Directed numbers are something that a lot of people find confusing. I hope that this article will help you to get your head round them!

What are directed numbers?

“Directed numbers” means positive and negative numbers.

“Positive” or “negative” describes a number’s direction – either where it’s located in relation to zero, or which way it’s making us face (more on that when we look at adding and subtracting).

So positive 9 (+9 or just 9) is at the right-hand (high) end of this number line, and negative 9 (-9) is at the left-hand (low) end.

Number line with directed numbers going from -9 to 9

If you think of it in terms of money then positive numbers are like having money, and negative ones are like owing money to somebody. If your bank account goes overdrawn (because you’ve spent more money than you had), you have a negative amount of money in the bank and you have to pay in enough to climb back up to zero before you can get back to actually having any money in your account.

Comparing directed numbers

A number that’s further down the number line is less, or lower, than one that’s further up it. Looking at the number line above, the numbers get lower and lower as we move to the left, and higher and higher as we move to the right.

When you’re only dealing with positive numbers that’s easy to understand, but when negative numbers come in, it gets more confusing.
We shouldn’t really describe numbers as “bigger” and “smaller” once we start dealing with directed numbers, because any negative number, even a BIG negative number, is ALWAYS less than a positive number!
So -9 is LESS (or LOWER) than 2, and 2 is GREATER (or HIGHER) than-9, because -9 is further to the left on the number line, even though 9 is a bigger number than 2.

Your turn 1

Decide whether “GREATER” or “LESS” should go in each of the gaps below. If you’re not sure then use the number line to decide whether the first number is to the right or the left of the second one.

-3 is …………… than 0
2 is …………… than -7
-4 is …………… than 5
-6 is …………… than -1

Click here for the answers

Adding and subtracting directed numbers

As we said before, “positive” or “negative” can be thought of as describing which way a number is making us face, as well as where it’s located in relation to zero.

“Plus” or “add” describes a forwards movement along the number line, and “minus” or”add” describes a backwards movement. But of course, whether “forwards” is up or down, depends on which way you’re facing!

Here are some examples:

3 + 4 means we start at 3. (The first number is always the starting point.)
The second number is positive – it wants us to move upwards – so we’re facing upwards, or to the right, on the number line below (green arrow).
We’re adding, so we move forwards in the direction that we’re facing (blue arrow) – i.e. 4 places up the line to 7.
So 3 + 4 = 7.

For 3 – 4, we start at 3 again.
The second number is positive – it wants us to move upwards – so we’re facing upwards, or to the right, on the number line below (green arrow).
We’re subtracting, so we move backwards (blue arrow) – i.e. 4 places down the line to -1.
So 3 – 4 = -1.

For 3 + -4, we start at 3 again.
The second number is negative – it wants us to move downwards – so we’re facing downwards, or to the left, on the number line below (red arrow).
We’re adding, so we move forwards in the direction that we’re facing (blue arrow) – i.e. 4 places down the line to -1.
So 3 + -4 = -1, which is the same as 3 – 4.

For 3 – -4, we start at 3 again.
The second number is negative – it wants us to move downwards – so we’re facing downwards, or to the left, on the number line below (red arrow).
We’re subtracting, so we move backwards (blue arrow) – i.e. 4 places up the line to 7.
So 3 – -4 = 7, which is the same as 3 + 4.

So in summary:
💡 Adding a negative number is the same as subtracting a positive number.
💡 Subtracting a negative number is the same as adding a positive number.

If you’re starting with a negative number, for example -2 + 5, then you can start at -2 and work in the same way.
Sometimes it can make it easier if you swap the numbers over, keeping the right sign in front of each number… so in this case you get 5 + -2 (the 5 stays positive and the 2 stays negative), which of course is the same as 5 – 2.
Try it both ways and check that you get the same answer (which should be 3).

Your turn 2

Work out the answers: Don’t be scared to sketch out a number line if it helps you!

2 + -7
3 – 5
-3 + 2
4 – -1
-1 – -6

Click here for the answers

Multiplying and dividing directed numbers

💡 If you multiply or divide any number by a negative, its sign changes.

If you multiply or divide by a positive then the sign stays the same as it was to begin with.

For example:

3 × -2
Start with 3, multiply it by 2 and change its sign
so we get -6.

-3 × -2
Start with -3, multiply it by 2 and change its sign
so we get (+)6.

-3 × 2
Start with -3, multiply it by 2 and leave the sign the same
so we get -6.

12 ÷ -3
Start with 12, divide it by 3 and change its sign
so we get -4.

-12 ÷ -3
Start with -12, divide it by 3 and change its sign
so we get (+)4.

-12 ÷ 3
Start with -12, divide it by 3 and leave the sign the same
so we get -4.

Your turn 3

Work out the answers:

2 × -7
15 ÷ -5
-3 × 2
-14 ÷ -2
-3 × -6

Click here for the answers

Note: Be careful when raising negative numbers to a power!
If you square negative 3 then that should always be written as (-3)², NOT -3².
That’s because you’re multiplying the whole -3 by itself, not just the “3” part.
(-3)² = -3 × -3 = 9
But -3² = 3 × 3 with a negative sign in front of it, i.e. -9.
You can NEVER get a negative answer if you square something!
(Well, not until you get into the realms of imaginary numbers, anyway, and that’s way beyond GCSE!)

When you’re working with directed numbers, forget the “two minuses make a plus” that you might have heard before; it’s too ambiguous to be very helpful. Instead, just remember the points marked with light bulbs, and you won’t go wrong:

💡 Any negative number is less than any positive number.
💡 Adding a negative number is the same as subtracting a positive number.
💡 Subtracting a negative number is the same as adding a positive number. (If you face backwards and walk backwards then you end up ahead of where you started!)
💡 If you multiply or divide any number by a negative, its sign changes.

I hope this article has helped to clear up any confusion you might have had over positive and negative numbers. If you’ve found it helpful then please share it!

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