Understanding Integers 1

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

For the second problem, a normal &ldquo;immediate&rdquo; response from students could be &ldquo;2 steps&rdquo;. 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.
 * 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?

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

This is because the 2ndproblem is about &ldquo;movement&rdquo; 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 &ldquo;direction&rdquo; had to be integrated into the meaning of a number! Such numbers are called integers. The above &ldquo;story&rdquo; 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, &ldquo;what is left when 3 apples are taken away from 4 apples?&rdquo;. We would never ask the question &ldquo;what is left when 4 apples are taken away from 3 apples?&rdquo;. 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 &ldquo;what is 3 – 4?&rdquo;. For this they had to invent integers, numbers which could be negative also!

Integers – a change in the understanding of &ldquo;understanding&rdquo;

Integers brought with them a whole lot of issues, both in representing them and operating with them.

The numbers which came before integers; whole numbers, fractions &amp; decimals; could be represented and the result of operations on them derived, by relating them to our daily life events &amp; transactions. We have seen this in earlier chapters. For these numbers &ldquo;relating to daily life&rdquo; was the meaning of &ldquo;understanding&rdquo; them.

Integers were the first instance of numbers which could not be represented easily with daily-use materials. The results operations with them could also not be derived solely through the logic of daily events &amp; transactions.

In a sense, integers were numbers, totally unlike their predecessors. Arithmetic seemed to have gone out of the sphere of physical reality! It was like an aircraft, which after running on the tarmac faster and faster, has taken off and now has no physical contact with the ground! These issues created a lot of confusions and discussions among mathematicians. The end result was an effort by mathematicians to restructure arithmetic like geometry!

Euclid had shown the way, in Elements, for deriving the entire discipline of plane geometry, through logical arguments, starting with a few axioms whose truth had to be accepted without proof. Axioms were called &ldquo;self-evident&rdquo; truths.

In the 19thcentury, mathematicians similarly wrote down the axioms of arithmetic. (See chapter of Axioms of Arithmetic). The fundamental laws of arithmetic were a part of this axiomatic structure. These became the logic for developing arithmetic

The rules of operations on integers were derived by applying these axioms in a logical manner. The famous rule that the product of two negatives is positive is a result of the application of the axioms. It is not possible to &ldquo;understand&rdquo; this rule through our daily experiences. They have to be &ldquo;understood&rdquo; only through the axioms of arithmetic.

Similarly, the rules of operations on any newly discovered number (imaginary and complex numbers had not been discovered then) could also be discovered by applying these laws.

Hence, we can say that the invention of integers was a watershed moment in the development of arithmetic. In terms of &ldquo;understanding&rdquo; in mathematics, we had a &ldquo;before integers&rdquo; and &ldquo;after integers&rdquo; separation.

< 18.10 Math of Proportions | Topic Index | 19.2 Understanding Integers 2 >