Positive and Negative Numbers
Standard numbers, anything greater than zero, are described as ‘positive’ numbers. We don’t put a plus sign (+) in front of them because we don’t need to since the general understanding is that numbers without a sign are positive.
Numbers that are less than zero are known as 'negative' numbers. These have a minus sign (−) in front of them to indicate that they are less than zero (for example, -10 or 'minus 10').
Visualising Negative and Positive Numbers
Probably the easiest way to visualise negative and positive numbers is using a number line, a tool with which you may well be familiar, especially if you have children at primary school.
It looks something like this:
A number line can help you to visualise both positive and negative numbers and the operations (adding and subtracting) that you can do with them.
When you have a sum, you start at the first number and move the second number of places either to the right (for an addition) or left (for a subtraction).
This number line is a simplified version, but you can draw them with every number included if you wish. The big advantage of a number line is that it is very easy to draw for yourself on the back of an envelope or a piece of scrap paper, and it is also quite hard to go wrong with the sum. As long as you are careful to count the number of places that you are moving, you will reach the correct answer.
What is 10 − 25?
Starting at 10, you move 25 numbers to the left, and you see immediately that the answer is −15.
What is −17 + 23?
This time you start at -17 and move 23 places to the right. You can see straight away that the answer is 6.
Subtracting Negative Numbers
If you subtract a negative number, the two negatives combine to make a positive.
−10−(−10) is not −20. Instead, you can think of it as turning one of the negative signs upright, to cross over the other, and make a plus. The sum would then be −10+10 = 0.
A Quick Note on Brackets
For clarity, you would never write two negative signs side by side without brackets.
So if you are asked to subtract a negative number, it will always have brackets around it so that you can see that the use of two negative signs was intentional.
-10--10 is incorrect (and confusing)
-10-(-10) is correct (and clearer)
Multiplying and Dividing with Positive and Negative Numbers
Your first act when multiplying or dividing with signs is to ignore the signs and just multiply or divide the numbers as if they were both positive. Once you have the numerical answer, then you can apply a very simple rule to decide the sign of the answer:
- When the signs of the two numbers are the same, the answer will be positive.
- When the signs of the two numbers are different, the answer will be negative.
(positive number) × (positive number) = positive number
(negative number) × (negative number) = positive number
(positive number) × (negative number) = negative number
As a side issue, this goes some way to explaining why you cannot have the square root of a negative number, of which more in our page on Special Numbers and Concepts. The square root is the number that is multiplied by itself to get the number. You cannot multiply a number by itself to get a negative number. To get a negative number, you need one negative and one positive number.
The rule works the same way when you have more than two numbers to multiply or divide. An even number of negative numbers will give a positive answer. An odd number of negative numbers will give a negative answer.
What is −5 × 25?
5 x 25 is 125. But here you have one negative and one positive number, so the sign of the answer will be negative. The answer is therefore −125.
What is −40 ÷ 8?
40 ÷ 8 is 5. Again, you have one positive and one negative number, so the sign of the answer will be negative. The answer is −5.
What is −50 ÷ −5?
50 ÷ 5 is 10. This time, you have two negative numbers, so the sign of the answer will be positive. The answer is 10.
What is −100 × −2?
100 x 2 is 200. Again, you have two negative numbers, so the answer is positive. It is 200.
What is 10 x −2 × 3?
Take the first part of the sum first. 10 x 2 is 20. You have one positive and one negative number, so the sign of the answer will be negative, making it −20.
Now take the second part of the sum: −20 × 3. 20 × 3 is 60. Again, you have a negative and a positive number, so the answer will be negative: −60.
Why does multiplying two negatives give a positive answer?
The fact that a negative number multiplied by another negative number produces a positive result can often confuse and seem counter-intuitive.
To explain why this is the case, think back to the number lines used earlier in this article since these help to explain this visually.
- First, imagine standing on the number line at point zero and facing towards the positive direction, ie. towards 1, 2 and so on. You take two steps forwards, pause, then take two steps more. You have moved 2 × 2 steps = 4 steps.
Hence positive × positive = positive
- Now go back to zero, and face in the negative direction, ie. towards −1, −2, etc. Take two steps forwards, then another two. You are now standing on −4. You have moved 2 × −2 steps = −4 steps.
Hence negative × positive = negative
In both of these examples, you have moved forwards (ie. the direction you were facing), a positive move.
- Go back to zero again, but this time you are going to walk backwards (a negative move). Face the positive direction again and take two steps backwards. You are now standing on −2. A positive (the direction you are facing) and a negative (the direction you are moving) result in a negative move.
Hence positive × negative = negative
- Finally, back at zero again, face in the negative direction. Now take two steps backwards, and then another two backwards. You are standing on +4. By facing in the negative direction, and walking backwards (two negatives), you have achieved a positive result.
Hence negative × negative = positive
- Two negatives cancel each other out. You can see this in speech:
- “Just do it!” is a positive encouragement to do something.
- “Don’t do that!” is asking someone not to do something. It’s a negative.
- “Don’t not do it” means “please do it”. Two negatives cancel out and make a positive, in maths as well as in speech.
- The signs add together physically. When you have two negative signs, one turns over, and they add together to make a positive. If you have a positive and a negative, there is one dash left over, and the answer is negative. This is a simple and visual aide-mémoire, despite not necessarily being satisfying to those who want to understand the rule.
Negative signs can look a bit daunting, but the rules that govern their use are simple and straightforward. Keep these in mind, and you will have no problems.