Operations on irrational numbers — lesson. Mathematics State Board, Class 9.

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There are many techniques to produce more and more irrational numbers between two numbers.

But sometimes it is easy to identify the provided number is irrational or not by checking the following properties.

Properties of irrational numbers:

Addition, subtraction, multiplication and division of two irrational number is may or may not be irrational.
Addition of rational and irrational number is always irrational.
Subtraction of rational and irrational number is always irrational.
Multiplication of a non-zeror rational and irrational is always irrational.
Division of a non-zero rational and irrational is always irrational.

1. The first property says :

i) Irrational number $+$ Irrational number $=$ Irrational$/$Rational number.

Example:

\begin{array}{l} 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} and 2 - \sqrt{3} . \\ Adding the number will result in, \\ 2 + \sqrt{3} + 2 - \sqrt{3} = 4. This is a rational number . \end{array}

ii) Irrational number $-$ Irrational number $=$ Irrational$/$Rational number.

Example:

\begin{array}{l} Consider two irrational numbers 2 + \sqrt{3} and 5 + \sqrt{3} \\ Subtracting the number will result in, \\ 2 + \sqrt{3} - (5 + \sqrt{3}) = - 3. It is a rational number . \\ Now consider another two irrational numbers 2 + \sqrt{3} and 2 - \sqrt{3} . \\ Subtracting the numbers will result in, \\ 2 + \sqrt{3} - 2 + \sqrt{3} = 2 \sqrt{3} . It is an irrational number . \end{array}

iii) Irrational number $×$ Irrational number $=$ Irrational$/$Rational number.

Example:

\begin{array}{l} Consider two irrational numbers 2 \sqrt{3} and \sqrt{3} \\ Multiplying the number will result in, \\ 2 \sqrt{3} \times (\sqrt{3}) = 2 \times 3 = 6. It is a rational number . \\ Now consider another two irrational numbers 2 \sqrt{3} and \sqrt{2} . \\ Multiplying the numbers will result in, \\ 2 \sqrt{3} \times (\sqrt{2}) = 2 \sqrt{6} . It is an irrational number . \end{array}

iv) Irrational number

\div

Irrational number $=$ Irrational$/$Rational number.

Example:

\begin{array}{l} Consider two irrational numbers \sqrt{12} and \sqrt{3} \\ Dividing the number will result in, \\ \sqrt{12} \div (\sqrt{3}) = 2 \sqrt{3} \div \sqrt{3} = 2. It is a rational number . \\ Now consider another two irrational numbers \sqrt{15} and \sqrt{5} . \\ Dividing the numbers will result in, \\ 15 \div (\sqrt{5}) = \sqrt{3} . It is an irrational number . \end{array}

2. Rational number $+$ Irrational number $=$ Irrational number.

Example:

\begin{array}{l} Consider a rational number 5 and an irrational number e . \\ Adding the rational and irrational numbers becomes, \\ 5 + e . \end{array}

Adding a rational number with an irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.

3. Rational number $-$ Irrational number $=$ Irrational number.

Example:

\begin{array}{l} Consider a rational number - 8 and an irrational number \sqrt{19} . \\ Subtracting the rational and irrational numbers becomes, \\ - 8 - \sqrt{19} . \end{array}

Subtracting a rational number from the irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.

4. Non-zero rational number $×$ Irrational number $=$ Irrational number.

Example:

\begin{array}{l} Consider a non - zero rational number 3 and an irrational number \sqrt{7} . \\ Multiplying the rational and irrational numbers becomes, \\ 3 \times \sqrt{7} = 3 \sqrt{7} . \end{array}

Multiplication of a non-zero rational number and irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.

5. Non-zero rational number $\div$ Irrational number $=$ Irrational number.

Example:

\begin{array}{l} Consider a non - zero rational number - 3.2 and an irrational number π . \\ Dividing the non - zero rational and irrational numbers becomes, \\ - 3.2 \div π = - 3 / 2 π . \end{array}

Division of a non-zero rational number and irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.

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7. Operations on irrational numbers

Theory: