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Irrational numbers

Here you will learn about irrational numbers, including what an irrational number is, examples of irrational numbers and how to identify irrational numbers.

Students will first learn about irrational numbers as part of the number system in 8 th grade.

What is an irrational number?

An irrational number is a real number or set of real numbers that cannot be written as a fraction of two integers (whole numbers). It is a non-terminating decimal that cannot be expressed as a fraction.

For example,

\sqrt{3}=1.73205… is a non-terminating decimal number which is irrational because it cannot be expressed as a fraction in the form \cfrac{a}{b} where a and b are integers.

Properties of irrational numbers

Irrational numbers have several properties that distinguish them from rational numbers. A set of irrational numbers can have the following properties:

• The decimal representation, or decimal expansion of an irrational number continues on forever, without repeating.
• Irrational numbers cannot be expressed in the form of a ratio of integers.
• The square roots of non-perfect squares are always irrational.
• The least common multiple (LCM) of any two irrational numbers may or may not exist.

Famous irrational numbers

There are several famous irrational numbers. These include,

Surds are types of irrational numbers. A surd is the root of a number which produces a non-terminating decimal.

Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8: The Number System (8.NS.A.1) Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

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How to identify if a number is an irrational number

In order to identify if a number is an irrational number:

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.

Identify what type of root it is and write a list of corresponding powers.

Identify where the integer inside the root falls on the list.

Irrational numbers examples

Example 1: identifying if the root of an integer is rational or irrational without a calculator.

Is \sqrt{30} a rational or an irrational number?

30 is an integer.

2 Identify what type of root it is and write a list of corresponding powers.

This is a square root.

The list of square numbers is 1,4,9,16,25,36,49…

3 Identify where the integer inside the root falls on the list.

30 lies between two square numbers.

\sqrt{30} falls between two numbers on the list, which means it is a surd and is an irrational number.

\sqrt{30}= is an irrational number between 5 and 6.

Example 2: identifying if the root of an integer is rational or irrational without a calculator

Is \sqrt[3]{56} a rational or an irrational number?

56 is an integer.

This is a cube root.

The list of cube numbers: 1, 8, 27, 64, 125…

56 lies between two cube numbers.

\sqrt[3]{27}=3

\sqrt[3]{64}=4

\sqrt[3]{56} falls between two numbers on the list, which means it is a surd and is an irrational number.

\sqrt[3]{56}= is an irrational number between 3 and 4.

Example 3: identifying if the root of a fraction is rational or irrational without a calculator

Is \sqrt{\cfrac{36}{121}} a rational or an irrational number?

\cfrac{36}{121} is a fraction.

The list of square numbers is 1,4,9,16,25,36,49,64,81,100,121,144…

Both 36 and 121 are square numbers.

\sqrt{36}=6

\sqrt{121}=11

Therefore, \sqrt{\cfrac{36}{121}}=\cfrac{\sqrt{36}}{\sqrt{121}}=\cfrac{6}{11} \, .

\sqrt{\cfrac{36}{121}} is equal to a number on the list, which means that this is not a surd and is therefore a rational number.

\sqrt{\cfrac{36}{121}} is a rational number.

Example 4: identifying if the root of a decimal is rational or irrational without a calculator

Is \sqrt{2.5} a rational or an irrational number?

2.5 is a decimal, so we must convert this to a fraction.

2.5=\cfrac{25}{10}

10 lies between two square numbers so \sqrt{10}= \text{ is an irrational number.}

25 is a square number so \sqrt{25}=5 \text{ which is a rational number.}

However, both numbers need to be rational for the fraction to be rational.

Because one of the numbers is irrational then the fraction will be irrational.

Example 5: identifying if the root of a decimal is rational or irrational without a calculator

Is \sqrt[5]{0.00005} a rational or an irrational number?

0.00005 is a decimal, so we must convert this to a fraction.

0.00005=\cfrac{5}{100000}

This is a 5 th root.

The list of integers to the power of 5,

10^{5}=100000

5 lies between 1^{5} and 2^{5}, therefore \sqrt[5]{5}= \text{ is an irrational number.}

100000 is 10^{5} therefore \sqrt[5]{100000}=10 which is a rational number.

Example 6: estimating the value of a surd

Estimate the value of \sqrt{50} to one decimal place without using a calculator.

50 is an integer.

The list of square numbers is 1,4,9,16,25,36,49,64,…

50 lies between two square numbers.

\sqrt{49}=7

\sqrt{64}=8

\sqrt{50}= \text{ is an irrational number between } 7 \text{ and } 8.

As 50 is very close to 49, then \sqrt{50} will be very close to 7. Picturing a number line may also help you to estimate your answer.

Estimate \sqrt{50} \approx 7.1

(A calculator gives the answer 7.071067812… )

Teaching tips for irrational numbers

• Students should have a solid understanding of rational numbers before being introduced to irrational numbers.
• Provide students with real life examples of irrational numbers, including the diagonal of a unit square or the ratio of the circumference to the diameter of a circle (\pi).
• While worksheets have their place within the math classroom, consider using interactive technology, like educational apps or online tools, to allow students to explore and manipulate irrational numbers.

Easy mistakes to make

• Assuming all fractions are rational numbers All rational numbers can be written as fractions but not all fractions are rational numbers. If the fraction is in the form \cfrac{a}{b} \, , \; a and b are integers, and b ≠ 0 , then the number is rational. However, if a or b are not integers then the fraction could represent an irrational number. For example, \cfrac{\sqrt{2}}{3} is a fraction which is irrational.
• Assuming all non-terminating decimals are irrational All irrational numbers are non-repeating decimals. However not all non-terminating decimals are irrational numbers. Recurring decimals are non-terminating decimals which are rational. For example, \sqrt{5}=2.23606… is a non terminating decimal which is an irrational number. \cfrac{8}{9}=0.888…=0.\dot{8} is a repeating decimal; a non terminating decimal which is a rational number.
• Writing a negative irrational number incorrectly Here is an example of a positive irrational number, \sqrt{3}=1.73205… To write the negative irrational number of the same magnitude you must write -\sqrt{3} and not \sqrt{-3}. Try both of these on your calculator. – \sqrt{3}=-1.73205… \sqrt{-3}= you will get an error message on your calculator.
• Not all fractions are rational numbers The definition of an irrational number is a number that cannot be expressed as a fraction in the form \cfrac{a}{b} , where a and b are integers and b ≠ 0. But this does not mean that all fractions are rational. For example, \cfrac{\sqrt{2}}{2} is a fraction which is also an irrational number because the numerator is irrational. The same would be true if the denominator was irrational.
• The square root of a positive number that is not a square number is an irrational number Square numbers (also known as perfect squares) are the numbers produced by the square of an integer. 1, 4, 9, 16, 25, 36… When we take the square root of a square number, the answer is rational; when we take the square root of any other positive integer, the answer is irrational. For example, \sqrt{72}=8.48528… is an irrational number because 72 is NOT a square number. \sqrt{12}= 3.46410… is an irrational number because 12 is NOT a square number. Note that \sqrt{12} can be simplified as \sqrt{4}\times\sqrt{3}=2\sqrt{3}, but as 3 is not a square number then 2\sqrt{3} produces an irrational number. When we take the square root of the quotient of two square numbers, the answer is rational. For example, \sqrt{0.09}= \sqrt{\cfrac{9}{100}} = \cfrac{ \sqrt{9}}{ \sqrt{100}} = \cfrac{3}{10} = 0.3.

Related types of numbers lessons

• Even numbers
• Odd numbers
• Number sets
• Whole numbers
• Prime and composite numbers
• Prime numbers
• Natural numbers
• Rational numbers
• Absolute value

Practice irrational numbers questions

1. Which of the following is irrational?

2. Which of the following is irrational?

None of these answers are irrational.

\sqrt[4]{16}=2 and \sqrt[4]{81}=3 so

\sqrt[4]{64}= is an irrational number between 2 and 3.

3. Which of the following calculations gives an irrational answer?

\sqrt{16}=4 and \sqrt{25}=5, so \sqrt{21} lies between these and is therefore irrational.

