Understanding Integers 1

Let us consider 2 problems in addition/subtraction which use the same numbers and look similar at first glance.

1. Ram had 6 chocolates. He gave 4 chocolates to Shyam. How many chocolates does Ram have now?

2. Ram is standing 6 steps away from Shyam. Then he takes 4 steps. How many steps away from is he now from Shyam?

For the second problem, a normal “immediate” response from students could be “2 steps”. A few minutes later another student may say that it depends on whether Ram was coming towards Shyam or away from him. After some discussions, the class will agree that there could be 2 answers to the problem: 2 or 10.

Why is there an ambiguity in the 2ndproblem which was not there in the 1stproblem?

This is because the 2ndproblem is about “movement” i.e taking steps. Steps could be taken in 2 directions: towards or away. Hence in the 2ndproblem, the steps taken i.e 4 have to be further qualified by the direction in which they are taken. This kind of situation where the direction of a number had to be taken into account was a new experience even to mathematicians. Additional information related to “direction” had to be integrated into the meaning of a number! Such numbers are called integers. The above “story” explains the need to invent integers using our daily life experiences.

But mathematicians also had to invent integers for a totally different reason.

Invention of Negative Numbers

In real life situations, it makes sense when to say, “what is left when 3 apples are taken away from 4 apples?”. We would never ask the question “what is left when 4 apples are taken away from 3 apples?”. That would be considered impossible.

But mathematicians had stripped the problem of any real-life context and symbolized it as 4 – 3 = 1. Hence some mathematician asked an audacious question “what is 3 – 4?”. For this they had to invent integers, numbers which could be negative also!