These are sets of whole numbers ranging from negative infinity to positive infinity

Example  

Z = (………., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5)

Z⁺ = (1, 2, 3, 4, 5, …….)

Z^- = (……., -4, -3, -2, -1)

NB:

Z⁺∩Z^- = ∅

Z⁺∪ Z^- ∪ {0} = Z 

Z⁺ ⊂Z,  Z^- ⊂ Z

Z⁺=N, N ⊂Z

Compering integers

The negative integers closer to zero are greater than those further from zero, ex: -1 is greater than -2, -2 is greater than -3, -1 is greater than-50, etc…

The positive integers further away from zero are greater than those closer to zero, ex: 2 is greater than 1

Ex: compare the following using symbol < or >

a)-2 and -3, b)-4 and -7, c) 5 and 12

Solution:

a)-2>-3, b)-4>-7, c)5<12

Arithmetic operations on integers

1. Addition of integers

In adding integers, take note on the signs (+ or-). If all the signs are the same, add and maintain the sign

Ex: a)-4+-3=-4-3=7   b) +5+4=+(5+4) =+9=9

If the signs are different, subtract the smaller from bigger numbers in the magnitude and maintain the sign of the bigger numbers.

Ex: a)-7+3=-(7-3) =-4 This is because 7 is greater than 3   b) +8+-6=+(8-6) =+2=2 

2. Subtraction of integers

If A and B are 2 integers, then A-B=A+-B. Now, if the sign of B was positive, it becomes A+B but as it is negative, it becomes A+-B. We then apply the addition rule.

Ex: evaluate a) +3-+2, b) +3--2, c)-3-+2 d)-3--2

Solution: a) +3-+2=+3-2=+1, b) +3--2=+3+2=+5, c) -3-+2=-3-2=-5, d) -3--2=-3+2=-1

3. Multiplication and division of integers

When multiplying 2 integers, the following rules must be respected;

• Positive number * positive number = positive number    ex:4*4=16
• Negative number * negative number = positive number   ex: -4*-4=16
• If one is positive and the other negative, multiply them and leave your answer negative   ex:8*-2=-16

All the above rules also apply for division

Ex: a)20 / +5 = 4, b) -20 / -5 = 4, c) -20 / +5 = -4, d)20 / -5 = -4

Commutative properties of integers

Let M and N be 2 integers

• M + N = N + M and we say addition of integers is commutative

Ex: 4+-5=-5+4=1

• M * N = N * M and we say multiplication of integers is commutative

Ex: 3*-2=-2*3=-6

• M - N ≠ N - M and so, we say subtraction of integers is not commutative

Ex: 5-3=2 ≠ 3-5=-2

• M / N ≠ N / M and so, we say division is not commutative

Ex: 16 / 8=2 ≠ 8 / 16=1/2

NB: Division and subtraction of integers are not commutative except when M=N.