Number - Irrational Numbers

• What Is an Irrational Number?
• Definition of Irrational Numbers
• Properties of Irrational Numbers
• Operations on Two Irrational Numbers
• Finding an Irrational Number
• Concept of Numbers
• Solved Examples
• Practice Problems
• Frequently Asked Questions

What Is an Irrational Number?

Irrational is derived from the word rational. Rational numbers are numbers that can be expressed in the form of ratios. Irrational numbers are those that cannot be expressed as ratios. These numbers cannot be expressed in the form of quotients or ratios.

Definition of Irrational Numbers

An irrational number is a real number that cannot be expressed as a fraction or ratio of form `a/b` of two integers, where `a, b` are integers, and `b≠0`.

Irrational numbers cannot be represented by a terminating or repeating decimal. They cannot be expressed as simple fractions.

Existence of numbers

• Integers, `I=-`\(\infty\), .....,`-4,-3,-2,-1,0,1,2,3,4`,..., \(\infty\)
• Whole numbers, `W=0,1,2,3,4`,..., \(\infty\)
• Natural numbers, `N=1,2,3,4`,..., \(\infty\)
• Rational numbers `=Q`.

Rational numbers refer to the numbers that can be expressed in the form of `a/b`, where `a, b` are integers, and `b!=0`.

Rational and irrational numbers combined are called real numbers. Therefore, the irrational numbers can be defined as:

Irrational numbers are real numbers that cannot be expressed in the form of `a/b` where `a, b` are integers, and `b≠0`.

 "`P`" is used to represent irrational numbers, while real numbers are represented by `R`. `Q` is used to represent rational numbers. Therefore, irrational numbers can also be represented by `R-Q` .

Examples of irrational numbers: `sqrt2, sqrt3, pi`.

•  `pi`(pi) Pi is defined as the ratio of the circumference of the circle to its radius. The value `pi`  is `3.14159265358979`, where the decimal places continue up to infinity.
• `e` (Euler’s number) is the fundamental mathematical constant that is used in calculus. Its value is  `2.71828182845904`, where its decimal continues up to infinity.
• All the prime numbers’ square roots are irrational numbers, `sqrt2,sqrt3,sqrt5,sqrt7`etc.  For example, if we expand `sqrt2=1.41421356237` is non-termianting and non-recurring and it cannot be expressed in the form of a ratio.
• The golden ratio \(\Phi\) is another mathematical constant with a value of `1.61803398874989`, where its decimal continues up to infinity.
• All square roots of non-perfect squares are irrational numbers.

Properties of Irrational Numbers

1. Noninteger: Irrational numbers are not integers, and they do not represent whole numbers. 
2. Non-rational: Irrational numbers cannot be expressed as ratios, i.e., they cannot be expressed as `a/b`, where `b` cannot be equal to zero.
3. Non-repeating decimal: They do not repeat themselves after the decimal, and they are non-terminating. So, they have infinite non-repeating decimal values. 
4. Uncountable: Irrational numbers are infinite, which means that irrational numbers are more vast than natural numbers. 
5. Addition: When two irrational numbers are added together, the result can be a rational or an irrational number. For example, `sqrt5` when added to `sqrt2` gives `sqrt5+sqrt2`. 
   Add `sqrt5` and `5-sqrt5` i.e., `sqrt5+(5-sqrt5)=5`
   The addition of two irrational numbers can be rational or irrational. 
6. Multiplication: When two irrational numbers are multiplied the result can be a rational or an irrational number. For example, `sqrt5` when multiplied to `sqrt2` gives `sqrt10`.
   Multiply `sqrt5` and `sqrt5` i.e., `sqrt5xxsqrt5=5`
   The multiplication of two irrational numbers can be rational or irrational. 
7. LCM (Lowest Common Multiple): The LCM of two irrational numbers may or may not exist.

Operations on Two Irrational Numbers

`1`. Addition: The addition of two rational numbers can be a rational or irrational number.

Example `1`: Add `sqrt3` and `2sqrt3`.

Solution: `sqrt3+2sqrt3=3sqrt3` is an irrational number.

Example `2`: Add `5+sqrt3` and `2-sqrt3`.

Solution: `5+sqrt3+2-sqrt3=5+2=7` is a rational number.

`2`. Subtraction: The subtraction of two rational numbers can be a rational or irrational number.

Example `1`: Subtract `3-sqrt3` and `4-2sqrt3`.

