Development of Numbers 1

This is a good time to look back briefly at the various types of numbers that we have studied. The history of the development of numbers, from intuitive Natural numbers to abstract Integers and Irrationals, is a good example of the journey from the concrete to the abstract. It is also the story of the development of math from simple ideas to a complex discipline involving abstract ideas & their relations.

Natural Numbers

The quest for numbers started with countable numbers. The numbers which could be related to groups of objects like 1, 5 & 3 etc were called Natural Numbers possibly because they were “seen” to exist in nature. It was also realized that they were infinite in number and there was no “biggest number”. These numbers were conceptualized as a set. The numbers 1, 2, 3 and so on formed the Set of Natural Numbers.

The Place Value System was invented to enable any magnitude to be written with just ten numerals. Numerals were symbols which represented numbers. Numbers embodied magnitudes.

The four arithmetic operations on these numbers were invented to model daily activities.

All numbers are abstract

Natural numbers were thought to be “real”.

All cultures found natural numbers to be useful to keep track of objects & events. But as societies developed, the natural numbers were not sufficient to keep track of objects & events. They had to invent many different kinds of number to meet the needs of the society. It also led to special classes of humans who studied numbers and developed into “mathematicians”.

Numbers, other than natural numbers, were considered a “product of our thinking”. The truth is that all numbers, including natural numbers are products of our mind and do not exist in our world in a concrete form.

But even today there are 2 groups among mathematicians – “Platonists” who believe that numbers exist in the world independently and those who hold that they are just creations of our mind.

Need for new Kinds of Numbers

While performing these four arithmetic operations with natural numbers, people came across situations where new kinds of numbers were generated. Mathematicians who generated these new numbers had to also formulate rules for operating on them. For this purpose they formulated 2 very important and fundamental concepts.

1. Closure Property through which new kinds of numbers could be discovered.

2. Fundamental Laws of Arithmetic which helped them to formulate rules of operation with them.

Idea of Set & Closure

Along with the ideas of a "set" of numbers, the idea of "closure" were formulated.

The idea was that the result of the 4 operations & exponentiation on any 2 numbers from a set, should also belong to the set. That is, a set of numbers should be "closed" with respect to the 5 operations.

For example, 4 & 6 add to 10. Both 4 ^ 6 and the result 10 belong to the set of natural numbers. Same was true of multiplication. But subtraction and division created numbers which were not natural numbers. Hence such new numbers had to be defined and the natural number set expanded.

Fundamental Laws of Arithmetic

We have studied the Fundamental Laws of Arithmetic. They consist of 5 properties. At an abstract level, any entity which obeys these laws, is accepted as a number. From another perspective, when any new type of number is invented, the rules of operation are determined so as to be consistent with these laws.

As human cultures invented different kinds of numbers to help them perform more sophisticated tasks, the criterion for accepting the new entities as numbers was to ensure that they obeyed these laws.

This is the mathematical basis of accepting fractions, integers, imaginary numbers & others also as "numbers" along with Natural numbers.

These laws helped to expand the idea of numbers to represent many areas other than magnitude. One example is that of representing Truth values in logical operations as numbers!

Zero as a number

The idea of “Nothing” as a number 0 (Zero) was invented as an answer to the question, what is 4 – 4? The rules for applying the four operations to 0 were formalized. But 0 was not part of the Natural Number set.

So the set of Natural numbers was expanded by adding 0 to it and the new set was called Set of Whole Numbers. Possibly this meant that they can be represented by the presence or absence of discrete objects.

The set of “Whole Numbers” was “closed” with respect to both addition & “limited cases” of subtraction.

Number Labels are Arbitrary

We also need to remember that the names given to different types of numbers - proper/ improper, imaginary & complex are mall just names. They have no mathematical, moral or ethical significance.

These names have just been given for convenience. Names familiar to us from the daily language used by us, have been used so that they are easier to remember.

This important fact should be clarified to students right from the lower classes. Otherwise many students would get the feeling that the names really indicate a "quality" of a number. They may feel that "improper" fractions really have something which is not "proper" in the sense we use it in daily language.