Section 1-1
1
Math 150 Lecture Notes
Real Numbers
The natural or counting numbers are 1, 2, 3, 4,…
The whole numbers consist of the natural numbers and 0:
0, 1, 2, 3, 4,…
The integers consist of the natural numbers together with the negatives and 0:
…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …
The rational numbers are numbers that can be written as an integer divided by an integer (or a
ratio of integers). Examples: ½ -¼ 0.19 4.27 31
The irrational numbers are numbers that cannot be written as an integer divided by an integer.
Examples: 3 π
3
5 e
Properties of Real Numbers
Commutative Property
for Addition: a + b = b + a
for Multiplication: ab = ba
Associative Property
for Addition: (a + b) + c = a + (b + c)
for Multiplication: (ab)c = a(bc)
Distributive Property
a(b + c) = ab + ac or (b + c)a = ab + ac
Additive Identity
a + 0 = 0 + a = a
Subtraction is the inverse operation for addition (“undoes” addition) and is the same as adding
the negative of the number to be subtracted: a – b = a + (-b)
Properties of Negatives
1. (-1)a = -a
2. –(-a) = a
3. (-a)b = a(-b) = -(ab)
4. (-a)(-b) = ab
5. –(a + b) = -a – b
6. –(a – b) = b – a