4. Which of the following numbers is not irrational?

Although \pi is an irrational number, the fraction can be simplified to cancel out \pi so this number is rational.

5. The value of \sqrt{76} lies between which two integers?

1,4,9,16,25,36,49,64 \, {\color{red}(76)} \, 81,100,121, 144,169

76 lies between 64 and 81 on the list of square numbers.

6. State an irrational number that lies between 11 and 12.

Therefore, the square root of any number between 121 and 144 will be an irrational number between 11 and 12.

\sqrt{130} is the correct answer from the multiple choice but there are many possible solutions.

For example, \sqrt{125} or \sqrt{142.576}.

Irrational numbers FAQs

First introduced by Leonhard Euler, the Euler’s number is often noted as an “e” and is a mathematical constant that is equal to approximately 2.71828.

The understanding of irrational numbers goes back to ancient Greek mathematicians from 6 th century BCE. The story is told that Hippasus of Metapontum, a member of the Pythagorean school, was the one who discovered the existence of irrational numbers. The discovery of irrational numbers was challenged at the time, because it challenged some of the Pythagorean beliefs at the time.

When any irrational numbers are multiplied by another nonzero rational number, the product will be an irrational number.

The next lessons are

• Rounding numbers
• Factors and multiples

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Irrational number

Because the rational numbers are countable while the reals are uncountable , one can say that the irrational numbers make up "almost all" of the real numbers.

There are two types of irrational numbers: algebraic and transcendental.

• Algebraic number
• Rational number
• Transcendental number
• Jan Gullberg, Mathematics

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Irrational numbers - practice problems

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Problems on Irrational Numbers

Till here we have learnt many concepts regarding irrational numbers. Under this topic we will be solving some problems related to irrational numbers. It will contain problems from all topics of irrational numbers.

Before moving to problems, one should look at the basic concepts regarding the comparison of irrational numbers.

For comparing them, we should always keep in mind that if square or cube roots of two numbers (‘a’ and ‘b’) are to be compared, such that ‘a’ is greater than ‘b’, then a\(^{2}\) will be greater than b\(^{2}\) and a\(^{3}\) will be greater than b\(^{2}\) and so on, i.e., n\(^{th}\) power of ‘a’ will be greater than n\(^{th}\) power of ‘b’. 

The same concept is to be applied for the comparison between rational and irrational numbers. 

So, now let’s have look at some problems given below:

1. Compare √11 and √21.

Solution: 

Since the given numbers are not the perfect square roots so the numbers are irrational numbers. To compare them let us first compare them into rational numbers. So,

(√11)\(^{2}\) = √11 × √11 = 11.

(√21)\(^{2}\) = √21 × √21 = 21.

Now it is easier to compare 11 and 21. 

Since, 21 > 11. So, √21 > √11.

2. Compare √39 and √19.

Since the given numbers are not the perfect square roots of any number, so they are irrational numbers. To compare them, we will first compare them into rational numbers and then perform the comparison. So,

(√39)\(^{2}\) = √39 × √39 = 39.

(√19)\(^{2}\) = √19 × √19 = 19

Now it is easier to compare 39 and 19. Since, 39 > 19.

So,√39 > √19.

3. Compare \(\sqrt[3]{15}\) and \(\sqrt[3]{11}\).

Solution:  

Since the given numbers are not the perfect cube roots. So, to make comparison between them e first need to convert them into rational numbers and then perform the comparison. So,

\((\sqrt[3]{15})^{3}\) = \(\sqrt[3]{15}\) × \(\sqrt[3]{15}\) × \(\sqrt[3]{15}\) = 15.

\((\sqrt[3]{11})^{3}\) = \(\sqrt[3]{11}\) × \(\sqrt[3]{11}\) × \(\sqrt[3]{11}\) = 11.

Since, 15 > 11. So, \(\sqrt[3]{15}\) > \(\sqrt[3]{11}\).

4.  Compare 5 and √17.

Among the numbers given, one of them is rational while other one is irrational. So, to make comparison between them, we will raise both of them to them to the same power such that the irrational one becomes rational. So,

(5)\(^{2}\) = 5 × 5 = 25.

(√17)\(^{2}\) = √17 x × √17 = 17.

Since, 25 > 17. So, 5 > √17.

5.  Compare 4 and \(\sqrt[3]{32}\).

Among the given numbers to make comparison, one of them is rational while other one is irrational. So, to make comparison both numbers will be raised to the same power such that the irrational one becomes rational. So,

4\(^{3}\)= 4 × 4 × 4 = 64.

\((\sqrt[3]{32})^{3}\) = \(\sqrt[3]{32}\) × \(\sqrt[3]{32}\) × \(\sqrt[3]{32}\) = 32.

Since, 64 > 32. So, 4 > \(\sqrt[3]{32}\).

6.  Rationalize \(\frac{1}{4 + \sqrt{2}}\).

Since the given fraction contains irrational denominator, so we need to convert it into a rational denominator so that calculations may become easier and simplified ones. To do so we will multiply both numerator and denominator by the conjugate of the denominator. So,

\(\frac{1}{4 + \sqrt{2}} \times (\frac{4 - \sqrt{2}}{4 - \sqrt{2}})\)

⟹ \(\frac{4 - \sqrt{2}}{4^{2} - \sqrt{2^{2}}}\)

⟹ \(\frac{4 - \sqrt{2}}{16 - 2}\)

⟹ \(\frac{4 - \sqrt{2}}{14}\)

So the rationalized fraction is: \(\frac{4 - \sqrt{2}}{14}\).

7.  Rationalize \(\frac{2}{14 - \sqrt{26}}\).

\(\frac{2}{14 - \sqrt{26}} \times \frac{14 + \sqrt{26}}{14 + \sqrt{26}}\)

⟹ \(\frac{2(14 - \sqrt{26})}{14^{2} - \sqrt{26^{2}}}\)

⟹ \(\frac{2(14 - \sqrt{26})}{196 - 26}\)

⟹ \(\frac{2(14 - \sqrt{26})}{170}\)

 So, the rationalized fraction is: \(\frac{2(14 - \sqrt{26})}{170}\).

Irrational Numbers

Definition of Irrational Numbers

Representation of Irrational Numbers on The Number Line

Comparison between Two Irrational Numbers

Comparison between Rational and Irrational Numbers

Rationalization

Problems on Rationalizing the Denominator

Worksheet on Irrational Numbers

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Irrational Numbers

An Irrational Number is a real number that cannot be written as a simple fraction:

  1.5 is rational, but π is irrational

Irrational means not Rational (no ratio)

Let's look at what makes a number rational or irrational ...

Rational Numbers

A Rational Number can be written as a Ratio of two integers (ie a simple fraction).

Example: 1.5 is rational, because it can be written as the ratio 3/2

Example: 7 is rational, because it can be written as the ratio 7/1

Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

But some numbers cannot be written as a ratio of two integers ...

...they are called Irrational Numbers .

Example: π (Pi) is a famous irrational number.

π  = 3.1415926535897932384626433832795... (and more)

We cannot write down a simple fraction that equals Pi.

The popular approximation of 22 / 7 = 3.1428571428571... is close but not accurate .

Another clue is that the decimal goes on forever without repeating.

Cannot Be Written as a Fraction

It is irrational because it cannot be written as a ratio (or fraction), not because it is crazy!

So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

Example: 9.5 can be written as a simple fraction like this:

So it is a rational number (and so is not irrational )

Here are some more examples:

Square Root of 2

Let's look at the square root of 2 more closely.

The answer is the square root of 2 , which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...

... in fact we cannot write the square root of 2 using a ratio of two numbers ...

... (you can learn why on the Is It Irrational? page) ...

... and so we know it is an irrational number .

Famous Irrational Numbers

But √4 = 2 is rational, and √9 = 3 is rational ...

... so not all roots are irrational.

Note on Multiplying Irrational Numbers

Have a look at this:

• π × π = π 2 is known to be irrational
• But √2 × √2 = 2 is rational

So be careful ... multiplying irrational numbers might result in a rational number!

