2.1 subsets of real numbers
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handoutsTRANSCRIPT
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Mathematics Division, IMSP, UPLB 1
Subsets of the Set
of Real Numbers
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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
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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
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Mathematics Division, IMSP, UPLB 4
Natural Numbers
All natural numbers are
truly natural. We find them
in nature.
Real Number System Natural numbers
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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,
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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
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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
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Mathematics Division, IMSP, UPLB 8
Whole Numbers
Real Number System Whole numbers
W= 0,1,2,3,4,5,
=N 0
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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
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Mathematics Division, IMSP, UPLB 10
Integers
Give some subsets of Z.
Real Number System Integers
Z= , 3, 2, 1,0,1,2,3,
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Mathematics Division, IMSP, UPLB 11
Even Integers
Real Number System Integers
E= , 4, 2,0,2,4,
= 2 Zk k
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Mathematics Division, IMSP, UPLB 12
Odd Integers
Real Number System Integers
O= , 5, 3, 1,1,3,5,
= 2 1 Zk k
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Mathematics Division, IMSP, UPLB 13
Time to think:
Is E = O? No Why?
Is E O? Yes Why?
Are E and O disjoint? Yes
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Mathematics Division, IMSP, UPLB 14
Negative Integers
Real Number System Negative Integers
Z = , 3, 2, 1
Non-positive Integers
Z = , 3, 2, 1,0
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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
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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
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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
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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
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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
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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
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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
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Mathematics Division, IMSP, UPLB 22
Subsets Real Number System
N 0
WZ
Zterminating or non-terminating but
repeating decimals
Q cQ
RSUMMARY
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Mathematics Division, IMSP, UPLB 23 22
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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
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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.
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Mathematics Division, IMSP, UPLB 26
Real Number Line
01 112
1
422
11 correspondence
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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
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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.
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Mathematics Division, IMSP, UPLB 29
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