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    NAME: - SUBJECT: Mathematics in The Modern World COURSE AND YEAR

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    Agyao Yam Faith

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    NAME: - SUBJECT: Mathematics in The Modern World COURSE AND YEAR

    Uploaded by

    Agyao Yam Faith
    1K views4 pages

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    Mathematics-in-modern-world-9-college

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    NAME: - SUBJECT: Mathematics in The Modern World COURSE AND YEAR

    Uploaded by

    Agyao Yam Faith

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    SAINT TONIS COLLEGE, INC.

    Formerly: Kalinga Christian Learning Center


    P4 Bulanao, Tabuk City, Kalinga
    3800 Philippines

    NAME: _________________________________SUBJECT: Mathematics in the Modern World

    COURSE AND YEAR: _______________________

    I. Course Code: Math A


    II. Course Title: Mathematics in the Modern World
    III. Module #: 9
    IV. Learning Content/ Topic: Modular Arithmetic

    V. Learning Outcomes
    At the end of this module, you must be able to:
    1.describe modular arithmetic ;
    2.perform the operations on modular arithmetic; and
    3. discuss group theory in mathematics;
    VI. Introduction
    In this module . you will learn modular arithmetic.
    VII. Lesson Proper
    a. Review
    Review the past lesson on mathematics of finance.

    b. Discussion

    Time-keeping on this clock uses arithmetic modulo 12.

    In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap
    around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic
    was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
    A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-
    hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7
    + 8 = 15, but clocks "wrap around" every 12 hours. Because the hour number starts over after it reaches
    12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so
    "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock.
    The expression a b(mod m) reads as a is congruent to b modulo m and that means a and b have the
    same remainder when they are divided by m where m
    Two integers a and b are said to be congruent modulo m, where m is a natural number , if

    Example;

    STCI COLLEGE DEPARTMENT |1


    1. 29 8 (mod 3)
    a= 29 b= 8 m= 3

    = =7

    Because 7 is an integer , 29 is true congruence


    Applying alternative method.
    29
    29
    8
    Since both a= 29and b= 8 have the same remainder 2, then it is a true congruence.
    Other examples:
    14 mod 3 14= 3 4 2

    9 mod 3 9=3 3 0
    -1 mod 3 -1= 3 -1 2

    -5 mod 3 -5 = 3 -2 1
    25 mod 7 25 = 7 3 4

    Residue
    We say that a is the modulo – m residue of n when n The set of all integers
    congruent to a modulo m is called the residue class [a].
    Examples:
    1 Residue classes mod 3: (when you are dividing any integer by 3, the remainders are either 0, 1, 2 only)
    [0] = { .., -6,-3, 0, 3, 6 …}
    [1] ={ …, -5, -2,1,4,7, …}
    [2] = {…, -4, -1, 2, 5, 8}
    1. 31 1 (mod 10) because 31-1=30 is multiple of 10.

    3. 43 ( because (43-22) /7 = 21/7 =3

    Operations on Modulo
    Addition Rule
    In general, when a, b, c, and d are integers and is a positive integer such that

    a (

    b (

    the following always true : a + b (

    Example :

    17 (
    + 15 (

    =( 17+15) ( 5 +3) (
    = 32 (

    = 32 (

    Subtraction Rule

    STCI COLLEGE DEPARTMENT |2


    When a, b, c, and d are integers and is a positive integer such that
    (
    (

    the following always true : – (

    Multiplication Rule
    When a, b, c, and d are integers and is a positive integer such that
    (

    the following always true : (

    Example :
    17 (

    15 (
    = (17 15) (5 ( mod 6 )

    =255 (

    255 (

    Congruence
    Integers a and b are congruent modulo n , and write a ( , if they have the same remainder on division
    by n. There is a mathematical way of saying that all of the integers are the same as one of the
    modulo 5 residues. For instance the symbol In other words, This means in base 5, these
    integers have the same residue modulo 5: 2 (

    Group Theory
    Group theory in modern algebra , is the study of groups, which are systems consisting of a set of elements and a
    binary operation that can be applied to two elements of the set , which together, satisfy
    certain axioms.
    A group is G, * is a set of G together with (closed) binary operation * G such that the following rules are
    satisfied :
    1. Closure : If a and b are in the group then a • b is also in the group.
    2. Associativity: If a, b and c are in the group then (a• b) •c= a •(b• c).
    3. Identity: There is an element of e of the group such that for any element of the group, a • e= e • a = a .
    4. Inverses : For any element of the group there is an element a- 1 such that a• a -1 = e and a-1• a= e.
    5. Commutativity: It is said to be commutative ( or Abelian) if g * h = h * g for all g, h *G.
    That’s it . Any mathematical system that obeys the four rules is a group. The study of system that obey these four
    rules is the basis of this theory.
    The order of finite group G, denoted by , is the number of elements in G.
    The subgroup is a group H contained within a bigger one , G.
    A cyclic group is a group of all whose element are powers of a particular element a.
    Example:
    A group of real numbers , R with addition.

    1. x, y
    2. ( x+y) + z = x + (y+ z)
    VIII. EVALUATION

    STCI COLLEGE DEPARTMENT |3


    Determine if the following statements are true or false.
    1. 17 is congruent to 5 modulo 6.
    2. 24 and 14 are congruent in modulo 6.
    3. -25 4 ( mod 3)
    4. 2 5 (mod 3)
    5. 38 (

    Fill in the blanks .


    1. 55 (
    2. 406 (
    3. 27 (

    Perform as indicated.
    Given 22 12 (mod 5) and
    -1 14 (mod 5),
    Find the sum, the difference and the product of these congruences

    IX. REFLECTION
    Give atleast 2 application of modular arithmetic in real life.
    X. REFERENCE
    Mathematics in the Modern World
    By: Romeo M. Daligdig, EdD
    Copyright 2018
    Lorimar Publishing , Inc.

    Mathematics in the Modern World


    By: Merlita Castillo Medallon, Ed.D.and
    Dr. Felix M. Calubaquib

    Copyright 2018
    Mindshapers Co. , Inc

    Prepared by:

    JOSELYN G. VARGAS

    STCI COLLEGE DEPARTMENT |4

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