Fun Facts ....

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!

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Irrational Numbers

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• Sandeep Bhardwaj
• Sravanth C.
• Mahindra Jain
• Mursalin Habib
• Peter Taylor
• Yasir Soltani
• Rajat Pathak

Irrational numbers are real numbers that cannot be expressed as the ratio of two integers . More formally, they cannot be expressed in the form of \(\frac pq\), where \(p\) and \(q\) are integers and \(q\neq 0\). This is in contrast with rational numbers , which can be expressed as the ratio of two integers. One characteristic of irrational numbers is that their decimal expansion does not repeat or terminate.

Examples of Irrational Numbers

Properties of irrational numbers, irrationality of \(\sqrt{2}\).

Main Article: History of Irrational Numbers

The first man to recognize the existence of irrational numbers might have died for his discovery. Hippassus of Metapontum was an ancient Greek philosopher of the Pythagorean school of thought. Supposedly, he tried to use his teacher's famous theorem \( a^{2}+b^{2}= c^{2}\) to find the length of the diagonal of a unit square. This revealed that a square's sides are incommensurable with its diagonal, and that this length cannot be expressed as the ratio of two integers. The other Pythagoreans believed dogmatically that only positive rational numbers could exist. They were so horrified by the idea of incommensurability that they threw Hippassus overboard on a sea voyage, and vowed to keep the existence of irrational numbers an official secret of their sect. However, there are good reasons to believe Hippassus's demise is merely an apocryphal myth. Historical documents referencing the incident are both sparse and written 800 years after the time of Pythagoras and Hippassus. It wasn't until approximately 300 years after Hippassus's time that Euclid would give his proof for the irrationality of \(\sqrt{2}.\)

The Pythagoreans had likely manually measured the diagonal of a unit square. However, they would have regarded such a measurement as an approximation close to a precise rational number that gave the true length of the diagonal. Before Hippassus, Pythagoreans had no reason to suspect that there were real numbers that in principle, not merely in practice, could not be measured or counted to. Numbers were the spiritual basis of their philosophy and religion for the Pythagoreans. Cosmology, physics, ethics, and spirituality were predicated on the premise that "all is number." They believed that all things--the number of stars in the sky, the pitches of musical scales, and the qualities of virtue--could all be described by and apprehended through rational numbers.

Irrational numbers arise in many circumstances in mathematics. Examples include the following:

• The hypotenuse of a right triangle with base sides of length 1 has length \( \sqrt{2}\), which is irrational.
• More generally, \( \sqrt{D}\) is irrational for any integer \( D\) that is not a perfect square. For demonstration, we will prove that \(\sqrt 2\) is an irrational number in a later section Irrationality of \(\sqrt 2\) .
• The ratio \(\pi\) of the circumference of a circle to its diameter is irrational.
• The base \(e\) of the natural logarithm is irrational.

Check out the following example for better understanding:

Is \( \frac{ 7 \sqrt{2+4} }{\sqrt{2} } \) rational or irrational? We have \[ \frac{ 7 \sqrt{2+4} }{\sqrt{2} } = \frac{ 7 \sqrt{6} }{\sqrt{2} } = 7 \sqrt{3}. \] By property 2 above, \(\sqrt{3}\) is an irrational number since 3 is not a perfect square. Therefore \( 7 \sqrt{3} \) is an irrational number. \(_\square\)

Try the following problem:

What can you say about \(\sqrt{2+7}\times \sqrt 7?\)

• Taking the sum of an irrational number and a rational number gives an irrational number. To see why this is true, suppose \(x\) is irrational, \(y\) is rational, and the sum \(x+y\) is a rational number \(z\). Then we have \(x = z-y\), and since the difference of two rational numbers is rational, this implies \(x\) is rational. This is a contradiction since \(x\) is irrational. Therefore, the sum \(x+y\) must be irrational.
• Multiplying an irrational number with any nonzero rational number gives an irrational number. We argue as above to show that if \(xy = z\) is rational, then \({x = \frac{z}{y}}\) is rational, contradicting the assumption that \(x\) is irrational. Therefore, the product \(xy\) must be irrational.
• The lowest common multiple (LCM) of two irrational numbers may or may not exist.
• The sum or the product of two irrational numbers may be rational; for example,

\[ \sqrt{2} \cdot \sqrt{2} = 2.\]

Therefore, unlike the set of rational numbers, the set of irrational numbers is not closed under multiplication.

Here are some examples based on the above properties:

Is \( \sqrt{36} \) rational or irrational? Since \( \sqrt{36} =6, \) this is a rational number. \(_\square\)

Show that \( \sqrt{2} + \sqrt{3}\) is not rational. We give a proof by contradiction. If \( \sqrt{2}+\sqrt{3}\) is rational, then \( (3-2) \times \frac {1}{\sqrt{3} + \sqrt{2}} = \sqrt{3}-\sqrt{2}\), implying \( \sqrt{3} - \sqrt{2}\) is also rational. Since \( \big(\sqrt{3} + \sqrt{2}\big) - \big(\sqrt{3}-\sqrt{2}\big) = 2 \sqrt{2}\), we obtain \( 2 \sqrt{2}\) is rational. Thus, \( 2 \sqrt{2} \times \frac {1}{2} = \sqrt{2}\) is also rational, which is a contradiction. \(_\square\)

We generalize the result above to show that \( \sqrt{D}\) is rational if and only if \( D\) is a perfect square.

Given integers \( n\) and \( m\), if \( n^{\frac {1}{m}}\) is rational, then \( n^{\frac {1}{m}}\) is an integer. In particular, the only rational \( m^\text{th}\) roots of integers \( n\) are the integers.

Let \( n^{\frac {1}{m}}= \frac {a}{b} \), where \( a\) and \( b\) are coprime integers. Then taking powers and clearing denominators gives \( b^m n = a^m \). If \( p\) is a prime that divides \( b\), then \( p\) divides \( b^m n\), so \( p \) divides \( a^m\) and thus \( p\) divides \( a\). Since \( a\) and \( b\) are coprime, there is no prime that divides both \( a\) and \( b\). Hence, no prime divides \( b\), implying \( b=1\). Therefore, \( n^{\frac {1}{m}} = a\) is an integer. \( _\square\)

Here are some problems to try:

True or False?

The sum of two irrational numbers is always an irrational number.

Read the following statements:

1) \(\frac { e }{ \pi } \) is a rational number. 2) \(\frac { \pi }{ e } \) is an irrational number. 3) \(\frac { \pi+e }{ e } \) is a rational number.

Give your answer as the mean of the serial numbers of the statements which are true. \(\big(\)E.g., if all statements are true, the answer is \(\frac { 1+2+3 }{ 3 } =2.\big)\)

\(\) Details and Assumptions:

• \(e\) may not necessarily be the exponential constant and \(\pi \) may not necessarily be equal to 3.14159...

\[\Large \color{blue}{e},~~ \color{green}{\pi}\]

Find the lowest common multiple (LCM) of the two numbers above.

\(\) Details and Assumptions :

• If you think that the existence of this LCM is unknown to humans, submit "Does not exist" as your answer.
• If you think that it is extremely close to zero, but not zero, then you may press "0".
• If you think that it is extremely close to one, but not one, then you may press "1".

\[\large \color{orange}{-6\pi} \color{black},~~ \color{green}{\pi}\]

To clarify, the LCM of two irrational numbers exists if and only if their ratio is rational.

