This page covers the basics of arithmetic, the simplest way of manipulating numbers through subtraction (−).

See our other arithmetic pages, for discussion and examples of: Addition (+), Multiplication (×) and Division (÷).

Subtraction

Subtraction is the term used to describe how we ‘take away’ one or more numbers from another.

Subtraction is also used to find the difference between two numbers. Subtraction is the opposite of addition. If you have not already done so, we recommend reading our addition page.

The minus sign ‘−‘ is used to denote a subtraction operation, such as 4 − 2 = 2. The '−' sign can be used multiple times as required: for example, 8 − 2 − 2 = 4.

This calculation is correct, but it can be simplified by adding together the numbers we are subtracting.  In our example, 8 − 2 − 2 = 4 can be simplified to 8 − 4 = 4 (the two 2s have been added together to give 4, which is then subtracted from 8).

Beware of Signs and Order in Subtraction

When we are performing an addition calculation, the order in which we add the numbers does not matter.

For example,
8 + 3 + 5 is the same as 3 + 8 + 5 and gives us the same answer, 16.

However, when we are performing a subtraction, we need to take extra care with the order of the numbers.

Usually with a subtraction, we write the number we are subtracting from first, and the numbers we are taking away in any order after that.

For example,
8 − 5 = 3
This is NOT the same as 5 − 8 = −3

We can see that we have the same numerical answer (3), but that its value is different: 3 in the first calculation, but minus 3 (−3) in the second.

Similarly 8 − 5 − 3 = 0, but 5 − 8 − 3 = −6, which is a completely different answer.

The reason that the answers are different is not because we have put the numbers in the ‘wrong’ order, but because we have not taken care to notice whether they are positive or negative.

In our example, 8 is a positive number, so we could write it as ‘+ 8’ and it would be correct, but convention says that we do not need to write the ‘+’ symbol. However, the ‘+’ symbol is very important if we change the order, as are the ‘−‘ symbols that precede 5 and 3.

Here is the last example re-written to give the correct answer:

8 − 5 − 3 = 0 as before, and − 5 + 8 − 3 = 0, giving the same answer. In this case we have written the numbers in the same order as before, but we have taken their positive or negative value into account.

For a more detailed explanation and examples, see the section on Subtraction in Special Situations: Zero and Negative Numbers below.

Performing subtraction

Simple subtraction can be carried out in the same way as addition, by counting or using a number line:

If Phoebe has 9 sweets and Luke has 5 sweets what is the difference?

Starting with the smaller number (5) and count up to the larger number (9).

6 (1), 7 (2), 8 (3), 9 (4).

Phoebe has 4 more sweets than Luke, the difference in sweets is 4.

So: 9 − 5 = 4.

For more complex subtraction, where using counting is not appropriate, it is useful to write our numbers in columns one above the other—similar to an addition calculation.

Suppose that Mike earns £755 a week and pays £180 a week for rent.  How much money does Mike have left after he has paid his rent?

In this example we are going to take £180 away from £755. We write the starting number first and the number we are taking away underneath, taking care to make sure the numbers are in the correct columns.

Hundreds	Tens	Units
7	5	5
1	8	0

Step 1: First we perform a subtraction on the numbers in the Units column on the right, then write the answer at the bottom in the same column. In this case, 5 − 0 = 5.

	Hundreds	Tens	Units
	7	5	5
	1	8	0
Total			5

Step 2: Using the same approach as an addition calculation, we work across the columns from right to left. Next we need to subtract the numbers in the tens column. In our example, we need to subtract eight from five (5 − 8), but  8 is larger than 5, so we cannot do this as we would end up with a negative number. We need to borrow a number from the hundreds column. This can be a tricky concept and we look at it in greater detail below: We have 7 in the hundreds column, so we ‘borrow’ 1 for the tens column, leaving us with 6 in the hundreds. Cross through the 7 and write 6 in the hundreds column to avoid mistakes later. Move the 1 to the tens column and write it in front of the 5. We are not adding ‘1’ to the tens, we are lending ‘1 lot of 10’. So instead of 5 tens, we now have 15 tens.

15 is larger than eight, so we can perform our subtraction in the tens column. Take 8 from 15 and write the answer (7) at the bottom of the tens column.

	Hundreds	Tens	Units
	7 6	15	5
	1	8	0
Total		7	5

Step 3: Finally take 1 away from 6 in the hundreds column. 6 − 1 = 5, so put a 5 in the answer of the hundreds column to give our final answer. Mike has £575 left after he has paid his rent.

	Hundreds	Tens	Units
	7 6	15	5
	1	8	0
Total	5	7	5

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Borrowing in Subtraction

Borrowing, as in the example above, can be confusing in subtraction calculations. It is similar to ‘carrying over’ in addition calculations, but in reverse, because subtraction is the reverse (opposite) of addition.

Repeated borrowing may occur in a subtraction calculation.
Suppose we have £10.01 and we want to take away £9.99. We can work this out without having to write anything down – the answer is £0.02 or 2p. However if we write this calculation out formally then the concept of borrowing becomes clearer.

