Representation of directed numbers

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.

Example 1

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.

 

Directed numbers

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 represent positive 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.

 

Examples

Example 1

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.

Example 2

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 Represent Negative Numbers

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.

 

example 3

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.

 

example 4

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.

 

Worked Examples

Question 1

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