Inspiration

Below you can see the proof of the irrationality of \(\sqrt{2}.\)

Let's use the method known as proof by contradiction . Let's say \(\sqrt{2}\) is a rational number: \[\sqrt { 2 } =\frac { a }{ b},\] where \(a\) and \(b\) are coprime integers, i.e. \(a\) and \(b\) do not have prime factors in common and \(\gcd\left( a,b \right) =1\). In other words, \(\frac { a }{ b }\) is an irreducible fraction. Solving the equation above gives \[\begin{align} \sqrt { 2 } &=\frac { a }{ b }\\ 2&=\frac { { a }^{ 2 } }{ { b }^{ 2 } } \\ \\ \Rightarrow a^2&=2{ b }^{ 2 }. \end{align}\] As you can see, \({a}^{2}\) is even because it is the double of \({b}^{2}\). If \({a}^{2}\) is even \(a\) is also even because the square of an odd number is an odd number, and the square of an even number is an even number. If \(a\) is even, we can write it as \(2c=a,\) which implies \[\begin{align} 2{ b }^{ 2 } &={ a }^{ 2 }\\ &={ \left( 2c \right) }^{ 2 }\\ &={ 4c }^{ 2 }\\ \\ \Rightarrow { b }^{ 2 }&={ 2c }^{ 2 }. \end{align}\] As you can see, \(b\) is also even by the same reason as \(a\) is. Wait! We have a contradiction here. We say that \(a\) and \(b\) are coprime integers, i.e \(\frac{a}{b}\) is irreducible, but \(a\) and \(b\) are both even numbers which can't form an irreducible fraction. Thus, this is an impossible fraction, so \(\sqrt{2}\) can't be written as the ratio of 2 integers. \(_\square\)

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Irrational Numbers

What are irrational numbers, irrational numbers symbol, solved examples, practice problems, frequently asked questions, irrational numbers – introduction.

We use numbers in daily life for a variety of reasons. Also, we use different types of numbers for different purposes, such as natural numbers for counting, fractions for describing portions or parts of a whole, decimals for precision, etc. Today we will explore ‘Irrational Numbers’ in math, their applications, examples, operations. Let’s begin!

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Irrational numbers are the type of real numbers that cannot be expressed in the rational form $\frac{p}{q}$, where $p, q$ are integers and $q \neq 0$. 

In simple words, all the real numbers that are not rational numbers are irrational.

We see numbers everywhere around us and use them on a daily basis. Let’s quickly revise.

• Natural Numbers $= N = {1, 2, 3, 4, . . .}$ 
• Whole Numbers $= W = {0, 1, 2, 3, 4, . . .}$ 
• Integers $= Z = {. . . – 3,-2, -1, 0, 1, 2, 3, . . .}$
• Rational Numbers $= Q$

They include all the numbers of the form $\frac{p}{q}$, where $p, q$ are integers and $q \neq 0$. 

Decimal expansions for rational numbers can be either terminating or repeating decimals.

Examples: $\frac{1}{2} , \frac{11}{3}, \frac{5}{1}$, 3.25, 0.252525 . . .

• Irrational Numbers $= P$

Irrational numbers are the type of real numbers that cannot be expressed in the form $\frac{p}{q}, q \neq 0$. These numbers include non-terminating, non-repeating decimals

• Real Numbers $= R$

           Rational and irrational numbers together make real numbers. 

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Irrational Numbers Definition

Irrational numbers can be defined as real numbers that cannot be expressed in the form of $\frac{p}{q}$, where p and q are integers and the denominator $q \neq 0$. 

Example: 

The decimal expansion of an irrational number is non-terminating and non-recurring/non-repeating. So, all non-terminating and non-recurring decimal numbers are “irrational numbers.”

Example: Suppose a square has an area of 5 square meters. Calculate the length of each side.

The area of the square = 5 sq. meters

The length of each side is $\sqrt{5}$  meter $=$ 2.23606797749 . . . meter

This answer is in irrational form since we cannot express $\sqrt{5}$ in rational form. Also, the decimal expansion is non-terminating, non-repeating.

Irrational Numbers Examples

The following are examples of a few specific irrational numbers that are commonly used. 

• In math, we know “pi” as the circumference to diameter ratio. Is pi an irrational number? 

Yes! Pi or is an irrational number. The decimal expansion of $π =$ 3.14159265 . . . is neither terminating nor repeating decimal.

 (Understand that we use pi as 3.14 or $\frac{22}{7}$ to make calculations easier.)

• $\sqrt{2}$ is an irrational number.
• Euler’s number e is an irrational number, where $e$ $=$ 2.718281 . . .
• Golden ratio, $\varphi =$ 1.61803398874989 . . .
• Square root of non-perfect squares like $\sqrt{26}, \sqrt{63}$, etc.
• Square root of a prime numbers like $\sqrt{2}, \sqrt{3}$, etc.
• All non-terminating and non-recurring decimals.

Irrational Numbers List

Here’s a list of some common and frequently used irrational numbers.

• Pi or $\Pi=$ 3.14159265358979…
• Euler’s Number e $=$ 2.71828182845904…
• Golden ratio $\Theta =$ 1.61803398874989….

Generally, we use the symbol “P” to represent an irrational number, since  the set of real numbers is denoted by R and the set of rational numbers is denoted by Q. We can also represent irrational numbers using the set difference of the real minus rationals, in a way $\text{R} – \text{Q}$ or $\frac{R}{Q}$.

Are all Irrational Numbers Real Numbers?

Rational numbers and irrational numbers together form real numbers. So, all irrational numbers are considered to be real numbers. The real numbers which are not rational numbers are irrational numbers. Irrational numbers cannot be expressed as the ratio of two numbers. However, every real number is not an irrational number.

Properties of Irrational Numbers

The irrational numbers, being a type of real numbers, follow all the properties of real numbers. The following are the properties of irrational numbers: 

• When we add an irrational number and a rational number, it will always give an irrational number.  Example: $\sqrt{3} + \frac{2}{5}$
• When we multiply an irrational number with a non-zero rational number, it will result in an irrational number. Example: $\frac{2}{5} \times \sqrt{3}$
• The LCM (Least Common Multiple) of any two irrational numbers may or may not exist. 
• The addition or the multiplication of two irrational numbers may be rational. 

For example, $\sqrt{3} \times \sqrt{3} = 3$. Here, $\sqrt{3}$ is an irrational number. If we multiply it twice, then the final product obtained is a rational number, 3. 

Also, consider the example of addition of two irrational numbers that gives a rational number. $\sqrt{3} + (2$ $-$ $\sqrt{3} ) = 2$  

It means the set of irrational numbers is not closed under the multiplication process and addition process, unlike the set of rational numbers.

Operations on Two Irrational Numbers

We can do some operations on two or more irrational numbers like addition, subtraction, multiplication, and division. 

Addition of two irrational numbers may or may not be irrational.

Example 1: $\sqrt{2}$ and $3\sqrt{2}$

$\sqrt{2} + \sqrt{3}2 = \sqrt{2} + 3\sqrt{2} = 3\sqrt{2}$ is an irrational number.

Example 2: $6 + \sqrt{2}$ and $2$ $-$ $\sqrt{2}$

$6 + \sqrt{2} + 2$ $-$ $\sqrt{2} = (6 + 2) + (\sqrt{2}$ $-$ $\sqrt{2}) = 8$ is a rational number.

Subtraction

Subtraction of two irrational numbers may or may not be irrational. 

Example 1: $5 + \sqrt{3}$ and $2$ $-$ $3\sqrt{3}$

$5 + \sqrt{3}$ $-$ $(2$ $-$ $3\sqrt{3}) = (5$ $-$ $2) + (\sqrt{3}$ $-$ $3\sqrt{3}) = 3$ $-$ $2\sqrt{3}$ is an irrational number.

Example 2: $5 + 3\sqrt{3}$ and $4 + 3\sqrt{3}$

$5 + 3\sqrt{3}$ $-$ $( 4 + 3\sqrt{3}) = ( 5$ $-$ $4 ) + ( 3\sqrt{3}$ $-$ $3\sqrt{3}) = 1$ is a rational number.

Multiplication

The product of two irrational numbers may or may not be irrational. 

Example 1: $2\sqrt{3}$ and $6\sqrt{3}$

$2\sqrt{3} \times 6\sqrt{3} =2 \times 3 \times 6 = 36$ is a rational number.

Example 2: $6+\sqrt{2}$ and $2$

$(6 + \sqrt{2}) \times 2 = (6 \times 2) + (\sqrt{2} \times 2) = 12 + \sqrt{2}2$ is an irrational number.

The division of two irrational numbers may or may not be irrational. 

