## Directed Numbers

Directed numbers are numbers which have a direction and a size. Once a direction is chosen as positive (+), the opposite direction is taken as negative (-).

A negative number is a real number that is less than zero. Whilst a positive number is a real number greater than zero. If positive represents a movement to the right, negative represents a movement to the left.

In everyday life, if positive represents height above sea level, then negative represents depth below sea level.

In banking and finance if positive represents a deposit then negative represents a withdrawal.

Negative numbers are also used to describe values on a scale that goes below zero. Negative numbers are usually written with a minus sign in front. For example,−3 degress celsius means 3 degress celcius below zero.

−3 is pronounced "minus three" or "negative three".

Zero is neither positive nor negative.

The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. Lack of a sign in front of number also means its a positive number, e.g. 3 = +3.

Every real number other than zero is either positive or negative and the non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3…), while the positive and negative whole numbers (together with zero) are referred to as integers.

## Operations on directed numbers

Directed numbers can be added, subtracted, multiplied or divide as shown in the following examples.

#### Adding negative numbers

When adding negative numbers, you add their magnitudes and keep the answer negative.

- (-2)+(-3) = -5
- (-10)+(-8) = -18
- (-1)+(-2) = -3

**As you can see, if you add two negative numbers the answer is always negative.**

Same applies even if you add more than two numbers.

- (-1)+(-1)+(-1) = -3
- (-2)+(-2)+(-2)+(-2) = -8

**In some cases you might find (-2)+(-3) written as -2+-3 without the brackets. Don’t worry about that, its the same thing.**

*Two consecutive opposite signs resolve to a negative sign. For example (-7)+(-3) is the same as -7+-3 or (-7)-(+3) or -7-3 because consecutive opposite signs (+ and -) resolve to -.*

- -2-3 = -5
- -10-8 = -18
- -1-2 = -3

#### Subtracting negative numbers

When you subtract two negative numbers, the easiest way is to deal the consecutive negative sign formed. For example, (-2)-(-8) without brackets is -2 **—** 8. Notice those consecutive negative signs.

*Two consecutive negative signs resolve to a positive sign. This means -2–8 resolves to -2+8.*

Examples:

- (-2)-(-8) = -2+8 which when rearranged becomes 8-2 = 2.
- (-3)-(-7) = -3+7 = 7-3 = 4
- (-1)-(-1) = -1+1 = 1-1 = 0

#### Smaller number subtracting bigger number

A smaller number minus a bigger number = negative number. For example: for 3-8, just do 8-3 and make the answer negative. So 3-8 = -5.

- 2-23 (Again just do 23-2, get 21 and make it negative) = -21.
- 5-8 = -3
- 1-2 = -1

*In addition and subtraction, the "Commutative Law" say we can swap numbers over and still get the same answer, i.e. 1+2 = 2+1:*

Which means -8+5 = 5-8 = -3.

This simplifies a lot of calculations which involve long chains of directed numbers. For example:

-3+5-6-10+6–5

First resolve consecutive negative signs to get:

= -3+5-6-10+6+5

Then rearrange the numbers and start with positive numbers.

= 6+5+5-10-6-3

Add the positive numbers:

= 16-10-6-3

Add the negative numbers and subtract their total from the first number:

= 16-(10+6+3)

= 16-19

= -3

#### Multiplying and dividing directed numbers

Multiplying or dividing numbers with the same signs gives a positive answer:

- 3 × 4 = 12
- -3 × -4 = 12
- 10 ÷ 2 = 5
- -10 ÷ -2 = 5

Multiplying or dividing numbers with the different signs gives a negative answer:

- 3 × -4 = -12
- -3 × 4 = -12
- 10 ÷ -2 = -5
- -10 ÷ 2 = -5