan

Decrees, declaring that any ships en route to Great Britain would be subject
to capture, for France had also blockaded Great Britain. But as neither
blockade could be fully enforced, they were both to a large degree
disregarded. Both Great Britain and Napoleon were attempting to cut off
each other's trade and not primarily trying to disregard the rights of neutrals.
All goods attempting to run these blockades were subject to capture.

The principal losers through these captures were the New England
traders, but they preferred losing occasional ships to joining in a war which
would involve them with their principal customer, Great Britain. There had
been no serious crisis since 1807, five years before war was declared.
Napoleon was then losing fast and it seemed evident that it would be only a
short while before the causes of friction would be over. The flagrant
disregard of the "rights" of neutral trade had taken place before 1807. In
1812, the solution or end of the problem was in sight. In 1810, our
registered tonnage in foreign trade was 981,019 tons, which high mark it
was not to reach again till 1847. Our foreign trade was not ruined, and the
New England merchants who sustained the loss wanted nothing done. They
were Federalists and would have preferred a war with France rather than a
war with England, because they regarded Napoleon as the real cause of all
the trouble. The Federalists were pro-British, while the Democrat-
Republicans were pro-French. Early in 1811 our minister, William Pinkney,
left London, and thus the United States was cut off from a knowledge of the
movements in England. England was attempting to avoid war with America
because such a war would naturally hurt her foreign trade and domestic
prosperity. By the spring of 1812 England was ready to revoke the Orders
in Council as soon as it could be done with dignity, but this fact was
unknown to America. On June 23, 1812, the orders were revoked. But this
was five days after the war of 1812 had been declared. England did not
know war was declared when the orders were revoked, and the United
States did not know till a good while later in the season that the orders had
been revoked. Perhaps modern cable communication would have prevented
this war.

Another source of friction lay in the impressment of seamen and sailors.

During this period Great Britain was hard pressed for men in her naval
campaign against Napoleon. Many sailors deserted English ships and came
to America because of the higher wages paid by the owners of American
ships. Every British warship anchoring in American waters would lose a
good part of its crew, who would secure positions on American ships. Great
Britain demanded the return of these deserters, who would often become
naturalized citizens. Great Britain, however, at that time regarded
citizenship as a contract between citizens and government which could not
be broken without the consent of both, disqualifying the sailor from
citizenship in the United States, without her consent. This European custom
has now disappeared, of course, and one can change citizenship at will.

When the United States refused to return these men, the British ships
would search American vessels on the high seas to see if any British sailors
were on board. This policy of impressment waned, however, after 1805,
because Napoleon had been defeated on the sea and Great Britain was not
in such great need of sailors. Impressment was not made a cause of war
until after the war had begun and President Madison had learned that the
Orders in Council had been revoked. President Madison in 1812 estimated
the number of impressments at 6,057, but the Massachusetts legislature
appointed a committee to investigate the situation, which reported that the
Madison estimate was "three or four times too large." Great Britain took the
position that the United States was acting as a harbor for her deserters from
the British navy and merchant ships, and that therefore the search was
warranted as a defensive measure.

The British "Orders in Council" prohibiting the trading of neutral

powers with France, and the British impressment of fugitive sailors from
English ships, were the maritime controversies which resulted in the War of
1812. Both policies on the part of Great Britain were adopted as necessary
measures in her conflict with Napoleon.

The New England Federalists were the people principally concerned in

the United States, but they opposed the war. War was declared by a vote of
79 to 49 in the House, and 19 to 13 in the Senate. There was open
discouragement of enlistment in New England. The Governors of
Massachusetts and Connecticut refused to honor President Madison's call
for the militia. Henry Adams estimated that the New England bankers
loaned more money to Great Britain than to the United States for war
purposes. Of the $17,000,000 in specie in the country in 1812, about
$10,000,000 was in the hands of the New England Federalists. They
subscribed less than $3,000,000 to the United States war loan. Thus,
strangely, enough, the War of 1812 was fought in spite of the protest of
those for whom it was presumably fought.

But in recent years another cause of the war and the chief cause has
been discovered. This was land hunger.