Solution: `3-sqrt3-(4-2sqrt3)=3-sqrt3-4+2sqrt3=-1+sqrt3` is an irrational number.

Example `2`: Subtract `4+sqrt3` and `2+sqrt3`.

Solution: `4+sqrt3-(2+sqrt3)=4+sqrt3-2-sqrt3=2` is a rational number.

`3`. Multiplication: The multiplication of two rational numbers can be a rational or irrational number.

 Example `1`: Multiply `3-sqrt3` and `5`.

Solution: `(3-sqrt3)xx5=15-5sqrt3` is an irrational number.

Example `2`: Subtract `4sqrt3` and `6sqrt3`.

Solution: `4sqrt3 xx 6sqrt3=4xx6xx3=72` is a rational number.

`4`. Division: The division of two rational numbers can be a rational or irrational number.

Example `1`: Divide `4sqrt3` by `2sqrt3`.

Solution: `4sqrt3÷2sqrt3=(4sqrt3)/(2sqrt3)=2` is a rational number.

Example `2`: Divide `4-sqrt3` by `sqrt3`.

Solution: `(4-sqrt3)-:sqrt3=(4-sqrt3)/sqrt3=(4/sqrt3)-1` is an irrational number.

Finding an Irrational Number

Let us find the irrational number between `2` and `3`.

`sqrt4=2` and `sqrt9=3`

Therefore, `sqrt5, sqrt6, sqrt7` etc. lies between `sqrt4` and `sqrt9` which are not square. These square roots yield decimal numbers which are non-repeating and non-terminating. 

Concept of Numbers

Solved Examples

Example `1`: Find the value of `sqrt3+sqrt2` and determine whether it is a rational or irrational number.

Solution: 

Here, `sqrt3=1.732….` and `sqrt2=1.414…..`

Therefore, `sqrt3+sqrt2=1.732….+1.414….=3.146….`

The value of `sqrt3+sqrt2` is a non-terminating and non-repeating decimal. Therefore, it is an irrational number.

Example `2`: Identify whether this `3.5` is a rational or irrational number.

Solution: 

A rational number is a number that can be expressed in the form of `a/b`, where `a, b` are integers, and `b≠0`. The number `3.5` can be expressed as `7/2` which is a fraction of two integers. Therefore, `3.5` is a rational number.

Example `3`: Can `sqrt3` and `sqrt3+5` be added. If yes, then determine whether it is rational or irrational.

Solution: 

Yes, `sqrt3` and `sqrt3+5` can be added as follows.

`sqrt3+sqrt3+5=2sqrt3+5`

The sum is an irrational number.

Example `4`: Is the square root of `25` rational or irrational?

Solution: 

The square root of `25` can be calculated as `sqrt25=5`.

`5` can be expressed as `5/1`. Therefore, `sqrt25` is a rational number.

Example `5`: Determine if `0.6666` repeating decimals is rational.

Solution: 

The repeating decimal `0.6666` can be expressed as `2/3`. Since, `2/3` is a fraction of two integers, it is a rational number.

Practice Problems

Q`1`. Find the product of `2+sqrt3` and `4` and determine whether it is rational or irrational.

1.  `8+4sqrt3`, Irrational
2.  `8+4sqrt3`, Rational
3.  `8+sqrt3`, Irrational
4. `8+sqrt3`, Rational

Answer: a

Q`2`. Which of the following is an irrational number?

1. `-2`
2. `sqrt5`
3. `0`
4. `4`

Answer: b

Q`3`. Solve `(4-sqrt3)(4+sqrt3)`.

1. `5`
2. `19`
3. `13`
4. `10`

Answer: c

Q`4`. Find the division of `24sqrt5` and `6sqrt5`.

1. `24/sqrt5`
2. `sqrt5`
3. `24`
4. `4`

Answer: d

Q`5`. The square root of which number is a rational number?

1. `12`
2. `14`
3. `16`
4. `18`

Answer: c

Frequently Asked Questions

Q`1`. Are irrational numbers real?

Answer: Both rational and irrational numbers are real, so they collectively form a set of real numbers on the number line.

Q`2`. Are all square roots irrational numbers?

Answer: No, all the square roots are not irrational numbers. The terminating and perfect square roots are rational numbers.

Q`3`. Does a rational number exist between integers?

Answer: There is an infinite rational numbers that exist between any two integers.

Q`4`. What are the practical examples of irrational numbers?

Answer: `sqrt2, sqrt3, pi, e`  some of the examples of irrational numbers. Other names of irrational numbers are surds.