What are irrational numbers?

Irrational numbers are numbers that cannot be written as the ratio of two integers. This means that they cannot be written as a fraction with an integer in the top and an integer in the bottom.

The Real Numbers are divided into two large subsets called “Rational Numbers” and “Irrational Numbers“. “Irrational” means not rational. Let's examine the Irrational Numbers. What are Irrational Numbers?
What are Irrational Numbers?  An irrational number is a number that is NOT rational. It cannot be expressed as a fraction with integer values in the numerator and denominator.
  • Examples:     What are Irrational Numbers?
  • When an irrational number is expressed in decimal form, it goes on forever without repeating.
  • Since irrational numbers are a subset of the real numbers, they possess all of the properties assigned to the real number system.
Regarding :
While it is popular to use 3.14 or to represent “pi”, these values are only estimates or approximations. Notice the differences in the decimal representations on the calculator screen at the right. What are Irrational Numbers?
What are Irrational Numbers? Pi (π) is an irrational number. Its decimal value goes on forever, and does not repeat.

There are certain radical values which fall into the irrational number category. For example, cannot be written as a “simple fraction”

which has integers in the numerator and the denominator. As a decimal, = 1.414213562373095048801688624 … which is a non-ending and non-repeating decimal, making irrational.

Consider: In a right triangle whose legs each measure 1 unit, the hypotenuse will measure units. By the Pythagorean Theorem, a2 + b2 = c2, we have:What are Irrational Numbers? But how does one measure a non-ending, non-repeating decimal such as in a diagram? What are Irrational Numbers?
A precise measurement of such a value is not possible. The best measurement is an approximation.

The Pythagoreans thought this idea of drawing a length that could never be precisely measured was absurd, crazy, not reasonable, not rational, thus the name “irrational”.

Such numbers were thought to be imperfections of mathematics and their existence was hidden by the Pythagoreans.

In fact, it is rumored that the Pythagorean (Hippassus), who first discovered this possibility, may have been thrown overboard during a sea voyage due to his discovery.

What are Irrational Numbers?

Irrational Numbers on a Number Line By definition, a number line is a straight line diagram on which every point corresponds to a real number.
What are Irrational Numbers? Since irrational numbers are a subset of the real numbers, and real numbers can be represented on a number line, one might assume that each irrational number has a “specific” location on the number line. NOPE! The best we can do to locate irrational numbers on a number line is to “estimate” their locations. This is the same “measurement” problem we saw demonstrated in the right triangle. You cannot measure a value that, as a decimal, is non-ending. And, you cannot locate the “exact” position of a non-ending decimal value on a number line.

“Estimates” of the locations of irrational numbers on number line:

Irrational Number — from Wolfram MathWorld

What are Irrational Numbers?
What are Irrational Numbers?

An irrational number is a number that cannot be expressed as a fraction for any integers and . Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.

There is no standard notation for the set of irrational numbers, but the notations , , or , where the bar, minus sign, or backslash indicates the set complement of the rational numbers over the reals , could all be used.

The most famous irrational number is , sometimes called Pythagoras's constant.

Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. Other examples include , , , etc. The Erdős-Borwein constant



What are Irrational Numbers?


(OEIS A065442; Erdős 1948, Guy 1994), where is the numbers of divisors of , and a set of generalizations (Borwein 1992) are also known to be irrational (Bailey and Crandall 2002).

Numbers of the form are irrational unless is the th power of an integer. Numbers of the form , where is the logarithm, are irrational if and are integers, one of which has a prime factor which the other lacks.

is irrational for rational . is irrational for every rational number (Niven 1956, Stevens 1999), and (for measured in degrees) is irrational for every rational with the exception of (Niven 1956).

is irrational for every rational (Stevens 1999).

The irrationality of e was proven by Euler in 1737; for the general case, see Hardy and Wright (1979, p. 46). is irrational for positive integral . The irrationality of pi itself was proven by Lambert in 1760; for the general case, see Hardy and Wright (1979, p. 47).

Apéry's constant (where is the Riemann zeta function) was proved irrational by Apéry (1979; van der Poorten 1979). In addition, T. Rivoal (2000) recently proved that there are infinitely many integers such that is irrational. Subsequently, he also showed that at least one of , , …

, is irrational (Rivoal 2001).

