When students are first introduced to subtraction, they are taught to ‘take away’. They might be given an example such as: you have five apples, give three away, and now you have two apples. This might be written as 5 - 3 = 2 and read aloud as, “Five minus three equals two.”
Later, subtraction is often compared with addition and explained as the opposite operation. Rather than subtracting three apples, we might ask, “How many apples would we need if, by adding three more, we then have five?” It might be written as ☐ + 3 = 5 and read aloud as, “What plus three equals five?” Students then begin connecting subtraction to addition in their understanding.
However, these two ideas are not enough. In higher-level mathematics, we see subtraction as ‘difference’. Let’s say a table is 5 metres away and, between us and the table, is a chair which is 3 metres away. How far is the chair from the table? To answer this, we subtract 3 metres from 5 metres, leaving us with 2 metres. The difference in length between the table and the chair is 2 metres. We take the further distance and subtract the closer distance to find the length between the objects. This is often asked as, “What is the difference between 5 and 3?” The difference is the length between them; from three to five is 2 units.
This idea is crucial in coordinate geometry, particularly with the distance formula. To find the distance between two points, we use Pythagoras' theorem, which relies on knowing the lengths of two sides of a right triangle. These two sides are the horizontal and vertical lengths. Each of these lengths is found by using the concept of difference. Subtracting the x-value on the left from the x-value furthest to the right gives the horizontal distance. Similarly, subtracting the y-value below from the y-value furthest up gives the vertical distance. Other areas of Mathematics that use ‘difference’ include the Fundamental Theorem of Calculus, the Fundamental Theorem of Algebra, differentiation by first principles, and more.
In summary, recognising how subtraction provides the distance between two values is fundamental to higher-level maths. Do you think about subtraction as difference?
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