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Irrational numbers

  • Last Updated : 03 Mar, 2021

The numbers we usually study and have been concerned about up until now are the positive whole numbers, 1,2,3, and so on, used for counting. These are called natural numbers and have been with us for so many millennia. 

A famous mathematician Kronecker reputedly said: 

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“God created the natural numbers; all the rest is the work of man.”



Then, the basic necessities of life led to the creation of fractions like 3/4, 1/5, and so on. These numbers were called rational numbers. 

Note: These numbers are not called rational numbers because they are “reasonable”, they are called rational because they are ratios of whole numbers. 

All of these numbers can be represented at a particular position in the number line. 

But around 2500 years ago, Greeks discovered something else while studying geometry. They noticed to their surprise that there were some numbers which they couldn’t express as ratio whole numbers. 

For example: 

They noticed while making a square of side length unit 1. The length of the diagonal cannot be represented by any ratio of whole numbers. 

In modern mathematical terms, such a number is called an irrational number. Geometrically it means that there is no common unit (however small it may be), such that the side and diagonal of the square are a whole number multiple of it. Similarly, the circumference of a circle is an irrational multiple of diameter. That multiple is “pi”. 



Now let’s formally define rational numbers and irrational numbers. 

Rational Numbers

A number “x” is a rational number if it can be written in the form of “p/q” where p and q are integers (q ≠ 0). 

Example: 25 is a rational number. How? 25 can be written as \frac{25}{1}  where p = 25 and q = 1. Similarly, 5.5 is also a rational number, it can be expressed as \frac{11}{2}  and so on. 

Irrational Numbers

A number “x” is called irrational if cannot be written in the form of “p/q” where p and q are integers (q ≠ 0). Just like there are infinitely many rational numbers, there are infinitely many irrational numbers. 

Example: √2, √3, \pi

√2 = 1.41421356237309504880…

\pi = 3.14159265358979323846264338327950…

Another clue in recognizing an irrational number is that the decimal goes on without repeating. 

Popular Irrational Numbers:

  • Pi, π = 3.14159265358979323846264338327950…. This is a really famous irrational number. People have calculated its value up to quadrillion decimal but still haven’t found any pattern yet.
  • Euler’s number “e”. It is also very popular in math. In this case, also, people tried calculating it up to a lot of decimals but still, no pattern was found. e = 2.7182818284590452353602874713527 (and more …).
  • The golden ratio. This is an irrational number found in many fields like computer science, design, art, architecture.

Is √2 an Irrational number?

Proof:



To begin with, let’s suppose it is a Rational Number

Since it is Rational, It can be represented in the most simplified form P/Q where P/Q are Integers (Q≠ 0) and P/Q can not be further simplified,

Which means that the fraction P/Q is irreducible.

                                                       P/Q = √2

 Squaring both Sides:

                                                      (P/Q)2 = 2

                                                       P2/Q2 = 2

                                                       P2 = 2 Q2

Here, It is clear that P2 is divisible by 2. Therefore, P2 is an Even Number.

Since, P2 is Even, P has to be an Even number too.



Therefore, P can be written as 2A as it is divisible by 2.

By putting the value P = 2A, we will get:

                                                       (2A)2 = 2Q2

                                                                            4A2 = 2Q2

                                                               2A2 = Q2

Here, It is observed that Q2 is also an Even number as it is divisible by 2.

Since Q2 is Even, Q has to be Even too. Therefore, Both P and Q turned out to be Even numbers, which means they can be Simplified further.

It is a Contradiction as it was already defined to be a fraction in its simplest form.

Therefore, √2 cannot be Rational, It is an Irrational number.

Properties of Irrational Numbers

Addition of an Irrational and Rational number:

The addition of an irrational and a rational number always gives an irrational number. 

For example, it’s known that √2 = 1.41421356237309504880… Now √2 + 1 = 2.41421356237309504880…. This is still irrational. 

Multiplication of an irrational number with a non-zero rational number.

Multiplication of any irrational number with any nonzero rational number results in an irrational number. 

Proof: Let x be an irrational number and y be a non-zero rational number. 

We want to know if z = xy is irrational or rational ? 

This will be proved by contradiction. Let’s assume z is rational number. 

If z is rational, then x = z/y where both z and y are rational number. This makes x as rational number. This is a contradiction, which means that the previous assumption was wrong. So, z will always be irrational number. 

Product of two Irrational Numbers:

When multiplying a rational number, it is not necessary that the resulting number is always irrational. 



  • π × π = π2 is irrational.
  • But √2 × √2 = 2 is rational.

Sum of Two Irrational Numbers:

The answer to this is also similar to the above property. The Sum of two irrational numbers is sometimes rational sometimes irrational.  

  • 3√2 + 4√3 is irrational.
  • (3√2 + 6) + (- 3√2) = 6, this is rational.

Fun Fact: Apparently Hippasus (one of Pythagoras’ students) discovered irrational numbers when trying to write the square root of 2 as a fraction (using geometry, it is thought). Instead, he proved the square root of 2 could not be written as a fraction, so it is irrational.

But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!

Mystery of Pi

Let’s take a circle, measure its circumference, and divide it by its diameter. It will always be a constant if measured accurately.  

This constant ratio is denoted by the greek symbol \pi  (read as pi). That is, 

\frac{circumference}{diameter} = \pi \\

It is an important universal constant, it occurs at lots of places in our universe and daily lives. It was not created by humans, it was discovered. We discovered one of the places where pi occurred, it was geometry. 

So, what is the value of Pi? 

π = 3.14159265358979323846264338327950…



It is not infinite, it is an irrational number. 

Note: We often take \frac{22}{7}  as value of Pi, but it is an approximation. 

Now one might think, how is pi irrational? One can measure the circumference, One can measure the diameter, and then take their ratio. So it must be rational. 

Actually, no one can encounter such a case, if the diameter is measured and it is rational. Then circumference must be irrational and vice versa. So either diameter or circumference. One of them will always be irrational. Usually, the measuring instruments are not precise enough. If there existed a perfect measuring scale, it would tell that at least one of the numbers in the fraction is irrational.

Some Sample Problems on Irrational Numbers

Question 1: Do these Numbers come under the category of Irrational numbers: 5, 3.45, 4.444444…, √9.

Answer: 

These numbers mentioned above are not Irrational number.

  • 5 is a Whole number and therefore, is Rational
  • 3.45 is a number with Terminating Decimal and therefore, it is Rational too.
  • 4.444444… is a Number with Repeating Decimal Expansion, It is Rational.
  • √9 is 3 i.e; the square root of 9 is 3 and 3 is a Whole number. Therefore, √9 is Rational.

Question 2: “Every real number is an irrational number”. True or False?

Answer

False, All numbers are real numbers and all non-terminating real numbers are irrational number. For example 2, 3, 4, etc. are some example of real numbers and these are not irrational.

Question 3: Determine whether the following numbers are Rational or Irrational.

√3, 74, 8.432432432…, 3.14159265358979…, √11, 55/5.

Answer

74, 8.432432432…, and 55/5(11) are Rational numbers as either they are Integers or their decimal expansions are terminating, repeating.

√3, 3.14159265358979…, and √11 are Irrational numbers as their decimal Expansions are Non-terminating, Non-repeating.

Question 4: Why Integers are not Irrational numbers?

Answer

Integers (Either Positive, Negative or Zero) are not Irrational but Rational numbers since they can be represented in the simplest fraction form P/Q (where Q ≠ 0).




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