How does simple addition and reduction work graphically?


wiseGEEK Writing Contest

Addition and reduction by themselves are simple procedures made to numbers that have the same unit. It is easy to see that 2+3=5 or that 6-4=2. But what does it mean graphically?

Graphically, the numbers should be considered as an axis – for example a horizontal axis going from left to right, where the approximate midpoint is 0, the numbers on the line left of 0 are negative and the numbers right of 0 are positive.

(-9)---(-8)---(-7)---(-6)---(-5)---(-4)---(-3)---(-2)---(-1)----(0)----(1)----(2)----(3)----(4)----(5)----(6)----(7)----(8)----(9)

An addition such as 2+3 would be handled like so: First the first number, 2, is considered. It is positive, so it is found right from 0. The second number indicates the steps we have to make to get to the result. The sign, plus (+), indicates that it's an addition, so whatever number is added, the direction of the steps will be to the right. Now the number to be added is 3, making the steps like this: from 2 to 3 (first step), then 4 (second step) and finally 5 (third step), for a total of three steps, as the second number indicated.

A reduction, 2-4, would go like this: As above, we find the number 2 on the axis. The sign, which is a minus (-), indicates that the direction of the steps will be to the left. Thus, we go from 2 to 1, then 0, -1 and finally -2, for a total of four steps.

However, it isn't always as easy. Sometimes it's important to keep the sign of the number that will be added or reduced from something: 2+(-2) – two plus negative of two. Directions will apply here as well: a positive sign of a number doesn't change the direction where the steps will be taken, while a negative one will. Therefore, when we add the negative of two to two, we again first find the number 2 on the axis. The plus sign indicates that the steps will be taken to the right, but the negative sign of the number changes that and the steps will be taken to the opposite direction, left, instead: from 2 to 1 and then to 0.

The following reduction is graphically the most complicated to understand: (-1)-(-3). We start from -1, just left of 0. The sign indicates a reduction, so the direction of steps will be to the left. However, the second number has a negative sign, changing the situation around – the steps will be taken to the opposite direction, right, instead: from -1 to 0, 1 and finally 2. The calculation could be simplified to (-1)+3, or even 3-1, if the order and original signs of the numbers aren't that important.

For some people, understanding mathematics graphically (visually) is easier than the abstract way, so diagrams and graphs play a big part in learning certain if not all mathematical subjects. Others, then again, won't remarkably benefit from them, while for some it's just a bit more insight into the issue.