Rational and Irrational

This is one pair of terms that we encounter both in Mathematics and in everyday English. We often hear statements such as, "She is a very rational person" or "Don't be irrational!"

So, is there a connection between the mathematical meaning and the English one?

Let's begin with the Mathematical point of view.

Rational Numbers

A rational number is any number that can be written in the form:

p/q

where p and q are integers and q ≠ 0.

Rational numbers have one of the following decimal forms:

• Terminating decimals (e.g., 0.25, 3.75)

• Repeating decimals (e.g., 0.333..., 1.272727...)

In other words, rational numbers follow a pattern.

As fractions, they appear neat and organized.As decimals, they either come to an end or settle into a repeating sequence.

Irrational Numbers

Irrational numbers cannot be expressed as a ratio of two integers.

Their decimal representations are:

• Non-terminating

• Non-repeating
  

The digits continue indefinitely without forming any predictable pattern.

Many square roots are irrational. For example:

√3

√7

√11

One of the most famous irrational numbers is π (Pi), the ratio of a circle's circumference to its diameter.

Note: in school books when π is considered as 22/7, it is only to ease the calculations. In true sense, representng an irrational number like π as a fraction of two integers is fundamentally incorrect.

Other well-known irrational numbers include:

φ (Phi), the Golden Ratio

e, Euler's Number

Now for the English Point of View...

By now, you may already see the resemblance.

A rational person is someone whose thoughts and actions follow a logical pattern. Their decisions can usually be explained and understood.

An irrational person, on the other hand, may appear to have no discernible pattern at all. Their next move can be as unpredictable as the digits of π.

Mathematicians describe irrational numbers as numbers that never repeat themselves.

English teachers might describe irrational people in much the same way.

And if you've ever argued with an irrational person, you'll know that—just like π—the conversation can seem to go on forever!