Example 1: $6 \sqrt{2} \div \sqrt{2}$ 

$\frac{6\sqrt{2}}{\sqrt{2}} = 6$ is a rational number.

Example 2: $(5 – \sqrt{2} ) \div \sqrt{2}$

$\frac{5-\sqrt{2}}{2} = \frac{5}{\sqrt{2}}$ $-$ $\frac{\sqrt{2}}{\sqrt{2}} = \frac{5} {\sqrt{2}}$ $- 1$ is an irrational number.

How to Find an Irrational Number between Two Numbers?

Let us find the irrational numbers between 3 and 4.

We know, square root of 9 is 3; $\sqrt{9} = \sqrt{3}$

and the square root of 16 is 4; $\sqrt16 = \sqrt4$

Therefore, $\sqrt{10},\sqrt{11}, \sqrt{12}$, etc., are irrational numbers between 3 and 4. These are not perfect squares and cannot be simplified further. Also, all the non-repeating, non-terminating decimals between 3 and 4 like 3.12537 . . . are irrational.

Rational Numbers vs. Irrational Numbers

The table illustrates the difference between rational numbers and irrational numbers.

• $\sqrt{2}$ is the first invented irrational number!

Hippasus, the Greek mathematician and the student of the great mathematician Pythagoras proved that “root two” could never be expressed as a fraction.

It was invented while calculating the length of the isosceles right-angled triangle using Pythagoras’ theorem.

$\text{AC}^2 = \text{AB}^2 + \text{BC}^2$

$1^2 + 1^ 2 = \text{AC}^2 \Rightarrow \text{AC} = \sqrt{2}$

$\sqrt{2}$ lies between the rational numbers 1 and 2 as the value of $\sqrt{2}$ is 1.41421 . . .

14 March or 3/14 is celebrated as Pi day since the date matches the first three digits of the decimal expansion of pi. 

In this article, we learned about irrational numbers. An irrational number is a number that cannot be expressed in the form of a fraction or ratio. To read more such informative articles on other concepts, do visit our website . We, at SplashLearn, are on a mission to make learning fun and interactive for all students. 

1. Find two irrational numbers between 3.14 and 3.2.

Solution: The decimal expansion of an irrational number is non-terminating and non-repeating. The two irrational numbers between 3.14 and 3.2 can be 3.15155155515555 . . . and 3.19876543 . . .

2. Identify rational and irrational numbers from the following numbers.

 $\sqrt{5}$, 2, $\sqrt{11}$, 3.56, 1.3333 . . ., 100, 4.5346782 . . .

Solution:  

Rational numbers:  2, 3.56, 100, 1.3333 . . .

Irrational numbers: $\sqrt{5}, \sqrt{11}$, 4.5346782 . . .

3. Add $(2 + \sqrt{3} )$ and $( 3$ $- \sqrt{3} )$ . Is the sum irrational?

 $( 2 + \sqrt{3} ) + ( 3$ $-\sqrt{3} ) = 2 + 3 +\sqrt{3} – \sqrt{3} = 5$

The sum is a rational number.

4. Compare irrational numbers $\sqrt{12}$ and $\sqrt{21}$ .

Solution: Since $12 \lt 21$, we can say that $\sqrt{12} < \sqrt{21}$.

5. Which decimals are irrational numbers?Solution: The decimals that are non-repeating and non-recurring are irrational numbers.

Attend this quiz & Test your knowledge.

Which of the following is an irrational number?

Find the value of $(\sqrt{2} + \sqrt{3}) (\sqrt{2} - \sqrt{3})$., on multiplying 0 and irrational number, we always get:, on dividing $16\sqrt{3}$ by $2\sqrt{3}$, we get:, the decimal expansion of $\sqrt{6}$ is.

Why are irrational numbers included in the set of real numbers?

The numbers we can express in the form of decimals are known as real numbers. We can express the irrational numbers in terms of decimals and we can also represent them on a real number line, hence irrational numbers are in the set of real numbers.

What is another name for irrational numbers?

Another name for irrational numbers is surds.

Are all real numbers irrational?

No, real numbers are divided into rational and irrational numbers. Every irrational number is a real number, but every real number is not an irrational number.

What is the difference between integers and irrational numbers?

Integers are a type of rational numbers and can be expressed in the form of fraction. For example: – 2, 9, etc. On the other hand, irrational numbers are a type of numbers that cannot be expressed in the form of a ratio or fraction. For example: $\sqrt{3}$, 𝛑

How many irrational numbers are there between two rational numbers?

There are infinite irrational numbers between any two rational numbers.

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Irrational Numbers

What are Irrational Numbers?

An irrational number is a real number that cannot be expressed as a ratio of integers; for example, √2 is an irrational number.  We cannot express any irrational number in the form of a ratio, such as p/q, where p and q are integers, q≠0. Again, the decimal expansion of an irrational number is neither terminating nor recurring . 

Irrational Meaning: The meaning of irrational is not having a ratio or no ratio can be written for that number. In other words, we can say that irrational numbers cannot be represented as the ratio of two integers.

What are the examples of Irrational Numbers?

The common examples of irrational numbers are pi(π=3⋅14159265…), √2, √3, √5, Euler’s number (e = 2⋅718281…..), 2.010010001….,etc.

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. are irrational. 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.

Is Pi an irrational number?

Yes, Pi (π) is an irrational number because it is neither terminating nor repeating decimal. Also, Pi is not equal to 22/7 as 22/7 is a rational number while pi is an irrational number. The value of π is 3.141592653589………..

Note- Rational numbers (Q) and Irrational numbers (P or Q’ ) are always alternate with each other.

Therefore, 22/7 ≠  π  but they are alternate or next to each other.

Irrational Number Symbol

Generally, the symbol used to represent the irrational symbol is “P”.  Since 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 R\Q.

Properties of Irrational numbers

Since irrational numbers are the subsets of real numbers, irrational numbers will obey all the properties of the real number system. The following are the properties of irrational 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 an irrational number z.
• Multiplication of 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.
• The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.

List of Irrational Numbers

The famous irrational numbers consist of Pi, Euler’s number, and Golden ratio. Many square roots and cube root numbers are also irrational, but not all of them. For example, √3 is an irrational number, but √4 is a rational number. Because 4 is a perfect square, such as 4 = 2 x 2 and √4 = 2, which is a rational number.  It should be noted that there are infinite irrational numbers between any two real numbers. For example, say 1 and 2, there are infinitely many irrational numbers between 1 and 2. Now, let us look at famous irrational numbers’ values.

Note – √prime number always gives an irrational number.

Are Irrational Numbers Real Numbers?

In Mathematics, all irrational numbers are considered real numbers, which should not be rational numbers. It means irrational numbers cannot be expressed as the ratio of two numbers. For example, the square roots that are not perfect will always result in an irrational number. 

Sum and Product of Two Irrational Numbers

Now, let us discuss the sum and the product of irrational numbers.

Product of Two Irrational Numbers

Statement: The product of two irrational numbers is sometimes rational or irrational

For example, √2 is an irrational number, but when √2  is multiplied by √2, we get the result 2, which is a rational number.

(i.e.,) √2 x √2 = 2 

We know that π is also an irrational number, but if π is multiplied by π, the result is π 2 , which is also an irrational number.

(i.e..) π x π = π 2

It should be noted that while multiplying the two irrational numbers, it may result in an irrational number or a rational number. 

Sum of Two Irrational Numbers

Statement: The sum of two irrational numbers may be rational or irrational.

Like the product of two irrational numbers, the sum of two irrational numbers will also result in a rational or irrational number.

For example, if we add two irrational numbers, say 3 √2+ 4√3, a sum is an irrational number.

But, let us consider another example, (3+4√2) + (-4√2 ), the sum is 3, which is a rational number.

So, we should be very careful while adding and multiplying two irrational numbers, because it might result in an irrational number or a rational number.

Irrational Number Theorem and Proof

The following theorem is used to prove the above statement

Theorem : Given p is a prime number and a 2  is divisible by p,  (where a is any positive integer), then it can be concluded that p also divides a .