From Gelfond's theorem, a number of the form is transcendental (and therefore irrational) if is algebraic , 1 and is irrational and algebraic. This establishes the irrationality of Gelfond's constant (since What are Irrational Numbers?), and . Nesterenko (1996) proved that is irrational. In fact, he proved that , and are algebraically independent, but it was not previously known that was irrational.

Given a polynomial equation

What are Irrational Numbers?


where are integers, the roots are either integral or irrational. If is irrational, then so are , , and .

  • Irrationality has not yet been established for , , , or (where is the Euler-Mascheroni constant).
  • Quadratic surds are irrational numbers which have
    periodic continued fractions.
  • Hurwitz's irrational number theorem
    gives bounds of the form
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for the best rational approximation possible for an arbitrary irrational number , where the are called Lagrange numbers and get steadily larger for each “bad” set of irrational numbers which is excluded.

The series


where is the divisor function, is irrational for and 2.

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What Are Irrational Numbers?

What are irrational numbers? Good question, but don’t worry! Irrational numbers aren’t unreasonable, though they may seem a bit mysterious. After all, one of my favorite numbers, pi (π = 3.14159…), is irrational! Irrational numbers are just particular kinds of real numbers — specifically, those numbers that are not rational.

What are Irrational Numbers?

In this article, we’ll review rational and irrational numbers, focusing on the unique properties of the irrationals. We’ll learn how important and ubiquitous these fascinating numbers are. And we’ll even talk about a famous murder allegedly caused by an irrational number!

Real, Rational, and Irrational Numbers

By definition, a real number is irrational if it is not rational.

But what exactly is a real number? And what are the rational numbers?

We’d better start at the beginning!

The Ever Expanding Number System

  • People have been using numbers since as long as we have records and artifacts to prove it.
  • At first, people only used the counting (or natural) numbers, 1, 2, 3, 4, 5, and so on.
  • So for most of human history, the only “real” numbers were those that you could count things with.

The quipu is the ancient Peruvian equivalent of a thumb drive — not quite as high capacity of course! Numbers and other information can be encoded using the positions and numbers of knots on the strings.

What are Irrational Numbers? (Definition, Examples & List)

Irrational numbers are the numbers that cannot be represented as a simple fraction. It is a contradiction of rational numbers but is a type of real numbers. Hence, we can represent it as RQ, where the backward slash symbol denotes ‘set minus’ or it can also be denoted as R – Q, which means set of real numbers minus set of rational numbers.

The calculations based on these numbers are a bit complicated. For example, √5, √11, √21, etc are irrational. If such numbers are used in arithmetic operations, then first we need to evaluate the values under root. These values could be sometimes recurring also. Now let us find out its definition, lists of irrational numbers, how to find them, etc., in this article.

Table of Contents:

Irrational Numbers Definition

An irrational number is a real number that cannot be expressed as a ratio of integers, for example, √ 2 is an irrational number. Again, the decimal expansion of an irrational number is neither terminating nor recurring.

How do you know a number is irrational? The real numbers which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0 are known as irrational numbers. For Example  √ 2 and √ 3 etc.

Whereas any number which can be represented in the form of p/q, such that, p and q are integers and q ≠ 0 is known as a rational number

What are Irrational Numbers?

Irrational Number symbol

Generally, the symbol used to represent the irrational symbol is “P”.  Since the irrational numbers are defined negatively, the set of real numbers (R) that are not the rational number (Q), is called an irrational number.

The symbol P is often used because of the association with the real and rational number. (i.e) because of the alphabetic sequence P, Q, R.

But mostly, it is represented using the set difference of the real minus rationals, in a way R- Q or RQ.

Irrational Number Properties

The following are the properties of rational numbers:

  • The addition of an irrational number and a rational number gives an irrational number.  For example, let us assume that x is an irrational number, y is a rational number, and the addition of both the numbers x +y gives a rational number z.
  • While Multiplying any irrational number with any nonzero rational number results in an irrational number. Let us assume that if xy=z is rational, then x =z/y is rational, contradicting the assumption that x is irrational. Thus, the product xy must be irrational.
  • The least common multiple (LCM) of any two irrational numbers may or may not exist.
  • The addition or the multiplication of two irrational numbers may be rational; for example, √2. √2 = 2. Here, √2 is an irrational number. If it is multiplied twice, then the final product obtained is a rational number. (i.e) 2.

