Unit 1  Objective 2  Fundamental Laws of Algebra
Fundamental Laws of Algebra
Commutative Law of Addition: a + b = b
+ a
Example: 2 + 3 =
3 + 2
5 = 5 
Associative Law of Addition: a + (b +
c) = (a + b) + c
Example: 2 + (3 +
4) = (2 + 3)+ 4
2 + 7 = 5 + 4
9 = 9 
Commutative Law of Multiplication: ab
= ba
Example: (3) (4)
= (4) (3)
12 = 12 
Associative Law of Multiplication: a
(bc) = (ab) c
Example: 2 x (3 x
4) = (2 x 3) x 4
2 x 12 = 6 x 4
24 = 24 
Distributive Law: a (b + c) = ab + ac
Example: 2 (3 +
4) = 2 (3) + 2 (4)
2 (7) = 6 + 8
14 = 14 
Basic Operations
Addition of Integers
To add two real numbers with the same sign, add the numbers and give
to the sum the sign of the original number.
Example:
1. +8 + (+7) = 15
2. 9 + (24) = 33


To add two real numbers with different signs, take the absolute
value of both numbers, subtract the smaller absolute value from the larger, and give to
the answer the sign of the number with the larger absolute value.
Example:
1. 
8 + (+5) 
8 = 8
+5 = 5 

8 > 5 so the answer is negative
8  5 = 3 so 8 + (+5) = 3 
2. 
(9) + 14 
+14 = 14
9 = 9 

14 > 9 so the answer is positive
14  9 = 5 so 14 + (9) = +5 

Subtraction of Integers
To subtract one real number from another, change the sign of the
number being subtracted and then add.
Example:
1. 
6  (9) 
= 
6 + 9 


= 
3 
2. 
13  (+7) 
= 
13 + (7) 


= 
20 

Multiplication and Division of Integers
The product or quotient of two real numbers with the same sign is
the product or quotient of their absolute values.
Example:
1. 
(8) (9) 
= 
8 x 9 


= 
8 x 9 


= 
72 
2. 
27 / (3) 
= 
27 / 3 


= 
27 / 3 


= 
9 

The product or quotient of two real numbers with different signs is
the additive inverse of the product or quotient of their absolute values.
Example:
1. 
(8) (9) 
= 
72 
2. 
81 / (3) 
= 
27 

Order of Operations
 Operations inside parenthesis are done first.
 Multiplication and division in the order in which they appear
from left to right.
 Addition and subtraction in the order in which they appear
from left to right.
Example:
1. 
6 + 10 / 2 
(do the division first) 

= 6 + 5 


= 11 

2. 
8 + 9 x 2 + 16  12 / 4 
(multiply and divide from left to
right) 

= 8 + 18 + 16  3 
(add and subtract from left to
right) 

= 26 + 16  3 


= 42  3 


= 39 

3. 
28  (26  (3  (4  3))) 
(start with the innermost
parenthesis) 

= 28  (26  (3  1)) 


= 28  (26  2) 


= 28  24 


= 4 


Identity Elements
The identity element for addition is 0: a + 0 = 0 + a = a
Example: 3 + 0 =
3 or 0 + 3 = 3 
The identity element for multiplication is 1: a x 1 = 1 x a = a
Example: 7 x 1 =
7 or 1 x (7) = 7 
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