operations with Whole Numbers

When we have two whole numbers:

- if both have the same sign, we add their absolute values and put the common sign:

3 + 4 = 7                   - 3 - 4 = - 7

- If both have different signs, we subtract the bigger absolute value minus the lower absolute value and put the sign of the number which has bigger absolute value:

3 – 7 = - 4          11 – 7 = 4

When we have more whole numbers:

1) we add the positive numbers
2) separately, we add the absolute values of the negative numbers with the negative sign
3) at the end, we add these whole numbers with different sign:

-7+3-5+12 = 3+12-7-5 = 15-12 = 3

-11+2+3-150+23+34 = 2+3+23+34-11-150 = 62-161 = -99

NOTE:  If there is a negative sign before a bracket, we change the sign of the result of the bracket (or the sign of all the numbers inside the bracket) but we don´t change it if the sign before the bracket is positive: 2. MULTIPLICATION AND DIVISION

When we have two whole numbers, we multiply (or divide) their absolute values and:

- we write a positive sign, if they have the same sign
- or we write a negative sign, if they have the opposite sign

3·6 = 18      -7·3 = -21        -6·(-4) = 24       2·(-9) = -18

21:3 = 7      21:(-3) = -7     -21:(-3) = 7      -21:(-3) = 7

When we have more than two whole numbers, we use the associative property and we multiply (or divide) in pairs:

-3·(-5)·(-11)= 15·(-11) = -165

-120:(-5):12 = 24:12 = 2

NOTE: Remember the hierarchy of the operations:

- Brackets first
- Powers and square roots
- Divide and multiply (work from left to right)

3-5·(8-3·2)+25:5 = 3-5·(8-6)+5 = 3-5·2+5 = 3-10+5 = 3+5-10 = 8-10 = -2

You can practise the multiplication and division playing this game:

3. POWERS

A power of a positive whole number is always a positive number:

35 = 243

A power of a negative whole number is either a positive number if the index is an even number, or a negative number if the index is an odd number:

(-3)3 = -27      (-3)4 = 81

Properties of powers:

- Power of a product: (a · b)n = an · bn

Example:  25 · 55 = (2 · 5)5 = 105 = 100.000

- Power of a quotient: (a : b)n = an : bn

Example: 124 : 34 = (12 : 3)4 = 44 = 256

-  Multiplication of powers with the same base:

an · am = am+n

Example: 32 · 33 = (3 · 3) . (3 · 3 · 3) = 32+3 = 35 = 243

-  Division of powers with the same base:

an : am = am-n

Example: 1225 : 1223 = 1225-23 = 122 = 144

-  Power of a power:

(am)n = am·n

Example: (23)2 = 23 · 23 = (2 · 2 · 2)·(2 · 2 · 2) = 23·2 = 26 = 64

-  NOTE:  a0 = 1  a1 = a

Exercises:

1.- Calculate:

a) -5 + (7-3) - 8·3 + 2:2 =

b) 5 - 3·[3 + 2·2 - 5] + 15 - 12:(6-8) =

c) (-2)4 =

d) (-1)1110003238 =

2.- Reduce to one power:

a) [(-2)4]3:[(-2)3]3=

b) (x7⋅x6):x11=

c) (-12)5:(-3)5=

Solutions: 1.- a) -24; b) 20; c) 16; d) 1; 2.- a) (-2)3; b) x2; c) 45