For the purpose of this example we have ignored the decimal point and written the numbers as 1001 and 999.

1	0	0	1
	9	9	9

Starting in the units column on the right, we need to take 9 away from 1. In our subtraction calculations, the rule (as in the example above) is that we never take a larger number away from a smaller number because it would give us a negative answer.

In order to make the calculation work we need to 'borrow' a number from the next column on the left. The tens column has a 0 in it so there is nothing to borrow, so we have to move to the next column to the left. The hundreds column also has 0 so we can’t borrow from this column either, so we move to the next column on the left. The thousands column has 1, so we can borrow this and move it over to the next column on the right, the hundreds column. We cross through the 1 in the thousands column to avoid mistakes later.

One thousand is the same as 10 hundreds, so now we have 10 in the hundreds column where before we had zero:

Carried	0	10
	1	0	0	1
		9	9	9

However, this doesn’t help with 1 − 9 (in the units column) because we still have zero to borrow from in the tens column, but it is the first step in the process.

Now that we have 10 hundreds, we can borrow one of these for the tens column. One hundred is the same as 10 tens, so we carry 10 across to the tens column. We must not forget to adjust the hundreds column, so we cross through the 10 and write 9 instead.

Carried		9	10
Carried	0	10
	1	0	0	1
		9	9	9

Finally, we can perform our subtraction in the units column by borrowing 1 ten from the tens column. This leaves 9 tens in the tens column, and 10 + the 1 we already had in the units column, giving us 11 units.

Carried			9	10
Carried		9	10
Carried	0	10
	1	0	0	1
		9	9	9

We can now carry out the complete calculation, starting in the units column, 10 + 1 = 11 − 9 = 2. Then in the tens column 9 − 9 = 0. The same for the hundreds column 9 − 9 = 0. Finally in the thousands column 0 − 0 = 0.

Carried			9	10
Carried		9	10
Carried	0	10
	1	0	0	1
		9	9	9
Total	0	0	0	2

Having borrowed multiple times we have arrived at our answer of 2. When we replace the decimal point we have £0.02.

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Subtraction in Special Situations: Zero and Negative Numbers

If we were doing a simple addition calculation, we might count up in our heads or perhaps on our fingers. When we are doing subtraction, especially if it involves negative numbers, it helps to imagine ourselves walking along a line. Each step is a number on that line. If we start at zero, each step forward adds a number, each step backwards takes one away. The most important thing to remember is that we always face the positive direction. You might find it helpful to think of your line as climbing up and down a ladder, with each rung being a number. Or perhaps you are more familiar with travelling up and down a high-rise block in a lift, where zero is the ground floor, positive numbers are above ground and negative numbers are in the basement.

If we were to draw that line on a piece of paper, it would look like a ruler. We can move our pen backwards and forwards along the line in the same way as imagining our steps backwards and forwards. This is called a number line, and is a very useful tool for addition and subtraction.

We are going to use this analogy to help us understand the following examples.

When numbers of equal value are subtracted from each other the result is always zero: 19−19 = 0.

Using our analogy, starting at zero, if we walk 19 steps forwards along the line, then 19 steps backwards, we end up back at zero.

When subtracting zero from any number, the number remains unchanged: 19−0 = 19.

Using our number line, we are starting at 19 and walk backwards zero steps - we don’t move and remain at 19.

When we subtract any positive number from zero, the answer is negative: 0 – 15 = –15

Remember from our earlier examples, a positive number doesn’t usually need to be written with a positive sign. When we see the number ‘67’, mathematical convention tells us that it is positive, i.e. ‘+67’.

In this example, we subtract +15 from zero: 0 – (+15) = –15. Using our analogy, we start at zero and walk 15 steps backwards.

When we subtract any positive number from a negative number, the answer becomes ‘more negative’.

For example, if we start with our answer from above (–15) and subtract 6, we have: –15 – 6 = –21. Remember ‘6’ is positive, so we could write –15 – (+6) = –21 and it means the same. Using our number line to help us understand, we start by standing at –15. We walk backwards six steps, still facing in the positive direction. We end up 21 steps backwards from zero, i.e. –21.

But what happens if we need to subtract a negative number from any other number?

Let’s start with an example: 15 – (–6) = 15 + 6 = 21

The rule is two negatives make a positive, i.e. subtraction of a negative number becomes an addition.

Let’s go back to our number line to help us understand more easily: Starting at 15, we know we need to move backwards (in a negative direction) because we are doing a subtraction. But we have a negative number to subtract, so to illustrate this we must turn around. Then we move backwards 6 places to arrive at our answer. By turning around and then moving backwards (two negatives), our overall direction of travel is in a positive direction, i.e. we have performed an addition.

Subtracting a negative number is an abstract concept and you may think it doesn’t really occur in daily life. After all, we can’t hold a negative number of apples or pour a negative volume of coffee. However, it is very important when it comes to mathematical concepts such as vectors. A vector has direction as well as magnitude, so for example, it is not just important how far a boat has sailed, but we also need to know the direction in which it has travelled.

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