We often use positive and negative numbers when telling the temperature. In winter, for example, a temperature ℃ is ℃ below zero.
When doing arithmetic, it is very helpful to regard positive numbers as something we have in our possession and the negative numbers as something we owe (a debt) and need to pay back.
A negative number has a minus sign in front of it, as in ℃. Positive numbers are written with or without the positive sign in front of the number.
In , we regard the 10 as something we possess and the as something we possess. So in total, we possess . As .
In -10 + 3, we regard the as a debt and the as something we have. So, after selling the account, we shall still have as a debt. Therefore, .
In , we regard as a debt and the again as a debt. So in total, we have as a debt. Therefore, we should have .
Another way to calculate is by using the number line. Starting from the first number, we move to the right if we are adding and to the left if we are subtracting.
For , start at on the number line
Since we are adding, we move to the right from by steps. This gives .
For , start at and move steps to the right, since we are adding.
For , start at and move steps to the left since we are subtracting.
This take us to . Hence,
For , we start at and we move steps to the right
We are at , so .
We should also know some rules:
With bigger numbers, using the number line becomes impractical. We should then visualise what is happening on the number line. For example, consider .
On the number line we will start at , because we are adding . We should move steps to the right of . With some efforts we see that this will take us to . Therefore .
To calculate , we start at on the number line and because we are subtracting , we should move to the left of by steps. Again, with some effort, we see that this will take us to on the number line. Now . Therefore .
With practice, we shall become more fluent at these and will not even need to visualise the number line.
When we are multiplying numbers (positive, negative), we should remember these rules:
A positivea positivea positive
A positivea negativea positive
A negativea positivea negative
A negativea negativea positive
The same rules apply for division