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How to rationalise imaginary denominators ?

Last updated date: 24th Feb 2024
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IVSAT 2024
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Hint:An imaginary number is a number which is multiple of ‘i’. Here, ‘i’ is called iota and is equal to $\sqrt{-1}$. Check what happens to an imaginary number when we multiply it by a ‘i’ or any other imaginary number.

Complete step by step answer:
Let us first understand what is an imaginary number. Before that we will understand what complex number. Complex numbers is a set of those numbers that can be real or non-real numbers. We already know the real number. Every number that is present on a number line is a real number. Whereas an imaginary number is a number which is multiple of ‘i’. Here, ‘i’ is called iota and is equal to $\sqrt{-1}$. Therefore, if we have a number say ‘5i’ then, this number is an imaginary number.

We know that a rational number is a number that can be expressed in the form of a fraction of two integers where the denominator is not equal to zero. Therefore, when we say about rationalising an imaginary number, we mean to make the number a rational number.Suppose we have an imaginary number ‘xi’, where x is a real number. We know that $xi=x\times \sqrt{-1}$.

Now, see the fun part. If we multiple ‘i’ with ‘i’, then we get
$\Rightarrow x\times \sqrt{-1}\times \sqrt{-1}\\
\Rightarrow x\times {{\left( \sqrt{(-1)} \right)}^{2}}\\
\Rightarrow x\times (-1)\\
\therefore -x$
Therefore, we rationalised the imaginary number by multiplying it by i. Similarly, when we have an imaginary denominator, we multiply and divide the numerator and denominator by ‘i’ to make the denominator rationalise.

Note: When we rationalise the denominator of an imaginary number, it does not mean that the number becomes a real or a rational number. It is still an imaginary number but with a rational denominator and imaginary numerator.