The United States entered the conflict at the insistence of the south and
west, despite the opposition of the northeastern states. The inland section
overruled the opposition of the maritime section. At that time, there was an
ardent expansionist sentiment along the whole western and southern border
looking towards the annexation of Canada and Florida, with a vaguer idea
of seizing all of the Spanish possessions of North America. Spain then
owned Florida. Spain and Great Britain were allies against Napoleon, and a
war with one was looked upon as a war with both. The belief that the
United States would some day annex Canada had existed continuously
since the Revolution. Benjamin Franklin had advocated the buying of
Canada by the United States, since we failed to take it during the
Revolution. The Continental Congress made an effort to capture Canada,
but our armies were repulsed. Washington had objected to leaving Canada
in British hands. In 1803 Governor Morris of Pennsylvania wrote that at the
time of the Constitutional Convention he knew "that all North America
must at length be annexed to us—happy indeed if the lust of dominion stop
there." This idea, however, was a vague dream till about 1810.

There had been friction in the northwest between the Americans and
British. The British retained trading posts in the northwest after they had
agreed to give them up by the treaty of 1783 recognizing the independence
of the United States. These were held to compensate the Tories for their
property confiscated during the Revolutionary War, which had not been
done. For this reason, the British held the northwest posts until 1796, when
they were given up by the Jay Treaty. All the Indian trouble in that section
was attributed to British propaganda, which incited the Indians against the
United States. The Canadian traders made friends with the Indians to get
their trade while the Americans were aggressively pushing them back from
their land. The result was that the Indian was more friendly to the British in
Canada than to the United States.
 The idea of annexing Canada was intensified after 1810 because of the
belief that the Indians were being turned against the United States by the
British. The south was almost unanimous in its demand for the annexation
of Florida, while the southwest was taking a lively interest in Mexico. This
land hunger was making its appearance rapidly, but it was several years
later that the phrase "manifest destiny" was to come into general use.

President Madison and Secretary of State James Monroe were eager to

annex Florida. Thomas Jefferson was interested in the annexation of
Canada, Florida and Cuba. Jefferson considered the acquisition of Canada
only a "question of marching," with Florida and Cuba easy prey from Spain.
These expansionists were in favor of declaring war, while the rest of the
country opposed the idea.

When Congress met in 1811, Henry Clay was elected Speaker of the
House. He was leader of the war group known as "war hawks." Clay was
the first Speaker of the House of Representatives to recognize the great
power he could exercise over legislation through his appointment of
committees. He was the first "Czar" of the House. On the Foreign Relations
Committee, Clay appointed Peter B. Porter, Chairman, Calhoun of South
Carolina, Grundy of Tennessee, Harper of New Hampshire, and Desha of
Kentucky. All these were ardent expansionists and reliable war men. They
represented the frontier section of 1812, and Clay had been chosen Speaker
by the representatives from that section. In December, 1812, while on the
Foreign Relations Committee, Porter said in discussing trouble with Great
Britain, "We could deprive her of her extensive provinces lying along our
border to the north." Grundy and Rhea, ardent expansionists from
Tennessee, agreed.

R. M. Johnson of Kentucky during the same session made the

statement, "I shall never die contented until I see her (Great Britain's)
expulsion from North America, and her territories incorporated with the
United States," and Harper of New Hampshire said in Congress: "To me,
sir, it appears that the Author of Nature has marked our limits in the South
by the Gulf of Mexico, and in the North by the regions of eternal frost."

These statements were representative of the sentiments of the members

in Congress from the western section. The Federalist Party consisted chiefly
of the mercantile and financial interests of the coast towns. They were
solidly against expansion, which would give the economic advantage to the
western section of the country.

The winter of 1811-1812 saw a great expansionist wave sweep over the
west, clamoring for the annexation of Canada. Contemporary newspapers
were filled with editorials demanding annexation. The cry came up from the
whole frontier, New Hampshire to Kentucky, to expel the British from
Canada. At a Washington's birthday dinner given at Lexington, Ky., on
February 22, 1812, the toast proposed was "Canada and our arms."
Although the frontier claimed that the British were inciting the Indians
against the United States, L. M. Hacker in "Western Land Hunger and the
War of 1812" proves that the Indian menace was greatly exaggerated, but
that land hunger was the real motive.

Randolph, of Virginia, who was opposed to the war, said in 1812 on the
floor of Congress: "Ever since the report of the Committee on Foreign
Relations came into the House, we have heard but one word—like the
whippoorwill with but one eternal monotonous tune—Canada! Canada!
Canada!"