Proof: Using the Fundamental Theorem of Arithmetic, the positive integer can be expressed in the form of the product of its primes as:

a = p 1  × p 2  × p 3………..   × p n …..(1)

Where, p 1,  p 2 ,  p 3 , ……,  p n  represent all the prime factors of a .

Squaring both the sides of equation (1),

a 2 = ( p 1  × p 2  × p 3………..   × p n) ( p 1  × p 2  × p 3 ………..  × p n )

⇒a 2 = (p 1 ) 2  × (p 2 ) 2   × ( p 3   ) 2 ……….. × (p n ) 2

According to the Fundamental Theorem of Arithmetic , the prime factorization of a natural number is unique, except for the order of its factors.

The only prime factors of a 2 are p 1 , p 2,  p 3………..,  p n . If p is a prime number and a factor of a 2 , then p is one of  p 1 , p 2 ,  p 3………..,  p n . So, p will also be a factor of a .

Hence, if a 2  is divisible by p , then p also divides a .

Now, using this theorem, we can prove that √ 2 is irrational.

How to Find an Irrational Number?

Let us find the irrational numbers between 2 and 3. We know the square root of 4 is 2; √4 =2 and the square root of 9 is 3;  √9 = 3 Therefore, the number of irrational numbers between 2 and 3 are √ 5, √ 6, √ 7, and √ 8, as these are not perfect squares and cannot be simplified further. Similarly, you can also find irrational numbers, between any other two perfect square numbers.

Another case:

Let us assume a case of  √ 2. Now, how can we find if  √ 2 is an irrational number?

Suppose √ 2 is a rational number. Then, by the definition of rational numbers, it can be written that,

√ 2 =p/q    …….(1)

Where p and q are co-prime integers and q ≠ 0 (Co-prime numbers are those numbers whose common factor is 1).

Squaring both sides of equation (1), we have

2 = p 2 /q 2

⇒ p 2 = 2 q  2     ………. (2)

From the theorem stated above, if 2 is a prime factor of p 2 , then 2 is also a prime factor of p .

So, p  = 2 × c , where c is an integer.

Substituting this value of p in equation (3), we have

(2c) 2  = 2 q  2

⇒  q 2 = 2c  2  

This implies that 2 is a prime factor of q 2 also. Again from the theorem, it can be said that 2 is also a prime factor of q .

According to the initial assumption , p and q are co-primes but the result obtained above contradicts this assumption as p and q have 2 as a common prime factor other than 1. This contradiction arose due to the incorrect assumption that √ 2  is rational.

So, root 2 is irrational.

Similarly, we can justify the statement discussed in the beginning that if p is a prime number , then √  p  is an irrational number. Similarly, it can be proved that for any prime number p , √  p is irrational.

Irrational Numbers Solved Examples

Question 1 : Which of the following are Rational Numbers or Irrational Numbers?

2, -.45678…, 6.5,  √  3,  √ 2

Solution : Rational Numbers – 2, 6.5 as these have terminating decimals.

Irrational Numbers – -.45678…,  √  3,  √ 2 as these have a non-terminating non-repeating decimal expansion.

Question 2: Check if the below numbers are rational or irrational. 

2, 5/11, -5.12, 0.31 

Solution: Since the decimal expansion of a rational number either terminates or repeats. So, 2, 5/11, -5.12, 0.31 are all rational numbers.

Frequently Asked Questions (FAQs) on Irrational Numbers

What is an irrational number give an example., are integers irrational numbers, is an irrational number a real number, what are the five examples of irrational numbers, what are the main irrational numbers.

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Irrational Numbers

Irrational numbers are those real numbers that cannot be represented in the form of a ratio. In other words, those real numbers that are not rational numbers are known as irrational numbers. Hippasus, a Pythagorean philosopher, discovered irrational numbers in the 5th century BC. Unfortunately, his theory was ridiculed and he was thrown into the sea.

But irrational numbers exist, let's have a look at this page to get a better understanding of the concept, and trust us, you won't be thrown into the sea. Rather, by knowing the concept, you will also know the irrational number list, the difference between irrational and rational numbers, and whether or not irrational numbers are real numbers.

What are Irrational Numbers?

Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction , p/q where p and q are integers . The denominator q is not equal to zero (q ≠ 0). Also, the decimal expansion of an irrational number is neither terminating nor repeating.

Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio , such as p/q, where p and q are integers, q≠0. It is a contradiction of rational numbers.

Examples of Irrational Numbers

Given below are the few specific irrational numbers that are commonly used.

• ㄫ ( pi ) is an irrational number. π=3⋅14159265… The decimal value never stops at any point. Since the value of ㄫ is closer to the fraction 22/7, we take the value of pi as 22/7 or 3.14 (Note: 22/7 is a rational number.)
• √ 2 is an irrational number. Consider a right-angled isosceles triangle , with the two equal sides AB and BC of length 1 unit. By the Pythagoras theorem , the hypotenuse AC will be √2. √2=1⋅414213⋅⋅⋅⋅
• Euler's number e is an irrational number. e=2⋅718281⋅⋅⋅⋅
• Golden ratio, φ 1.61803398874989….

Properties of Irrational Numbers

Properties of irrational numbers help us to pick up irrational numbers out of a set of real numbers. Given below are some of the properties of irrational numbers:

• Irrational numbers consist of non-terminating and non-recurring decimals .
• These are real numbers only.
• When an irrational and a rational number are added, the result or their sum is an irrational number only. For an irrational number x, and a rational number y, their result, x+y = an irrational number.
• When any irrational numbers multiplied by any nonzero rational number, their product is an irrational number. For an irrational number x and a rational number y, their product xy = irrational.
• For any two irrational numbers, their least common multiple (LCM) may or may not exist.

How to Identify an Irrational Number?

We know that the irrational numbers are real numbers only which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. For example, √ 5 and √ 3, etc. are irrational numbers. On the other hand, the numbers which can be represented in the form of p/q, such that, p and q are integers and q ≠ 0, are rational numbers. Here are some tricks to identify irrational numbers.

• The numbers that are not perfect squares , perfect cubes , etc are irrational. For example √2, √3, √26, etc are irrational. But √25 (= 5), √0.04 (=0.2 = 2/10), etc are rational numbers.
• The numbers whose decimal value is non-terminating and non-repeating patterns are irrational. For example √2 = 1.4142135623730950488016887242097.... is irrational, whereas 1/7 = 0.14285714285714285714285714285714... is rational as we can observe that "142857" is keep getting repeated in the decimal portion.

Irrational Numbers Symbol

Before knowing the symbol of irrational numbers, let us discuss the symbols used for other types of numbers.

• N - Natural numbers
• I - Imaginary Numbers
• R - Real Numbers
• Q - Rational Numbers .

Real numbers consist of both rational and irrational numbers. (R-Q) defines that irrational numbers can be obtained by subtracting rational numbers (Q) from the real numbers (R). This can also be written as (R\Q). Hence Irrational Numbers Symbol = Q'.

Set of Irrational Numbers

Set of irrational numbers can be obtained by writing all irrational numbers within brackets. But we know that there are infinite number of irrational numbers. So we cannot list the entire set of irrational numbers. But here are a few subsets of set of irrational numbers.

• All square roots which are not a perfect squares are irrational numbers. Example: {√2, √3, √5, √8}
• Euler's number, Golden ratio , and Pi are some of the famous irrational numbers. Example: {e, ∅, ㄫ}
• The square root of any prime number is an irrational number. Example: {√2, √3, √5, √7, √11, √13, ...}

The table illustrates the list of some of the irrational numbers .

Differences Between Rational and Irrational Numbers

Any number which is defined in the form of a fraction p/q or ratio is called a rational number. This may consists of the numerator (p) and denominator (q), where q is not equal to zero. A rational number can be a whole number or an integer.

• 2/3 = 0.6666 = 0.67. Since the decimal value is recurring (repeating). So, we approximated it to 0.67
• √4 = 2 and -2, where both 2 and -2 are integers.

The table illustrates the difference between rational and irrational numbers.

