## Difference Between Rational and Irrational Numbers

The term “numbers” brings to our mind what are generally classified as positive integer values greater than zero. Other classes of numbers include **whole numbers** and **fractions**, **complex** and **real numbers** and also **negative integer values**.

Extending the classifications of numbers further, we encounter **rational** and **irrational** numbers. A rational number is a number that can be written as a fraction. In other words, the rational number can be written as a ratio of two numbers.

Consider, for example, the number ** 6**. It can be written as the ratio of two numbers viz.

**and**

*6***, leading to the ratio**

*1***. Likewise,**

*6/1***, which is written as a fraction, is a rational number.**

*2/3*We can, thus, define a rational number, as a number written in the form of a fraction, wherein both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. By definition, therefore, every whole number is also a rational number.

A ratio of two large numbers such as (*129,367,871**)**/**(**547,724,863*** )** would also constitute an example of a rational number for the simple reason that both the numerator and the denominator are whole numbers.

Conversely, any number that cannot be expressed in the form of a fraction or a ratio is termed as irrational. The most commonly cited example of an irrational number is *√**2** (**1.414213*** …)**. Another popular example of an irrational number is the numerical constant

*π*

*(*

*3.141592…***.**

*)*An irrational number can be written as a decimal, but not as a fraction. Irrational numbers are not often used in daily life although they do exist on the number line. There are an infinite number of irrational numbers between ** 0** and

**on the number line. An irrational number has endless non-repeating digits to the right of the decimal point.**

*1*Note that the oft-cited value of ** 22/7** for the constant

**is in fact only one the values of**

*π***. By definition, the circumference of a circle divided by twice its radius is the value of π. This leads to multiple values of**

*π***, including, but not limited to,**

*π***and so on1.**

*333/106, 355/113*Only the square roots of the square numbers; i.e., the square roots of the *perfect squares* are rational.

*√1*** = 1** (Rational)

** √2** (Irrational)

** √3** (Irrational)

*√4*** = 2** (Rational)

** √5, √6, √7, √8** (Irrational)

*√9*** = 3** (Rational) and so on.

Further, we note that, only the *n*th roots of *n*th powers are rational. Thus, the ** 6th** root of

**is rational, because**

*64***is a**

*64***power, namely the**

*6th***power of**

*6th***. But the**

*2***root of**

*6th***is irrational.**

*63***is not a perfect**

*63*

*6***power.**

*th*Inevitably, the decimal representation of irrationals comes into picture and poses some interesting results.

When we express a *rational* number as a decimal, then either the decimal will be ** exact** (as in

**=**

*1/5***or it will be**

*0.20)***(as in,**

*inexact*

*1/3 ≈***). In either case, there will be a predictable pattern of digits. Note that when an**

*0.3333**irrational*number is expressed as a decimal, then clearly it will inexact, because otherwise, the number would be rational.

Moreover, there will not be a predictable pattern of digits. For example,

*√2 ≈**1.4142135623730950488016887242097*

Now, with rational numbers, we occasionally encounter ** 1/11 = 0.0909090**.

The use of both the equal sign (** =**) and three dots (

**) implies that though is it not possible to express**

*ellipsis***exactly as a decimal, we can still approximate it with as many decimal digits as permitted to get close to**

*1/11***.**

*1/11*Thus, the decimal form of ** 1/11** is deemed inexact. By the same token, the decimal form of

**which is 0.25, is exact.**

*¼*Coming to the decimal form for irrational numbers, they are going to be always inexact. Continuing with the example of *√*** 2**, when we write

**. . . (note the use of ellipsis), it immediately implies that no decimal for**

*√2 = 1.41421356237***will be exact. Further, there will not be a predictable pattern of digits. Using concepts from numerical methods, again, we can rationally approximate for as many decimal digits as till such point that we are close to**

*√2***.**

*√2*Any note on rational and irrational numbers cannot end without the obligatory proof as to why √2 is irrational. In doing so, we also elucidate, the classic example of a *proof by cont**radiction.*

Suppose √2 is rational. This leads us to represent it as a ratio of two integers, say ** p** and

**.**

*q**√2 = p/q*

Needless to say, ** p** and

**have no common factors, for if there were to be any common factors, we would have cancelled them out from the numerator and the denominator.**

*q*Squaring both sides of the equation, we end up with,

*2 = p**2**/ q**2*

This can be conveniently written as,

*p**2 **= 2q**2*

The last equation suggests that *p*** 2** is even. This is possible only if

**itself is even. This in turn implies that**

*p*

*p***is divisible by**

*2***. Hence,**

*4*

*q***and consequently**

*2***must be even. So**

*q***and**

*p***are both even which is a contradiction to our initial assumption that they have no common factors. Thus,**

*q***cannot be rational. Q.E.D.**

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