Development of Numbers 2

As and when new types of numbers are defined, some of the properties of the numbers may have to be modified. Such modifications are accepted since there is a "value - added" by the new numbers.

We will see examples as we go through different types of numbers.

Integers

The set of whole numbers was “closed” with respect to subtraction only in limited cases where the result of the subtraction is 0. But there were problems like 4 -8!

The result of the above operation was conceived as “-4” and the concept of negative numbers or Integers was developed.

The set of negative numbers was added to the set of Whole Numbers and this set was called Set of Integers. The set of Integers was “closed” with respect to addition and subtraction.

The set of Integers loses the property of a minimum number, which Whole numbers had.

Fractions

Quantities which were less than a whole arose from the idea of sharing or from the division operation.

To take care of such transactions, mathematicians invented the idea of fractions. The fraction was written as a combination of 2 natural numbers. For example, implied that a sack of grains was divided into 5 equal piles and 2 of these piles are now under discussion.

This was the second instance of a single number (first one being the place value system) which was represented using more than one numeral.

The set of fractions loses the property of the "next number or preceding number" which whole numbers had.

Rational Numbers

The specific idea of a fraction was generalized into the concept of a Rational Number. Rational numbers were seen as entities represented as “a/b” where both a & b were integers and b was not 0. These numbers were called Rational Numbers as they were seen to represent “ratios”.

“a” & “b” could also be –ve. Hence all whole numbers, zero, integers and fractions were seen as rational numbers. It was called the Set of Rational Numbers.

It was again clear that the result of any of the 4 operations on any 2 or more numbers from the set of rational numbers was also a rational number. Hence Closure Property holds.

Irrational Numbers

Greeks thought that all numbers could be expressed as rational numbers. But, through geometry, they discovered a set of numbers which could not be expressed as Rational Numbers; I.e of the type. The simplest of them was.

Eventually it was discovered that such numbers formed a set which was even larger than the set of rational numbers! Such numbers were called Irrational Numbers, in the sense of not being representable in the form.

New types of numbers called Algebraic, Transcendental & Computable were also discovered. They were seen to be subsets of Irrational numbers. We will see these in some detail in a subsequent chapter.

Real Numbers

All the above numbers were called the Set of Real Numbers. The set of Real Numbers was seen to be closed with respect to the four operations.

Mathematicians also realised that the concept of a "real number" filled a number line "completely" without leaving any gaps. Every point on the number line represents a real number which is either rational or irrational.

Hence, with the concept of a real number, mathematics reached a stage where all the different types of numbers and their position on the number line were completely described and defined. The Number Line could now be called the Real Number Line.

Number Line

Real Numbers could also be plotted on a Number Line. We can say that a Real Number is one which can be plotted on the Number Line. Conversely, any point on the number line represents a real number.

Non-Real Numbers

We will briefly see in a subsequent chapter that new numbers were invented, which could not be plotted on the number line.

Non-real numbers lose the property of "being compared" which real numbers had.