types of numbers

types of numbers

The number which is obtained as a result of measurement may be:

1-integral

if the unit goes an integral number of times into the magnitude concerned.

2-fractional 

if another unit exists, which goes an integral number of times both into the measured magnitude and into the unit previously chosen or in short, when the measured magnitude is commensurable with the unit of measurement

3-irrational 

when no such common measure exists, i.e. the given magnitude proves incommensurable with the unit of measurement.

more details 

It is shown in elementary geometry, for instance, that the diagonal of a square is incommensurable with its side, so that, if we measure the diagonal of a square using the length of side as unit, the number  obtained by measurement is irrational. The number  . is similarly irrational, obtained on measuring the circumference of a circle, the diameter of which is taken as unit.

Reference can usefully be made to decimal fractions, in order to understand the idea of irrational numbers. As is known from arithmetic, every rational number can be represented in the form of either a finite or an infinite decimal fraction, the infinite fraction being periodic in the latter case (simple periodic or compound periodic). For instance, on carrying out division of the numerator by the denominator in accordance with the rule for division into decimal fractions, we obtain:

Conversely, as is known from arithmetic, every periodic decimal fraction expresses a rational number.

to every infinite non-periodic decimal fraction there corresponds a certain irrational number. 

If only a few of the first decimal places are retained in this infinite decimal fraction, an approximate value is obtained below the irrational number represented by this fraction. Thus, for example, on extracting the square root in accordance with the usual rule to the third decimal place, we obtain:

All rational and irrational numbers are arranged in a certain definite order, according to their magnitudes. All these numbers form the aggregate of real numbers.

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek