Irrational Numbers Have Infinite Continued Fractions

The irrational numbers do not exist in nature because they are constructed in building the real numbers by the axiom of completeness. The irrational numbers are all the real numbers that are not rational numbers. That is, irrational number cannot be expressed as the ratio of two integers.

In this article, we will study what is an irrational number, its definition and symbol with properties and a list of irrational numbers. Let's find answers to are irrational numbers real numbers, how to find the sum and product of two irrational numbers, How to Find Irrational Number, the decimal expansion, Rational vs Irrational Number, Solved Examples and FAQs.

Irrational Numbers

Natural Numbers are a part of the number system. An understanding of irrational number is basic mathematics and is an important topic of algebra. An irrational Number is a special case of numbers in the entire number system. Irrational number is kind of the opposite of rational. They are real numbers that we can't write as a ratio \({p\over{q}}\) where p and q are integers, where q cannot be equal to zero. For example, \(\sqrt{2} = 1.414213….\) is irrational because we can't write that as a fraction of integers. An irrational number is hence, recurring numbers. Let's now see how can we define the irrational number.

Irrational Numbers Definition

An irrational number is a real number that cannot be expressed as the ratio of two integers. Equivalently, an irrational number, when expressed in decimal notation, never terminates nor repeats. It is 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.

Types of Irrational Numbers

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

Algebraic Irrational Number : An algebraic Irrational Number is a normal irrational number that is resulted from mathematical operations. Algebraic ones are those which have roots of the algebraic equation as the square root of 2.

Transcendental Irrational Number : A transcendental number is a number that is not algebraic, in the sense that it is not the solution of an algebraic equation with rational number coefficients. Transcendental numbers are irrational, but not all irrational numbers are transcendental. In 1844 Joseph Liouville discovered the existence of transcendental numbers. Transcendental numbers are usually the most famous- \(\pi\), e, etc.

Irrational Number Symbol

The symbol "P" is used for the set of Rational Numbers. The symbol Q is used for rational numbers. There is no generally accepted symbol for the irrationals. This is most likely because the irrationals are defined negatively: the set of real numbers that are not rational. Real numbers are denoted by R and rational numbers are denoted by P.

Properties of Irrational Numbers

Irrational number show distinct properties. Let's see what are properties of irrational number are:

• An irrational number is always real numbers.
• An irrational number cannot be expressed as a fraction.
• An irrational number are non-repeating and non-terminating as the decimal part never ends and never repeats itself.
• The value of the square root of any prime number is an irrational number.
• The sum of a rational number and an irrational number is irrational. The product of a rational number and an irrational number is irrational. This means that any operation between a rational and an irrational number, be it addition, subtraction, multiplication or divisions will always result in an irrational number only.
• If r is one irrational number and s is another irrational number, then r + s and r – s may or may not be irrational numbers and rs and r/s are may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number.
• If a and b are two distinct positive irrational numbers, then \(\sqrt{ab}\) is an irrational number lying between a and b.
• For any two irrational numbers, their least common multiple (LCM) may or may not exist.
• Irrational number is simplifications of Surds. When we can't simplify a number to remove a square root or cube root etc. then it is a surd. For example, \(\sqrt{2}\) (square root of 2) can't be simplified further so it is a surd.

List of Irrational Numbers

Here's a list of famous irrational number that are commonly used

Square Root of Primes: \(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{11}, \sqrt{13}, \sqrt{17}, \sqrt{19}…\)

The first irrational to be discovered was \(\sqrt{2}\). The Pythagoreans- and ancient Greek philosophical university and religious brotherhood- stumbled upon \(\sqrt{2}\) as the length of a diagonal of a square with side lengths 1 in the sixth century B.C. It was Hippasus who was one of the students of Pythagoras who discovered irrational number. However, the other followers of Pythagoras could not accept the existence of irrational number, and it is said that Hippasus was drowned at sea as a punishment from the gods!

Logarithms of primes with prime base: \(log_23, log_25, log_27, log_35, log_37…\)

The logarithm of a prime number with a prime base, like log_35 or log_72, is irrational.

\(\begin{matrix}
\text{ Assume } log_35 = {x\over{y}}, \text{ x and y are integers }, y ≠ 0.\\
3{x\over{y}} = 5 ( 3 < 5 \text{ therefore } {x\over{y}} > 1)\\
(3{x\over{y}})y = 5y\\
3x = 5y
\end{matrix}\)

᠎3 and 5 are prime numbers. x and y are integers. So the above equation is not balanced. Our assumption has led us to a contradiction. Therefore the assumption \(log_35 = {x\over{y}}\) = rational number is false. So log35 is an irrational number.

Sum of Rational and Irrational: \(3 + \sqrt{2}, 4 + \sqrt{7}…\)

Adding a rational number to an irrational number is an easy way to create new irrational number. See the lists of numbers created using this method:

• \(1 + \sqrt{2}, 2 + \sqrt{2}, 3 + \sqrt{2}, …..\)
• 1 + π, 2 + π, 2 + π, ….
• \(1 + log_35, 2 + log_35, 3 + log_35, …\)

Learn the various concepts of the Binomial Theorem here.

Product of Rational & Irrational Numbers

What works for the sum of a rational and an irrational number, works for their product also. This provides yet another method to create examples of an irrational number. See the lists of such numbers below:

• \(2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, …\)
• 2π, 3π, 4π, …
• \(2log_35, 3log_35, 4log_35, …\)

Infinite Continued Fraction

This is one of the ways to represent irrational number. It takes the form:

\(a_0+{1\over{a_1+{b_0\over{a_2+{b_1\over{…}}}}}}\)

For irrational number, we can limit \(a_i,b_i\) to be integers.

Pi (π)

The number Pi originated from geometry. It is the ratio of the circumference and the diameter of a circle. It remains constant, independent of the size of the circle. If the circumference of a circle is rational, the radius is irrational. Hence, pi is an irrational number. In India, some interesting values of \(\pi\) began to emerge. In 499, Aryabatha published \(\pi = 3.1416…\); Born in 598, Brahmagupta published \(\pi = \sqrt{10} = 3.1622…\); and Bhaskara, born 1114, said that \(\pi = 3.14156…\) China beat them all with Liu Hui (3.141024 to 3.142764) and Tsu Chung- Chih \((\frac{355}{113} = 3.1415929)\)

Number e

The Number e is the sum of Infinite Quotients. The number e is a recent discovery by Jacob Bernoulli. He was trying to compute a continuously compounded interest growth in the 17th century. In short, he was evaluating \((1+1/n)^n\), as n grows to infinity. Later Euler calculated this number. Euler used the following formula, an endless summation, to calculate the value of e up to 18 digits.

\(e = 1 + {1\over{1!}} + {1\over{2!}} + {1\over{3!}} + {1\over{4!}} + {1\over{5!}} ….\)

Euler also found that e could be represented as a continuous, infinite fraction and proved that it is an irrational number.

Are Irrational Numbers Real Numbers?

This question is common in many exams and MCQs. It is a trick question. As mentioned earlier, all rational numbers are real numbers. An irrational number is a real number that cannot be expressed as the ratio of two integers.

Sum of Two Irrational Numbers

If r is one irrational number and s is another irrational number, then r + s and r – s may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number.

Irrational number + Irrational number = Irrational / Rational number.

Example: Consider two irrational numbers \(2 + \sqrt{3} and 5 + \sqrt{3}\). Adding the number will result in, \(2 + \sqrt{3} + 5 + \sqrt{3} = 7 + 2\sqrt{3}\). This is agian the irrational number. Now consider another two irrational numbers \(2 + \sqrt{3} \text{ and } 2 – \sqrt{3}\). Adding the number will result in, \(2 + \sqrt{3} + 2 – \sqrt{3} = 4\). This is a rational number. Similarly, Irrational number − Irrational number = Irrational / Rational number.

Product of Two Irrational Numbers

If r is one irrational number and s is another irrational number, then rs and r/s are may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number. Irrational number × Irrational number = Irrational / Rational number.

Consider two irrational numbers \(2\sqrt{3} \text{ and } \sqrt{3}\). Multiplying the number will result in, \(2\sqrt{3} × \sqrt{3} = 2 × 3 = 6\). It is a rational number. Now consider another two irrational numbers \(2\sqrt{3} \text{ and } \sqrt{2}\). Multiplying the numbers will result in, \(2\sqrt{3} × \sqrt{2} = 2\sqrt{6}\). It is an irrational number. Similarly, Irrational number ÷ Irrational number = Irrational / Rational number.

Check out this article on Arithmetic Mean.

How to Find Irrational Numbers

We often need to find irrational numbers between different types of numbers. Here are very easy and simple methods to find irrational number between any two whole numbers.

How to Find Irrational Number Between Two Rational Numbers

Assume that we have two rational numbers a and b, then the irrational number between the two will be \(\sqrt{ab}\).

Let's take an example of numbers are 3 and 4. There can be an infinite number of irrational numbers between these numbers. The numbers between the squares of 3 and 4, i.e., between 9 and 16 are 10, 11, …14, 15. The square root of any of these numbers is always an irrational number. Hence, we take the square root of the product of the two vectors \(\sqrt{3\times4} = \sqrt{12}\)

How to Find Irrational Number Between Two Irrational Numbers

Assume that we have two irrational numbers a and b, then the irrational number between the two will be the square of both the numbers and take the square root of their average. If the square root is irrational, then we get the number we want. If we do not have the number you are looking for, we can repeat the procedure using one of the original numbers and the newly generated number.

Find the rational numbers between \(\sqrt{2} \text{ and } \sqrt{3}\)

The difference between \(\sqrt{2} \text{ and } \sqrt{3}\). 6 is between \(4\sqrt{2}\text{ and } 4\sqrt{3} \text{ and } \frac{3}{2} \text{ is between } \sqrt{2} \text{ and } \sqrt{3}\).

How to Find Irrational Numbers Between Decimals

The Irrational numbers have non-recurring and non-Terminating decimals. It doesn't have the specific pattern of repeating the numbers after the decimal point, and these numbers are neverending. So we can write the numbers which are not repeated in any manner.

Decimal Expansion of Irrational Number

For all rationals of the form p/q (q ≠ 0).

On a division of p by q, two main things happen. Either the remainder becomes zero or never becomes zero and we get a repeating string of remainders.

Case I: The remainder becomes zero In this case, the decimal expansion terminates or ends after a finite number of steps. We call the decimal expansion of such numbers terminating.

e.g., 78 , 12 , 34, etc.

Case II: The remainder never becomes zero. In this case, we have a repeating block of digits in the quotient, this expansion is called nonterminating recurring.

e.g., 23 = 0.6666…..

22/7 = 3.142857142857……..

The repeated digits are written as \(23 = \bar{0.6}\)

\({22\over{7}} = \bar{3.142857}\)

Also, learn about Mean Deviation.

Rational Numbers vs Irrational Numbers

The difference between Rational and Irrational number is as follows:

Rational Number	Irrational Number
Rational numbers refer to a number that can be expressed in a ratio of two integers.	An irrational number is one that can't be written as a ratio of two integers.
Rational numbers are expressed in fraction, where denominator ≠ 0	They cannot be expressed in fraction.
Rational numbers are perfect squares	They are Surds
Rational numbers are finite or recurring decimals	They are non-finite or non-recurring decimals.
Example: 2, 3, 4, 5, 6…	Example: \(\sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, \sqrt{6}…\)

Solved Examples on Irrational Numbers

Q. Show that \(3\sqrt{2}\) is irrational.

Ans: Let us assume, to the contrary, that \(3\sqrt{2}\) is rational.

That is, you can find the coprime a and b(b≠0) such that \(3\sqrt{2} = {a\over{b}}\).

Rearranging, we get \(\sqrt{2} = \frac{a}{3b}\).

Thus 3, a and b are the integers, \(\frac{a}{3b}\) is the rational, and so \(\sqrt{2}\) is a rational number.

But this contradicts the fact that \(\sqrt{2}\) is irrational.

So, we conclude that \(3\sqrt{2}\) is irrational.

Q. Write the two irrational numbers between the given numbers 0.16 and 0.17.

Ans: Let a = 0.16 and b = 0.17.

Here, a and b are the rational numbers such that a < b.

We observe that the numbers a and b have a 1 in the first place of decimal. But in the second place of decimal aa has a 6 and b has 7. So, we consider the numbers.

c = 0.1601001000100001…..

And d = 0.17101001000100001…

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

Hope this article on Irrational Numbers was informative. Get some practice of the same on our free Testbook App. Download Now!

Irrational Numbers FAQs

Q.1 What are examples of irrational numbers?

Ans.1 The number Pi originated from geometry. It is the ratio of the circumference and the diameter of a circle. It remains constant, independent of the size of the circle. If the circumference of a circle is rational, the radius is irrational. The Number e is the sum of Infinite Quotients. The number e is a recent discovery by Jacob Bernoulli. The Square Root of Primes: \(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{11}, \sqrt{13}, \sqrt{17}, \sqrt{19}…\) The first irrational to be discovered was \sqrt{2}. The Pythagoreans- and ancient Greek philosophical university and religious brotherhood- stumbled upon \(\sqrt{2}\). Logarithms of primes with prime base: \(log_23, log_25, log_27, log_35, log_37…\).

Q.2 Which are the irrational numbers?

Ans.2 Irrational Number is a special case of numbers in the entire number system. Irrational numbers are kind of the opposite of rational. They are real numbers that we can't write as a ratio \({p\over{q}}\) where p and q are integers, where q cannot be equal to zero. For example, \(\sqrt{2} = 1.414213….\) is irrational because we can't write that as a fraction of integers. Irrational number is hence, recurring numbers.

Q.3 How do you know a number is irrational?

Ans.3 An irrational number is a real number that cannot be expressed as the ratio of two integers. Equivalently, an irrational number, when expressed in decimal notation, never terminates nor repeats. They are real numbers that we can't write as a ratio \({p\over{q}}\) where p and q are integers, where q cannot be equal to zero. For example, \(\sqrt{2} = 1.414213….\) is irrational because we can't write that as a fraction of integers.

Q.4 Is 5.676677666777 an irrational number?

Ans.4 No despite having decimal point, this is not an irrational number as the numbers after the decimal point is not going on forever. Hence, 5.676677666777 is a rational number.

Q.5 Is 1.75 a rational number?

Ans.5 No despite having decimal point, this is not an irrational number as the numbers after the decimal point is not going on forever. Hence, 1.75 is a rational number.

Q.6 How to find an irrational number between two rational numbers?

Ans.6 Assume that we have two rational numbers a and b, then the irrational number between the two will be \(\sqrt{ab}\). Let's take an example of numbers are 3 and 4. There can be an infinite number of an irrational number between these numbers. The numbers between the squares of 3 and 4, i.e., between 9 and 16 are 10, 11, …14, 15. The square root of any of these numbers is always an irrational number. Hence, we take the square root of the product of the two vectors \(\sqrt{3\times4} = \sqrt{12}\)

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