Real Number System

Real Number System
What is zero?
 What is zero? Is it a number? How can the number of nothing be a
number? Is zero nothing, or is it something?
 History
 Arab and Indian scholars were the first to use zero to develop the
place-value number system that we use today. The concept of zero as
a digit in the decimal place value notation was developed in India,
presumably as early as during the Gupta period around 5th century.
 Today, zero — both as a symbol (or numeral) and a concept meaning the
absence of any quantity — allows us to perform calculus, do complicated
equations, and to have invented computers.
 When we write a number, we use only the ten numerals 0, 1, 2, 3, 4,
5, 6, 7, 8, and 9. These numerals can stand for ones, tens, hundreds,
or whatever depending on their position in the number. In order for
this to work, we have to have a way to mark an empty place in a
number, or the place values won’t come out right. This is what the
numeral “0” does.
 Is zero a number?
 The number zero obeys most of the same rules of arithmetic that
ordinary numbers do, so we call it a number. It is a rather special
number, though, because it doesn’t quite obey all the same laws as
other numbers—you can’t divide by zero, for example.
 Zero is the additive identity, because adding zero to a number does

not change the number.
Negative Numbers
 Having less than zero means that you have to add some to it just to
get it up to zero. And if you take more out of it, it will be even
further less than zero, meaning that you will have to add even more
just to get it up to zero.
 Mathematical definition of negative numbers

 For every real number n, there exists its opposite, denoted – n, such
that the sum of n and – n is zero, or
 n + (– n) = 0
5
Rational numbers
𝑎
 All numbers of the form 𝑏 ,where a and b are integers (but b cannot be zero) are
called rational numbers.
 Rational numbers include what we usually call fractions
 The bottom of the fraction is called the denominator.
 The top of the fraction is called the numerator.
 RESTRICTION: The denominator cannot be zero! (But the numerator can)
 Fractions can be numbers smaller than 1, like 1/2 or 3/4 (called proper
fractions), or they can be numbers bigger than 1 (called improper
fractions), like two-and-a-half, which we could also write as 5/2.
 All integers can also be thought of as rational numbers, with a denominator of 1.
 This means that all the previous sets of numbers (natural numbers, whole
numbers, and integers) are subsets of the rational numbers.
Natural Numbers
 The term natural numbers commonly refers to the set ℕ = { 1,2,3,…}
Integers
 The term integers refers to the set ℤ = { …,-2,-1,0,1,2,…}
 The integers can be represented as equally spaced points on a line called the
number line.
Real Numbers
 The set of all points (not just those representing integers) on the number line
represents the real numbers.
 The set of real numbers is denoted by ℝ.
 Irrational Numbers
 Irrational numbers cannot be expressed as a ratio of integers.

 As decimals they never repeat or terminate.
 Some common examples of irrational numbers:
 In a computer the real numbers can be stored only to a limited

number of figures. This is the basic difference between the ways in
which computers treat integers and real numbers, and is the reason
why the computer languages commonly used by engineers distinguish
between integer values and variables on the one hand and real
number values and variables on the other.
The Number system
 The number system is said to be ‘to base ten’ and is called the decimal
system.
 For some applications it is more convenient to use a base other than ten
 Early electronic computers used binary numbers (to base two); modern
computers use hexadecimal numbers (to base sixteen).
 An ordered set
 The real numbers have the property that they are ordered, which means that
given any two different numbers we can always say that one is greater or less
than the other. A more formal way of saying this is:
 For any two real numbers a and b, one and only one of the following three
statements is true:
 1. a is less than b, (expressed as a < b)
 2. a is equal to b, (expressed as a = b)
 3. a is greater than b, (expressed as a > b)
 For any four numbers a,b,c,d
Rules of arithmetic
 The basic arithmetical operations of addition, subtraction, multiplication and
division are performed subject to the Fundamental Rules of Arithmetic. For
any three numbers a, b and c:
➢ The commutative law of addition
a+b=b+a
➢ The commutative law of multiplication
a×b=b×a
➢ The associative law of addition
(a + b) + c = a + (b + c)
➢ The associative law of multiplication
(a × b) × c = a × (b × c)
➢ The distributive law of multiplication over addition and subtraction
 (a + b) × c = (a × c) + (b × c)
 (a − b) × c = (a × c) − (b × c)
➢ The distributive law of division over addition and subtraction

 (a + b) ÷ c = (a ÷ c) + (b ÷ c)
 (a − b) ÷ c = (a ÷ c) − (b ÷ c)
Here the brackets indicate which operation is performed first. These operations
are called binary operations because they associate with every two members of
the set of real numbers a unique third member;
 Exercise:
1.Rewrite (a + b) × (c + d) as the sum of products.
A further operation used with real numbers is that of powering.For
example a X a is written as 𝒂𝟐 , and a X a X a is written as 𝒂𝟑 .
In general product of n ,a’s where n is a positive integer is written as
𝒂𝒏 .( n is called the index or exponent)
𝒂𝒏 X 𝒂𝒎 = 𝒂𝒏𝒎
𝒂𝒏 ÷ 𝒂𝒎 = 𝒂𝒏−𝒎
(𝒂𝒏 ) 𝒎=𝒂𝒏𝒎
𝒂𝟎 = 1
The process of powering can be extended to include the fractional powers.
𝟏
𝒂 Τ𝒏 = 𝒏 𝒂
𝟏
= 𝒂−𝒎 ,(a is not equal to zero)
𝒂𝒎
Note:
 Powering operation operates on just one element and is consequently called a
unary operation.(+, −,÷,× are all binary operations-operates on two
numbers)
𝑚Τ 𝑛
 𝑎 𝑛 = ( 𝑛 𝑎)𝒎 = 𝒂𝒎
 Simplify the following.

Order of precedence in arithmetic
operations
1.Parenthesization
2.Factorial
3.Exponentiation
4.Multiplication and division
5.Addition and subtraction
 Note:
 When two operators of equal precedence are adjacent in an expression the
left-hand operation is performed first.
 Modulus and intervals
 The size of a real number x is called its modulus and is denoted by
 | x | (or sometimes by mod (x)). Thus
 Geometrically | x | is the distance of the point representing x on the

number line from the point representing zero. Similarly | x − a | is
the distance of the point representing x on the number line from that
representing a.
 The set of numbers between two numbers ,a and b defines an ‘open interval’
on the real line. This is the set,
 { x : a < x< b ,x∈ ℝ } this is usually denoted by (a,b)
 Here the inequalities a <x and x < b apply simultaneously.
 An interval that includes end points is called a ‘closed interval’ and denoted
by [a,b] = { x : a ≤x ≤ b ,x∈ ℝ } .
 Note that distance between a and b might be either a-b or b-a depending on
which one is the larger. An immediate consequence of this is that ,
 |a-b|=|b-a|

 For any two real numbers x and y :

Real Number System

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Real Number System

 Zero is the additive identity, because adding zero to a number does

 Mathematical definition of negative numbers

 Irrational numbers cannot be expressed as a ratio of integers.

 In a computer the real numbers can be stored only to a limited

➢ The distributive law of division over addition and subtraction

 Simplify the following.

 Geometrically | x | is the distance of the point representing x on the

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