What Is Irrational Number

  1. A number is irrational if and only if its decimal representation is non-terminating and non-repeating. e.g.……………. etc.
  2. Rational number and irrational number taken together form the set of real numbers.
  3. If a and b are two real numbers, then either (i) a > b    or    (ii) a =  b    or       (iii)  a < b
  4. Negative of an irrational number is an irrational number.
  5. The sum of a rational number with an irrational number is always irrational.
  6. The product of a non-zero rational number with an irrational number is always an irrational number.
  7. The sum of two irrational numbers is not always an irrational number.
  8. The product of two irrational numbers is not always an irrational number.
  9. In division for all rationals of the form (q ≠ 0), p & q are integers, two things can happen either the remainder becomes zero or never becomes zero.
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Type (1) Example:   = 0.875 This decimal expansion 0.875 is called terminating. ∴ If remainder is zero then decimal expansion ends (terminates) after finite number of steps. These decimal expansion of such numbers terminating.

Type (2) Example:  = 0.333……… =  or   = 0.142857142857….. =   In both examples remainder is never becomes zero so the decimal expansion is never ends after some or infinite steps of division. These type of decimal expansions are called non terminating. In above examples, after Ist step & 6 steps of division (respectively) we get remainder equal to dividend so decimal expansion is repeating (recurring). So these are called non terminating recurring decimal expansions. Both the above types (1 & 2) are rational numbers.

Types (3) Example: The decimal expansion 0.

327172398……is not ends any where, also there is no arrangement of digits (not repeating) so these are called non terminating not recurring.

 These numbers are called irrational numbers. Example: 0.1279312793         rational          terminating 0.1279312793….    rational          non terminating

or                                       recurring

0.32777              rational          terminating

or                rational         non terminating

0.32777…….                           & recurring 0.5361279             rational         terminating 0.3712854043….   irrational       non terminating non recurring 0.10100100010000 rational       terminating

0.10100100010000….    irrational       non terminating non recurring.

Irrational Number Example Problems With Solutions

Example 1:    Insert a rational and an irrational number between 2 and 3. Sol.    If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then is an irrational number lying between a and b. Also, if a,b are rational numbers, then  is a rational number between them. ∴ A rational number between 2 and 3 is

= 2.5

An irrational number between 2 and 3 is

=  = 

Example 2:   Find two irrational numbers between 2 and 2.5. Sol.   If a and b are two distinct positive rational numbers such that ab is not a perfect square of a rational number, then  is an irrational number lying between a and b. ∴ Irrational number between 2 and 2.5 is

=  = 

Similarly, irrational number between 2 and   is  So, required numbers are  and 

Example 3:   Find two irrational numbers lying between and . Sol.   We know that, if a and b are two distinct positive irrational numbers, then  is an irrational number lying between a and b.

∴ Irrational number between  and is  =  = 61/4 Irrational number between   and 61/4 is  = 21/4 × 61/8. Hence required irrational number are 61/4 and 21/4 × 61/8.      Example 4:    Find two irrational numbers between 0.12 and 0.13. Sol.

   Let a = 0.12 and b = 0.13. Clearly, a and b are rational numbers such that a < b. We observe that the number a and b have a 1 in the first place of decimal. But in the second place of decimal a has a 2 and b has 3. So, we consider the numbers c = 0.

1201001000100001 …… and,     d = 0.12101001000100001…….

Clearly, c and d are irrational numbers such that a < c < d < b.

Example 5:   Prove that is  irrational number Sol.    Let us assume, to the contrary, that is rational. So, we can find integers r and s (≠0) such that . Suppose r and s not having a common factor other than 1.

Then, we divide by the common factor to get  where a and b are coprime. So,  b = a. Squaring on both sides and rearranging, we get 2b2 = a2. Therefore, 2 divides a2. Now, by Theorem  it following that 2 divides a.

So, we can write a = 2c for some integer c.

Substituting for a, we get 2b2 = 4c2, that is,

b2 = 2c2. This means that 2 divides b2, and so 2 divides b (again using Theorem  with p = 2). Therefore, a and b have at least 2 as a common factor. But this contradicts the fact that a and b have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that  is rational.

So, we conclude that  is irrational.

Example 6:   Prove that is  irrational number. Sol.    Let us assume, to contrary, that  is rational. That is, we can find integers a and b (≠0) such that .

 Suppose a and b not having a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime. So, b = a. Squaring on both sides, and rearranging, we get 3b2 = a2.

