2.1 subsets of real numbers

29
Mathematics Division, IMSP, UPLB 1 Subsets of the Set of Real Numbers

Upload: lorraine-lacuesta

Post on 11-Sep-2015

37 views

Category:

Documents


7 download

DESCRIPTION

handouts

TRANSCRIPT

  • Mathematics Division, IMSP, UPLB 1

    Subsets of the Set

    of Real Numbers

  • Mathematics Division, IMSP, UPLB 2

    Learning Objectives At the end of the lesson, you should be able

    to

    identify subsets of the set of real

    numbers

    recognize the various forms of rational

    numbers

    distinguish rational numbers from

    irrational numbers

    locate numbers on the real number line

  • Mathematics Division, IMSP, UPLB 3

    The Real Number System

    evolved over time by expanding the notion of what we mean by the word number.

    at first, number meant something you could count, like

    how many children a father sired

    how many legs an insect has

    These are called

  • Mathematics Division, IMSP, UPLB 4

    Natural Numbers

    All natural numbers are

    truly natural. We find them

    in nature.

    Real Number System Natural numbers

  • Mathematics Division, IMSP, UPLB 5

    N = the set of natural numbers,

    also called as the counting

    numbers

    Real Number System Natural numbers

    N= 1,2,3,4,5,

  • Mathematics Division, IMSP, UPLB 6

    P = the set of prime numbers, natural numbers with exactly two factors, 1 and itself What is the smallest prime number? Name some more prime numbers.

    Real Number System Natural numbers Subsets of Natural numbers

  • Mathematics Division, IMSP, UPLB 7

    C = the set of composite numbers, natural numbers that have more than 2 factors What is the smallest composite number? Name some more composite numbers. Are P and C disjoint?

    Real Number System Natural numbers Subsets of Natural numbers

  • Mathematics Division, IMSP, UPLB 8

    Whole Numbers

    Real Number System Whole numbers

    W= 0,1,2,3,4,5,

    =N 0

  • Mathematics Division, IMSP, UPLB 9

    What is N W ? W

    Are N and W disjoint? Yes

    What is W N? {0}

    Real Number System Whole numbers

    N W ? N

  • Mathematics Division, IMSP, UPLB 10

    Integers

    Give some subsets of Z.

    Real Number System Integers

    Z= , 3, 2, 1,0,1,2,3,

  • Mathematics Division, IMSP, UPLB 11

    Even Integers

    Real Number System Integers

    E= , 4, 2,0,2,4,

    = 2 Zk k

  • Mathematics Division, IMSP, UPLB 12

    Odd Integers

    Real Number System Integers

    O= , 5, 3, 1,1,3,5,

    = 2 1 Zk k

  • Mathematics Division, IMSP, UPLB 13

    Time to think:

    Is E = O? No Why?

    Is E O? Yes Why?

    Are E and O disjoint? Yes

  • Mathematics Division, IMSP, UPLB 14

    Negative Integers

    Real Number System Negative Integers

    Z = , 3, 2, 1

    Non-positive Integers

    Z = , 3, 2, 1,0

  • Mathematics Division, IMSP, UPLB 15

    Multiples of k Real Number System Negative Integers

    =

    =

    kN kx x N

    kZ kx x Z

    Example:

    3Z= 3

    , 6, 3,0,3,6,

    x x Z

  • Mathematics Division, IMSP, UPLB 16

    Rational Numbers, Q

    A rational number is a number that can be expressed as the ratio or quotient of two integers p and q where q 0.

    Q | , , 0p

    p q Z qq

    Real Number System Rational numbers

  • Mathematics Division, IMSP, UPLB 17

    Examples of Rational Numbers

    1a) 0.25

    4

    0b) 0

    7

    11c) 5.5

    2

    20d) 4

    5

    2e) 0.666...

    3

    Real Number System Rational numbers

  • Mathematics Division, IMSP, UPLB 18

    Forms of Rational Numbers

    Integers

    Fractions

    proper fraction

    improper fraction

    Decimals

    terminating

    non-terminating but repeating decimals

    Real Number System Rational numbers

  • Mathematics Division, IMSP, UPLB 19

    Irrational Numbers, Qc

    are those real numbers that can not be expressed as the ratio of two integers

    denote the set of irrational numbers as Qc (the complement of Q)

    can also be described as decimal numbers that neither repeat nor terminate

    Real Number System Irrational numbers

  • Mathematics Division, IMSP, UPLB 20

    Examples of Irrational

    Numbers Non-terminating, non-repeating

    decimals

    a) 1.01001000100001

    b) 1.414213562 2c) 3.141592653589

    d) 2.71828182845904523536

    Real Number System Irrational numbers

    e

  • Mathematics Division, IMSP, UPLB 21

    Set of Real Numbers, R

    R is the union of the

    set of rational numbers

    and the set of irrational

    numbers.

    Real Number System

  • Mathematics Division, IMSP, UPLB 22

    Subsets Real Number System

    N 0

    WZ

    Zterminating or non-terminating but

    repeating decimals

    Q cQ

    RSUMMARY

  • Mathematics Division, IMSP, UPLB 23 22

  • Mathematics Division, IMSP, UPLB 24

    Time to think!

    Real Number System

    c

    c

    c

    Given R is the universal set, determine the ff:

    1. N W 7. R Q Z

    2. W N 8. P C

    3. Z E 9. 2Z 3Z

    4. Q 10. Z O

    5. Q Q 11. O E

    6. N W 12. 0,1,2 C

  • Mathematics Division, IMSP, UPLB 25

    Real Number Line

    One-dimensional

    coordinate system

    There is a 11 correspondence

    between the set of points on a line

    and the set of real numbers.

  • Mathematics Division, IMSP, UPLB 26

    Real Number Line

    01 112

    1

    422

    11 correspondence

  • Mathematics Division, IMSP, UPLB 27

    Locating numbers in the number

    line:

    Find the following numbers in the

    number line shown below:

    1 2 2 3

    0

    )a b) c) d)

    2

    5

    3

  • Mathematics Division, IMSP, UPLB 28

    SUMMARY A real number is either rational or irrational.

    If it is a rational number, it is either an integer or a non-integer fraction.

    If it is an integer, it is either a whole number or a negative integer.

    If it is a whole number, it is either a counting number or zero.

    There is a 1-1 correspondence between the set of real numbers and the set of points on the line.

  • Mathematics Division, IMSP, UPLB 29

    FB group of Sir Arniel Roxas

    https://www.facebook.com/groups/math.arnielroxas/