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Subtraction is actually the same as adding a negative number .
In this example, $a$ and $b$ are positive numbers, but $-b$ is a negative number
So, this means that subtraction is simply a short-cut for adding a negative number and instead of writing $a+(-b)$ , we write $a-b$ . This also means that $-b+a$ is the same as $a-b$ . Now, which do you find easier to work out?
Most people find that the first way is a bit more difficult to work out than the second way. For example, most people find $12-3$ a lot easier to work out than $-3+12$ , even though they are the same thing. So $a-b$ , which looks neater and requires less writing is the accepted way of writing subtractions.
[link] shows how to calculate the sign of the answer when you multiply two numbers together. The first column shows the sign of the firstnumber, the second column gives the sign of the second number and the third column shows what sign the answer will be.
$a$ | $b$ | $a\times b$ or $a\xf7b$ |
$+$ | $+$ | $+$ |
$+$ | $-$ | $-$ |
$-$ | $+$ | $-$ |
$-$ | $-$ | $+$ |
So multiplying or dividing a negative number by a positive number always gives you a negative number, whereas multiplying or dividing numbers which have thesame sign always gives a positive number. For example, $2\times 3=6$ and $-2\times -3=6$ , but $-2\times 3=-6$ and $2\times -3=-6$ .
Adding numbers works slightly differently (see [link] ). The first column shows the sign of the first number, the second column gives the sign of the second number and the third column showswhat sign the answer will be.
$a$ | $b$ | $a+b$ |
$+$ | $+$ | $+$ |
$+$ | $-$ | ? |
$-$ | $+$ | ? |
$-$ | $-$ | $-$ |
If you add two positive numbers you will always get a positive number, but if you add two negative numbers you will always get a negative number. If thenumbers have a different sign, then the sign of the answer depends on which one isbigger.
The number line in [link] is a good way to visualise what negative numbers are, but it can get very inefficient to use it every timeyou want to add or subtract negative numbers. To keep things simple, we will write down three tips that you can use to make working with negative numbers alittle bit easier. These tips will let you work out what the answer is when you add or subtract numbers which may be negative, and will also help you keep yourwork tidy and easier to understand.
If you are given an expression like $-a+b$ , then it is easier to move the numbers around so that the expression looks easier. In this case, we have seen thatadding a negative number to a positive number is the same as subtracting the number from the positive number. So,
This makes the expression easier to understand. For example, a question like “What is $-7+11$ ?” looks a lot more complicated than “What is $11-7$ ?”, even though they are exactly the same question.
When you have two negative numbers like $-3-7$ , you can calculate the answer by simply adding together the numbers as if they were positive and then putting anegative sign in front.
In [link] we saw that the sign of two numbers added together depends on which one is bigger. This tip tells us that all we need todo is take the smaller number away from the larger one and remember to give the answer the sign of the larger number. In this equation, $F$ is bigger than $e$ .
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