The south and southwest were interested in the annexation of Florida

and possibly Texas. To them, a war with Great Britain meant a war with
Spain also, since the British and Spain were then in alliance.

President Madison and Secretary of State Monroe, in their eagerness to

acquire Florida, had helped a General George Mathews to instigate a
revolution in Florida. In 1812 General Mathews took American troops to
Florida, with the co-operation of the War Department and also the support
of Governor Mitchell of Georgia. This territory was held for a year,
although Congress twice refused to authorize the President to hold it.
Finally Madison was forced to repudiate the act because of the opposition
of the Federalists and the northern members of his own party. Senator
Crawford, of Georgia, was active in his support of southern expansion;
Jefferson wished to annex Cuba as a state, and Madison and Monroe were
eager to annex Florida although they were not concerned with the
appropriation of Canada.
 The interest of the southwest in Mexico was a spirited one. McCaleb, in
his book on "The Aaron Burr Conspiracy" points out that Burr simply
attempted to do in 1806 what the whole southwest was dreaming of. He was
conspiring against Spain in Mexico and not against the United States as is
usually supposed. "Lands, water-ways, and Indians" was the cry of men
desiring to drive out Spain.

In the Nashville Clarion of April 28, 1812, there appeared a long article
advocating the annexation of all America, closing with the statement:
"Where is it written in the book of fate that the American republic shall not
stretch her limits from the capes of the Chesapeake to Nootka Sound, from
the Isthmus of Panama to Hudson Bay?" The paper then editorially
commended the article to its readers and followed it up with a series of
historical and descriptive articles about Mexico.

The War of 1812 continued for two years. Troops were raised to invade
Canada but interest in the venture was slight. Many of the militia refused to
march out of American territory, as it was understood then that the militia
could not be ordered to foreign soil. The expansionists could have united to
declare war, but plans of expansion collapsed. The northern states opposed
the annexation of Florida without Canada. The troops could never take
Canada. Madison and Monroe were interested in Florida, not Canada. The
British repulsed the troops from Canada. The south had no desire to acquire
northern territory.

The War of 1812, in fact, was a complete failure from every angle. Our
troops were defeated. General Winfield Scott declared that the army officers
were "generally sunk in either sloth, ignorance, or habits of intemperate
drinking," "swaggerers, dependents, decayed gentlemen utterly unfit for any
military purpose whatever."

Muzzey in "The United States of America through the Civil War," Vol.
I, page 253, says: "The War of 1812 was a blunder. It was unnecessary,
impolitic, untimely, and rash." It was primarily the work of Henry Clay. If
the United States had been in any condition to fight, we should have been of
great aid to Napoleon who at that time was being defeated by Great Britain.
 In the peace treaty of 1814, which brought the war to a close, the causes
of the war were not mentioned. The War of 1812 was a war of paradoxes. It
was waged ostensibly in defense of maritime commercial interests, but the
merchant states themselves threatened to secede so as to stop it. The
English Orders in Council, the alleged cause of the war, were repealed five
days after war was declared and before news of its declaration reached
England. The most important battle of the war, the Battle of New Orleans,
was fought after the treaty of peace had been signed. The United States did
not get any of the desired territory; was defeated in nearly every campaign;
and the capitol was burned by the English. The land was not gained and the
rights on the sea were not granted. England never yielded the right of
impressment, which remained a diplomatic controversy as late as 1842.

In order to save its reputation, the Administration published an

"Exposition of the Causes and Character of the War," prepared by A. J.
Dallas, in which it was denied that the administration had ever tried to
acquire Canada. Madison was a great scholar but not a strong executive,
and it was the war hawks led by Clay who forced the war upon him and the
nation.

BIBLIOGRAPHY

Adams, Henry—John Randolph.

Hocker, L. M.—"Western Land Hunger and the War of 1812; A

Conjecture". Mississippi Valley Historical Review, Vol. X, pages 363-395.

Johnson, Allen—Union and Democracy. Chapter 11.

Lewis, H. J.—"A Re-analysis of the Causes of the War of 1812." American

Historical Magazine. Vol. VI, pages 306-316, 577-584.

Muzzey, D. S.—The United States of America Through the Civil War. Vol.
I, chapter 5.

Pratt, J. W.—The Expansionists of 1812.