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Lecture 1: SETS of REAL NUMBERS P a g e | 1
College Algebra
Lecture 2: SET OF REAL NUMBERS
Rational Number
Set of real numbers which can
be expressed as a ratio or
quotient of two integers.
Irrational Numbers Consists of real numbers which
cannot be expressed as a ratio of
two integers like, and
PROPERTIES OF EQUALITY or EQUIVALENCE RELATION
EQUALITY is REFLEXIVE
If a is a real number,
then a = a.
Example:
3 = 3 and 4 = 4
EQUALITY is SYMMETRIC
If a and b are real
numbers, then a = b and
b = a
Example:
If A = bh, then bh = A
EQUALITY is TRANSITIVE
If a, b, c are real
numbers, and if a = b,
and b = c, then a = c
Example:
If x = y, and y = 3 then x =
3
ADDITION LAWof
EQUALITY
If a and b are realnumbers such that
a = b, then a + c = b + c.
Example:
If c = 3, and since a + c =
b + c, then,
a + 3 = b + 3
a = b
MULTIPLICATION LAWof
EQUALITY
If a and b are real
numbers such that a = b
then ac = bc for anynumber of c. If both
sides of the equation are
multiplied by the same
number, the equivalence
relation is retained.
Example:
If c = 4, and since ac =
bc, then,
a 4 = b 4
a = b
PROPERTIES OF REAL NUMBERS UNDER ADDITION
AND MULTIPLICATION
CLOSURE AXIOM
For real numbers a and b, there if a unique sum a + b,
and a unique product a b , both of which are real
numbers:
Addition If a and b are real numbers, then (a +
b) is a real number.
Example: 2 + 3 = 5
Multiplication If a and b are real numbers, then ab is
a real number
Example: 2 3 = 6
COMMUTATIVE RULE
The sum or product of any two real numbers a and b is
not affected by the order in which these numbers are
added or multiplied. The same sum or product is
obtained even if the order of addition or multiplication
is reversed.
Addition a + b = b + a
Example: 2 + 3 = 3 + 2
5 = 5
Multiplication a b = b a
Example: 2 3 = 3 2
6 = 6
ASSOCIATIVE RULE
The sum or product of any triple real numbers a, b and
c is not affected by the manner in which the numbers
are grouped for addition or multiplication.
Addition
(a + b) + c = a + (b + c)
Example: (4 + 5) + 6 = 4 + (5 + 6)
9 + 6 = 4 + 11
15 = 15
REAL NUMBERS
RATIONALNUMBERS
IRRATIONALNUMBERS
Integers Fractions
Common Fraction
Decimal Fractions
Negative
Zero
Positive
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Lecture 1: SETS of REAL NUMBERS P a g e | 2
Multiplication
(a b) c = a (b c)
Example: (2 3) 4= 2 (3 4)
6 4 = 2 12
24 = 24
DISTRIBUTIVE RULE
Multiplication is distributive over addition. This axiom
changes the product of two factors into a sum of two
terms.
a (b + c) = ab + ac
Example: 2 (4 + 6) = (2)(4) + (2)(6)
= 8 + 12
= 20
IDENTITY PROPERTY
Addition
a + 0 = 0 + a = a
By the definition of the identity
element, the real number, 0, is the
additive identityelementof the set of
real numbers.
Example: 3 + 0 = 0 + 3 = 3
Multiplication
a 1 = 1 a = a
The real number, 1, is the
multiplicative identityelementof the
set of real numbers.
Example: 3 1 = 1 3 = 3
INVERSE PROPERTY
Addition
a + (-a) = (-a) + a = 0
The numbera is called the additive
inverseor the negative of a.
Example: 2 + (-2) = (-2) + 2 = 0
Multiplication
The numberis called the
multiplicative inverse or the reciprocal
of a.
Example:
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