# Subtraction '-' | Basics of Arithmetic

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 taking away one or more numbers from another.

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

The minus sign ‘-‘ is used to denote a subtraction calculation, 4 - 2 = 2.  The '-' sign can be used multiple times as required, 8 - 2 - 2 = 4, for example; however, although this calculation is correct it is often simpler to add together multiple numbers for subtraction.  8 - 2 - 2 can be simplified to 8 - 4 (the two 2s were added together to give 4 which is then subtracted from the 8).  8 - 2 - 2 and 8 - 4 both equal 4.

Warning

Caution is needed when using the '-' sign as numbers with a negative value are written with a preceding ‘-‘ so minus 2 is written as -2.  This simply means 2 less than zero or 2 below zero.

When subtracting, unlike addition, the order of numbers does matter – the first number is the number you are subtracting from, subsequent numbers can be in any order.

10-5 is different to 5-10

10-5 = 5 whereas 5-10 = -5

When numbers of equal value are subtracted from each other the result is always 0.  19-19 = 0
When subtracting 0 from any number the number remains unchanged. 19-0 = 19

Simple subtraction can be easily learned in the same way as addition, by counting:

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 one above the other – similar to an addition sum.

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, the sum should be written with the starting number on top and the number we are taking away underneath.

 Hundreds Tens Units 7 5 5 1 8 0

Step 1: Subtract the numbers in the right (units) column. 5 - 0 = 5.

 Hundreds Tens Units 7 5 5 1 8 0 Total 5

Step 2: Subtract the numbers in the next (tens) column. 5 - 8 will not work as 8 is larger than 5 and we would end up with a negative number. We need to borrow a number from the hundreds column. Take 1 away from the 7 in the hundreds column leaving a 6, bring the 1 in front of the 5 in the tens column making 15. Take 8 away from 15, put the answer (7) in the total for 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.

## Borrowing in Subtraction

Borrowing, as in the example above, can be a little confusing in subtraction – it is essentially the same as carrying over in addition only in reverse, as subtraction is the reverse - or 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 easily 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 1,001 and 999.

 1 0 0 1 9 9 9

Starting in the right column we need to take 9 away from 1 – this doesn’t go, whenever we try to take a larger number away from a smaller number it doesn't go without producing a negative number.

In order to make the calculation work we need to 'borrow' a number from the next column on the right.  The tens column has a 0 in it so there is nothing to borrow, so we move to the next column to the right, the hundreds column, again this has a 0 so we can’t borrow. The thousands column has a 1 we can borrow this one and move it over to the next column on the left, the hundreds column. As 1000 (one thousand) = ten hundreds we can make the number in the hundreds column 10:

 Carried 0 10 1 0 0 1 9 9 9

This doesn’t help with 1 - 9 (in the units column) but is the first step in the process.  We can now borrow from the hundreds column as it now has 10 in it. Borrowing 1 leaves a 9 (10 - 1) and means we have 10 in the tens column.

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

Finally we can borrow one from the tens column giving 10 + the 1 we already had, 11.

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

We can now carry out the 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 or £0.02.

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