Therefore, a2 is divisible by 3, and by Theorem, it follows that a is also divisible by 3. So, we can write a = 3c for some integer c.

Substituting for a, we get 3b2 = 9c2, that is,

b2 = 3c2. This means that b2 is divisible by 3, and so b is also divisible by 3 (using Theorem with p = 3). Therefore, a and b have at least 3 as a common factor. But this contradicts the fact that a and b are coprime. This contradicts the fact that a and b are coprime.

  • This contradiction has arisen because of our incorrect assumption that  is rational.
  • Example 7:  Prove that  is irrational Sol.    Method I : Let  is rational number ∴  =    (p, q are integers, q ≠ 0) ∴ 7 –  =  ⇒ =  Here p, q are integers
  • ∴  is also integer
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So, we conclude that  is irrational. ∴ LHS =  is also integer but this  is contradiction that  is irrational so our assumption is wrong that   is rational ∴  is irrational proved. Method II : Let  is rational we know sum or difference of two rationals is also rational


= = rational but this is contradiction that  is irrational ∴  is irrational     proved.

Example 8:    Prove that  is irrational. Sol.    Let  is rational ∴ = is rational (∵ Q product of two rationals is also rational)

but this is contradiction that  is irrational

∴  is irrational proved.

Example 9:    Prove that  is irrational. Sol.    Let  is rational ∴ =  (∵ Q division of two rational no. is also rational)

∴ is rational

but this is contradiction that  is irrational

∴ is irrational

Example 10:    Find 3 irrational numbers between 3 & 5. Solution:    ∵ 3 and 5 both are rational The irrational are 3.127190385…………… 3.212325272930………



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Irrational Number

An irrational number is real number that cannot be expressed as a ratio of two integers. When an irrational number is written with a decimal point, the numbers after the decimal point continue infinitely with no repeatable pattern.

The number “pi” or π (3.14159…) is a common example of an irrational number since it has an infinite number of digits after the decimal point. Many square roots are also irrational since they cannot be reduced to fractions.

For example, the √2 is close to 1.414, but the exact value is indeterminate since the digits after the decimal point continue infinitely: 1.414213562373095…

This value cannot be expressed as a fraction, so the square root of 2 is irrational.

As of 2018, π has been calculated to 22 trillion digits and no pattern has been found.

If a number can be expressed as a ratio of two integers, it is rational. Below are some examples of irrational and rational numbers.

  • 2 – rational
  • √2 – irrational
  • 3.14 – rational
  • π – irrational
  • √3 – irrational
  • √4 – rational
  • 7/8 – rational
  • 1.333 (repeating) – rational
  • 1.567 (repeating) – rational
  • 1.567183906 (not repeating) – irrational

NOTE: When irrational numbers are encountered by a computer program, they must be estimated.

Updated: June 5, 2018

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Examples of Irrational Numbers

An irrational number cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.

  • Pi, which begins with 3.14, is one of the most common irrational numbers. Pi is determined by calculating the ratio of the circumference of a circle (the distance around the circle) to the diameter of that same circle (the distance across the circle). Pi has been calculated to over a quadrillion decimal places, but no pattern has ever been found; therefore it is an irrational number.
  • e, also known as Euler's number, is another common irrational number. The number is named for Leonard Euler, who first introduced e in 1731 in a letter he wrote; however, he had started using the number in 1727 or 1728. e is a universal number. The beginning of this number written out is 2.71828. e is the limit of (1 + 1/n)n as n approaches infinity. This expression is part of the discussion surrounding the subject of compound interest.
  • The Square Root of 2, written as √2, is also an irrational number. The first part of this number would be written as 1.41421356237…but the numbers go on into infinity and do not ever repeat, and they do not ever terminate. A square root is the opposite of squaring a number, meaning that the square root of two times the square root of two equals two. This means that 1.41421356237… multiplied by 1.41421356237… equals two, but it is difficult to be exact in showing this because the square root of two does not end, so when you actually do the multiplication, the resulting number will be close to two, but will not actually be two exactly. Because the square root of two never repeats and never ends, it is an irrational number. Many other square roots and cubed roots are irrational numbers; however, not all square roots are.The Golden Ratio, written as a symbol, is an irrational number that begins with 1.61803398874989484820…

These examples of different irrational numbers are provided to help you better understand what it means when a number is considered an irrational number.

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