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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 SubtractionSubtraction 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, 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). Warning Caution is needed when using the ‘−’ sign. Numbers that have a negative value are written with a For more information, see our page on Positive and Negative Numbers. Beware of Signs and Order in SubtractionWhen we are performing an addition calculation, the order in which we add the numbers does not matter. For example, 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, We can see that we have the same numerical answer (3), but that its 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 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 Performing subtractionSimple 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 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 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.
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.
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 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
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.
Borrowing in SubtractionBorrowing, 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. For the purpose of this example we have ignored the decimal point and written the numbers as 1001 and 999. 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 One thousand is the same as 10 hundreds, so now we have 10 in the hundreds column where before we had zero:
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.
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.
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.
Having borrowed multiple times we have arrived at our answer of 2. When we replace the decimal point we have £0.02. Subtraction in Special Situations: Zero and Negative NumbersIf 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 If we were to draw that  We are going to use this analogy to help us understand the following examples. When 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 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 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 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 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 Further Reading from Skills You Need Fundamentals of Numeracy This eBook provides worked examples and easy-to-understand explanations to show you how to use basic mathematical operations and start to manipulate numbers. It also includes real-world examples to make clear how these concepts are useful in real life. Whether you want to brush up on your basics, or help your children with their learning, this is the book for you. What is the least possible number must be subtracted from 9/11 and 15 so obtained numbers are in continue proportion?∴ The least possible number is 7. What number must be subtracted from each of 7/8 11 and 14 so that the remainders are proportional?′2 ′ is the least number to be subtracted so that the numbers will be in proportional. What number must be subtracted from each of the numbers 1011 7019 to them proportionate?Answer: 4 must be subtracted from each of the number. What number must be subtracted from 7 10 12 and 18 so that the resulting numbers are in proportion?1 Expert Answer The number that must be deducted is 2. |
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