Directed Numbers

Lesson

Numbers can be used to measure quantities that are found in everyday life, and in some situations we need to use negative numbers.

For example, if a company makes a profit of $\$5000$$5000 we can represent this as a positive number. But if a company makes a loss of $\$5000$$5000 we can represent this as a negative number.

We can represent positive and negative numbers on the number line:

Any number to the right of $0$0 is considered to be positive, and gets bigger as you move further to the right.

Any number to the left of $0$0 is considered negative, and gets smaller as you move further to the left.

The plus or minus sign in front of a number tells us whether it is to the right or left of $0$0.

a) $-2$−2 is $2$2 to the left of $0$0.

b) $+2$+2 is $2$2 to the right of $0$0, but we just write this as $2$2.

We use the term directed number to mean a number that has both direction and size.

For example, the number $-2$−2 is:

- $2$2 units away from zero, and
- to the left of $0$0.

'To the left' is the direction.

Let's have a look at some words that represent directed numbers.

Words that indicate growth or getting bigger are represented by positive numbers in maths.

This includes words like:

- "rise,"
- "profit,"
- "increase", and
- being "late" (i.e. after a due time).

For compass directions, North and East are typically represented by positive numbers.

Express the following statement as a directed number (i.e. positive or negative):

Driving $15$15km north.

We take North to be the "positive direction" so it can be represented by the number $+$+$15$15 which we can just write as $15$15.

Express the following statement as a directed number: Going up $4$4 flights of stairs.

"Going up" means "rising" so this can also be represented by the positive number $4$4.

Words that indicate a decrease or a decline are represented by negative numbers.

This includes words like

- "descending,"
- "loss," and
- being "early" (i.e. before a due time).

South and West directions on the compass are typically represented by negative numbers.

Express the following statement as a directed number: A weight loss of $2$2 kilograms.

"Loss" represents a decrease, so this would be represented by a negative number. Our answer would be $-2$−2.

Express the following statement as a directed number: Travelling west for $400$400 metres.

"West" is moving in a negative direction on our number line, so it would be represented by the number $-400$−400.

Express the following statement as a directed number: Losing $\$43$$43.