Interesting Facts about Irrational Numbers

There are some cool and interesting facts about irrational numbers that make us deeply understand the why behind the what.

1. Accidental Invention of √2

The square root of 2 or √2 was the first invented irrational number when calculating the length of the isosceles triangle. He used the famous Pythagoras formula a 2 = b 2 + c 2

AC 2 =AB 2 +BC 2 ⇒ AC 2 =1 2 +1 2 ⇒ AC = √ 2

√2 lies between numbers 1 and 2 as the value is 1.41421... So he revealed that the length AC cannot be expressed in the form of fractions or integers.

2. The value of π

The value of π is approximately calculated to over 22 trillion digits without an end. A computer took about 105 days, with 24 hard drives, to calculate the value of pi.

3. Invention of Euler's Number e

The Euler's number is first introduced by Leonhard Euler, a Swiss mathematician in the year 1731. This 'e' is also called a Napier Number which is mostly used in logarithm and trigonometry .

Proof of an Irrational number:

Let's understand how to prove that a given non-perfect square is irrational. Here is stepwise proof of the same.

To prove: √2 is an irrational number.

Suppose, √2 is a rational number. Then, by the definition of rational numbers, it can be written that,

√2=p/q ...(1) where p and q are co-prime integers and \(q ≠ 0\) (Co-primes are those numbers whose common factor is 1).

Squaring both the sides of equation (1), we have

\(\begin{align}2 &= p^2/q^2\\⇒ p^2 &= 2 * q^2\qquad \dots(2)\end{align}\)

From the theorem, that states “Given p is a prime number and a 2 is divisible by p, (where a is any positive integer), then it can be concluded that p also divides a”, if 2 is a prime factor of p 2 , then 2 is also a prime factor of p.

So, p= 2 x c where c is an integer.

Substituting this value of p in equation (3), we have

\(\begin{align}(2c)^2 &= 2q^2\\⇒ q^2 &= 2c^2\end{align}\)

This implies that 2 is a prime factor of q 2 also. Again from the theorem, it can be said that 2 is also a prime factor of q.

According to the initial assumption, p and q are co-primes but the result obtained above contradicts this assumption as p and q have 2 as a common prime factor other than 1. This contradiction arose due to the incorrect assumption that √2 is rational.

So, √2 is irrational.

• Prove that Root 2 is Irrational
• Prove that Root 3 is Irrational
• Prove that Root 5 is Irrational
• Prove that Root 6 is Irrational
• Prove that Root 7 is Irrational
• Prove that Root 11 is Irrational

Rational and Irrational Numbers Worksheets

Rational and irrational numbers worksheets can provide a better understanding of why rational and irrational numbers are part of real numbers. Rational and irrational numbers worksheets include a variety of problems and examples based on operations and properties of rational and irrational numbers. It consists of creative and engaging fun activities where a child can explore end-to-end concepts of rational and irrational numbers in detail with practical illustrations.

Important Points on Irrational Numbers:

• The product of any two irrational numbers can be either rational or irrational. Example (a): Multiply √2 and π ⇒ 4.4428829... is an irrational number. Example (b): Multiply √2 and √2 ⇒ 2 is a rational number.
• The same rule works for quotient of two irrational numbers as well.
• The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.
• The sum and difference of any two irrational numbers is always irrational.

☛Related Articles:

Check out a few more interesting articles related to irrational numbers.

• Decimal Representation of Irrational Numbers
• Rational Numbers
• Rationalize the Denominator
• Is pi a rational or Irrational Number

Irrational Numbers Examples

Example 1: John is playing "Roll a dice-Number game" with his friend. John takes a turn and rolls a dice. He gets 5. If he gets 5, he is supposed to collect all the irrational numbers from his friend. Help John to collect all the irrational numbers without missing even one. {e, -5, √9,√13, π, -2/8}

Among the given numbers:

-5 is an integer. √9 is a perfect square. -2/8 has a recurring terminating decimal value. These numbers are rational numbers. The irrational numbers are e, √13, π. Therefore, John collected all the irrational numbers and those are e, √13 and π.

Answer: √13 and π

Example 2: Jade has a box with four irrational numbers. Jade wants only one irrational number which is closest to 3 and should not exceed 3. Help Jade to find out the right one. The irrational numbers in the box are √3, √6, √10, √5.

First, we find the value of these irrational numbers. √3 = 1.732020.., √6 = 2.449489.., √10 = 3.162277.., √5 = 2.236067... Thus, √6 = 2.449489... comes closest to 3. Therefore, √6 is the closest number to 3.

Example 3: State whether each of the following statement is True/False.

(a) Every rational number is an integer

(b) Every rational number is a whole number

(c) Every irrational number is a real number

(d) π is a rational number

(e) every real number is an irrational number

(f) sum of a rational and an irrational number is an irrational number.

(a) False. Example: 1/2 is a rational number but not an integer.

(b) False. Example: 1/2 is a rational number but not a whole number.

(c) True as both rational and irrational numbers together form the real numbers set.

(d) False as π is irrational.

(e) False as a real number can be either rational or irrational.

(f) True. Example: 2 + √2 is irrational.

Answer: (a) False (b) False (c) True (d) False (e) False (f) True

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Practice Questions on Irrational Numbers

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FAQs on Irrational Numbers

What is the definition of irrational numbers in math.

Irrational numbers are a set of real numbers that cannot be expressed in the form of fractions or ratios made up of integers. Ex: π, √2, e, √5. Alternatively, an irrational number is a number whose decimal notation is non-terminating and non-recurring.

How can you Identify an Irrational Number?

• For any number which is not rational is considered irrational.
• Irrational numbers can be written as decimals, but definitely not as fractions.
• Also, these numbers tend to have endless non-repeating digits to the right of the decimal.

Are Rational Numbers and Irrational Numbers Same?

No, rational and irrational numbers are not the same. All the numbers are represented in the form of p/q where p and q are integers and q does not equal to 0 is a rational number. Examples of rational numbers are 1/2, -3/4, 0.3, or 3/10. Whereas, we cannot express irrational numbers such as √2, ∛3, etc in the form of p/q.

What is the Difference Between Rational and Irrational Numbers?

Rational numbers are those that are terminating or non-terminating repeating numbers, while irrational numbers are those that neither terminate nor repeat after a specific number of decimal places.

Is 2/3 an Irrational Number?

No, 2/3 is not an irrational number. 2/3 = 0.666666.... which is a recurring decimal. Therefore, 2/3 is a rational number.

Why Rational Numbers and Irrational Numbers Are in the Set of Real Numbers?

The numbers which can be expressed in the form of decimals are considered real numbers. If we talk about rational and irrational numbers both forms of numbers can be represented in terms of decimals, hence both rational numbers and irrational numbers are in the set of real numbers.

Why Pi is an Irrational Number?

Pi is defined as the ratio of a circle's circumference to its diameter. The value of Pi is always constant. Pi (π) approximately equals 3.14159265359... and is a non-terminating non-repeating decimal number. Hence 'pi' is an irrational number.

How Many Irrational Numbers Lies Between Root 2 and Root 3?

We can have infinitely many irrational numbers between root 2 and root 3 . A few examples of irrational numbers between root 2 and root 3 are 1.575775777..., 1.4243443..., 1.686970..., etc.

Are Irrational Numbers Non-Terminating and Non-Recurring?

Yes, irrational numbers are non-terminating and non-recurring. Terminating numbers are those decimals that end after a specific number of decimal places. For example, 1.5, 3.4, 0.25, etc are terminating numbers.

• All terminating numbers are rational numbers as they can be written in the form of p/q easily.
• Whereas non-terminating and non-recurring numbers are considered as the never-ending decimal expansion of irrational numbers.

Why are Irrational Numbers Called Surds ?

A surd refers to an expression that includes a square root, cube root, or other root symbols. Surds are used to write irrational numbers precisely. All surds are considered to be irrational numbers but all irrational numbers can't be considered surds. Irrational numbers, which are not the roots of algebraic expressions, like π and e, are not surds.

7.1 Rational and Irrational Numbers

Learning objectives.

