Natural Numbers and Integers

Subtraction
The use of parenthesis
Multiplication
The commutative property
The associative property
The distributive property of multiplication over addition and subtraction
Division
ab = ca and b are terms, and c is the sum.
To add integers having the same sign, keep the same sign and add the absolute value of each number.
To add integers with different signs, keep the sign of the number with larger absolute value and subtract smaller absolute value from the larger.
 The additive inverse or opposite of a is - a, so that a + (- a) = 0, and  a + 0 = a. The commutative property of addition, a + b = b + a. The associative property of addition, (a + b) + c = a + ( b + c).
Subtraction
a - b = ca - minuendb - subtrahend, and  c - difference
 Subtract an integer by adding its opposite, so that - (+ a) = - a,    - (- a) = + a
 Examples: 5 -  7 =  5 + (- 7) = - 2 - 1 -  8 =  - 1 - ( + 8) = - 1+ (- 8) = - 9 4 - ( - 3) =  4 + ( + 3) = 7 - 5 -  (- 9) =  - 5 + (+ 9) = - 5+ 9 = 4
The use of parenthesis:
 (a + b) + c = a + b + c (- 5 + 3) + 6 = - 5 + 3 + 6 = 4 a + (b + c) = a + b + c - 7 + (- 3 + 8) = - 7 - 3 + 8 = - 2 a - ( b + c) = a - b - c 6 - ( 5 - 12)  = 6 -  5 + 12 = 13 - (a + b) + c = - a - b + c - (- 2 + 7) + 3 = 2 - 7 + 3 = - 2
Multiplication
a · b = c,   a and  b are the factors  c is the product
The properties and rules:
 The commutative property: a · b = b · a 3 · (- 7) = - 7 · 3 = - 21 The associative property: ( a · b ) · c = a · ( b · c ) ( 2 · 5 ) · 7 = 2 · ( 5 · 7 ) = 70
 a · 0 = 0 7 · 0 = 0 a = 1 · a  and  - a = (- 1) · a - 18 = (- 1) · 18 (- a ) · (- b ) = a · b (- 3) · (- 4) = 12 (- a ) · b = a · (- b ) = - a · b 3 · (- 4) = -12
The distributive property of multiplication over addition and subtraction:
 (a + b) · c = a · c + b · c (- 3 + 5) · 6 = (- 3) · 6 + 5 · 6 = - 18 + 30 = 12 (a - b) · c = a · c - b · c ( 7 -  4) · (- 3) = 7 · (- 3) - 4 · (- 3) = - 21+ 12 = - 9 a · c + b · c = (a + b) · c (- 5 ) · 3 + 2 · 3 =  (- 5 + 2) · 3 = (- 3) · 3 = - 9 a · c - b · c = (a - b) · c 9 · (- 6) - 2 · (- 6) = (9 - 2) · (- 6) = 7 · (- 6) = - 42
 Examples: (- 7 + 4) · (- 2) = (- 7) · (- 2) + 4 · (- 2) = 14 -  8 = 6       or        (- 3) · (- 2) = 6
 - 9 · a + 5 · a = (- 9 + 5) · a = - 4a 3 · (1 - a) + 5 = 3 - 3 · a + 5 = 8 - 3a
Division
a ¸ b = c,   a is the dividend, b is the divisor and c is the quotient
 Basic identities: 0 ¸ a = 0   and   a ¸ 1 =  a 0 ¸ 5 = 0,     - 3 ¸ 1 =  - 3
If both the dividend and divisor signs are the same the quotient will be positive, if they are different, the quotient will be negative.
 (- a ) ¸ (- b ) = a ¸ b - 28 ¸ (- 4) = 28 ¸ 4 = 7 - a ¸ b = (- a ) ¸ b = - (a ¸ b) 45 ¸ (- 15) = - 45 ¸ 15 = - 3
Division is neither commutative nor associative, thus
 a ¸ b  is not the same as  b ¸ a (36 ¸ 6) ¸2 = 6 ¸ 2 = 3,  while  36 ¸(6 ¸ 2)= 36 ¸ 3 = 12 (a ¸ b) ¸ c  is not the same as  a ¸(b ¸ c) 18 ¸ 6 ¸ 3 = 3 ¸ 3 = 1,  while  18 ¸(6 ¸ 3) = 18 ¸ 2 = 9
Beginning Algebra Contents A