1.1
Properties of Real Numbers
Subsets of Real Numbers
Natural Numbers: Numbers used for counting. Ex: 1, 2, 3, . . .
Whole Numbers: The natural numbers and zero. Ex: 0, 1, 2, 3, . . .
Integers: The natural numbers (positive integers), their opposites (negative integers), and
zero. Ex: . . . , -3, -2, -1, 0, 1, 2, 3, . . .
Rational Numbers: All numbers that can be written as quotients (fractions) of integers- each quotient
must have a nonzero denominator. Ex: 7/5, -3/2, 0, 0.3, -1.2, 9
Some rational numbers can be written as terminating decimals. Ex: 1/8 = 0.125
All other rational numbers can be written as repeating decimals. Ex: 1/3 = 0.3
Irrational Numbers: Numbers that cannot be written as quotients of integers. Ex: 2, 7,
Their decimal representations neither terminate nor repeat.
If a positive rational number is not a perfect square such as 25 or 4/9, then it's square
root is irrational.
Real Numbers
Rational Numbers
1/2, 0.3, 1, -5/4, -1.07
Irrational Numbers
- 3, 40,
Integers
. . . , -3, -2, -1, 0, 1, 2, 3, . . .
Whole Numbers
0, 1, 2, 3, . . .
Natural Numbers
1, 2, 3, . . .
Properties of Real Numbers
Opposite of Additive Inverse: Of any number a is -a.
The sum of opposites is zero.
Reciprocal or Multiplicative Invers:e:
Of any nonzero number a is a
The product of reciprocals is 1.
Properties of Real Numbers
Let a, b, and c represent real numbers.
Property
Addition
Closure
Multiplication
a + b is a real number.
Commutative
ab is a real number.
a+b=b+a
ab = ba
Associative
(a + b) + c = a + (b + c)
Identity
a + 0 = a, 0 + a = a
Inverse
a + (-a) = 0
(ab)c = a(bc)
a * 1 = a, 1 * a = a
a * a = 1, a = 0
Distributive
a(b + c) = ab + ac
Absolute Value:
The distance from zero a number is on the number line.
In math, distance is never negative.