By the end of this section, you will be able to:

• Identify rational numbers and irrational numbers
• Classify different types of real numbers

Be Prepared 7.1

Before you get started, take this readiness quiz.

Write 3.19 3.19 as an improper fraction. If you missed this problem, review Example 5.4 .

Be Prepared 7.2

Write 5 11 5 11 as a decimal. If you missed this problem, review Example 5.30 .

Be Prepared 7.3

Simplify: 144 . 144 . If you missed this problem, review Example 5.69 .

Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six chapters of this book! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers , and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

In this chapter, we'll make sure your skills are firmly set. We'll take another look at the kinds of numbers we have worked with in all previous chapters. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.

We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

A rational number is a number that can be written in the form p q , p q , where p p and q q are integers and q ≠ 0 . q ≠ 0 .

All fractions, both positive and negative, are rational numbers. A few examples are

Each numerator and each denominator is an integer.

We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.

Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let's look at a few to see if we can write each of them as the ratio of two integers. We've already seen that integers are rational numbers. The integer −8 −8 could be written as the decimal −8.0 . −8.0 . So, clearly, some decimals are rational.

Think about the decimal 7.3 . 7.3 . Can we write it as a ratio of two integers? Because 7.3 7.3 means 7 3 10 , 7 3 10 , we can write it as an improper fraction, 73 10 . 73 10 . So 7.3 7.3 is the ratio of the integers 73 73 and 10 . 10 . It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 7.3 or −1.2684 ) −1.2684 ) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.

Example 7.1

Write each as the ratio of two integers: ⓐ −15 −15 ⓑ 6.81 6.81 ⓒ −3 6 7 . −3 6 7 .

Write each as the ratio of two integers: ⓐ −24 −24 ⓑ 3.57 . 3.57 .

Write each as the ratio of two integers: ⓐ −19 −19 ⓑ 8.41 . 8.41 .

Let's look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number , since a = a 1 a = a 1 for any integer, a . a . We can also change any integer to a decimal by adding a decimal point and a zero.

Integer −2 , −1 , 0 , 1 , 2 , 3 Decimal −2.0 , −1.0 , 0.0 , 1.0 , 2.0 , 3.0 These decimal numbers stop. Integer −2 , −1 , 0 , 1 , 2 , 3 Decimal −2.0 , −1.0 , 0.0 , 1.0 , 2.0 , 3.0 These decimal numbers stop.

We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.

Ratio of Integers 4 5 , − 7 8 , 13 4 , − 20 3 Decimal Forms 0.8 , −0.875 , 3.25 , −6.666… These decimals either stop or repeat. −6 . 66 — Ratio of Integers 4 5 , − 7 8 , 13 4 , − 20 3 Decimal Forms 0.8 , −0.875 , 3.25 , −6.666… These decimals either stop or repeat. −6 . 66 —

What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

Irrational Numbers

Are there any decimals that do not stop or repeat? Yes. The number π π (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat.

Similarly, the decimal representations of square roots of whole numbers that are not perfect squares never stop and never repeat. For example,

A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number .

Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

Let's summarize a method we can use to determine whether a number is rational or irrational.

If the decimal form of a number

• stops or repeats, the number is rational.
• does not stop and does not repeat, the number is irrational.

Example 7.2

Identify each of the following as rational or irrational:

• ⓐ 0.58 3 – 0.58 3 –
• ⓑ 0.475 0.475
• ⓒ 3.605551275… 3.605551275…

ⓐ 0.58 3 – 0.58 3 – The bar above the 3 3 indicates that it repeats. Therefore, 0.58 3 – 0.58 3 – is a repeating decimal, and is therefore a rational number.

ⓑ 0.475 0.475 This decimal stops after the 5 5 , so it is a rational number.

ⓒ 3.605551275… 3.605551275… The ellipsis (…) (…) means that this number does not stop. There is no repeating pattern of digits. Since the number doesn't stop and doesn't repeat, it is irrational.

ⓐ 0.29 0.29 ⓑ 0.81 6 – 0.81 6 – ⓒ 2.515115111… 2.515115111…

ⓐ 0.2 3 – 0.2 3 – ⓑ 0.125 0.125 ⓒ 0.418302… 0.418302…

Let's think about square roots now. Square roots of perfect squares are always whole numbers , so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

Example 7.3

ⓐ The number 36 36 is a perfect square, since 6 2 = 36 . 6 2 = 36 . So 36 = 6 . 36 = 6 . Therefore 36 36 is rational.

ⓑ Remember that 6 2 = 36 6 2 = 36 and 7 2 = 49 , 7 2 = 49 , so 44 44 is not a perfect square.

This means 44 44 is irrational.

Classify Real Numbers

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers . Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers , we get the set of real numbers .

Figure 7.2 illustrates how the number sets are related.

• Real Numbers

Real numbers are numbers that are either rational or irrational.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real”, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.

Example 7.4

Determine whether each of the numbers in the following list is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number.

ⓐ The whole numbers are 0 , 1 , 2 , 3 ,… 0 , 1 , 2 , 3 ,… The number 8 8 is the only whole number given.

ⓑ The integers are the whole numbers, their opposites, and 0 . 0 . From the given numbers, −7 −7 and 8 8 are integers. Also, notice that 64 64 is the square of 8 8 so − 64 = −8 . − 64 = −8 . So the integers are −7 , 8 , − 64 . −7 , 8 , − 64 .

ⓒ Since all integers are rational, the numbers −7 , 8 , and − 64 −7 , 8 , and − 64 are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so 14 5 and 5.9 14 5 and 5.9 are rational.

ⓓ The number 5 5 is not a perfect square, so 5 5 is irrational.

ⓔ All of the numbers listed are real.

We'll summarize the results in a table.

Determine whether each number is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number: −3 , − 2 , 0 . 3 – , 9 5 , 4 , 49 . −3 , − 2 , 0 . 3 – , 9 5 , 4 , 49 .

Determine whether each number is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number: − 25 , − 3 8 , −1 , 6 , 121 , 2.041975… − 25 , − 3 8 , −1 , 6 , 121 , 2.041975…

ACCESS ADDITIONAL ONLINE RESOURCES

• Sets of Real Numbers

Section 7.1 Exercises

Practice makes perfect.

In the following exercises, write as the ratio of two integers.

• ⓑ 3.19 3.19
• ⓑ −1.61 −1.61
• ⓑ 9.279 9.279
• ⓑ 4.399 4.399

In the following exercises, determine which of the given numbers are rational and which are irrational.

0.75 0.75 , 0.22 3 – 0.22 3 – , 1.39174… 1.39174…

0.36 0.36 , 0.94729… 0.94729… , 2.52 8 – 2.52 8 –

0 . 45 — 0 . 45 — , 1.919293… 1.919293… , 3.59 3.59

0.1 3 – , 0.42982… 0.1 3 – , 0.42982… , 1.875 1.875

In the following exercises, identify whether each number is rational or irrational.

Classifying Real Numbers

In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.

−8 −8 , 0 , 1.95286.... 0 , 1.95286.... , 12 5 12 5 , 36 36 , 9 9

−9 −9 , −3 4 9 −3 4 9 , − 9 − 9 , 0.4 09 — 0.4 09 — , 11 6 11 6 , 7 7

− 100 − 100 , −7 −7 , − 8 3 − 8 3 , −1 −1 , 0.77 0.77 , 3 1 4 3 1 4

Everyday Math

Field trip All the 5 th 5 th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 147 people. Each bus holds 44 44 people.

ⓐ How many buses will be needed?

ⓑ Why must the answer be a whole number?

ⓒ Why shouldn't you round the answer the usual way?

Child care Serena wants to open a licensed child care center. Her state requires that there be no more than 12 12 children for each teacher. She would like her child care center to serve 40 40 children.

ⓐ How many teachers will be needed?

Writing Exercises

In your own words, explain the difference between a rational number and an irrational number.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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• Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Prealgebra 2e
• Publication date: Mar 11, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/prealgebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/prealgebra-2e/pages/7-1-rational-and-irrational-numbers

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