The square root of a negative real number is called an imaginary number. It is also called as an imaginary quantity.

An imaginary number is actually formed when a negative real number appears under a square root symbol. The symbol of a square root is proposed to find a square root of any number but it is not possible in the case of negative numbers because, there is no negative numbers but we imagined them for our convenience.

However, the imaginary unit is useful to find square roots of negative real numbers. The square root of a negative real number is written as the square root of a product of $-1$ and a positive real number, and then it is written as the product of square root of $-1$ and square root of positive number. The square root of $-1$ is represented by the imaginary unit. Therefore, the square root of a negative real number is written as the product of imaginary unit and square root of the positive real number.

$\sqrt{-25}$

It is an example for an imaginary number. Actually, there is no meaning for finding square root of a negative number mathematically but the concept of imaginary unit made it possible in mathematics.

$\sqrt{-25} \,=\, \sqrt{-1 \times 25}$

$\implies \sqrt{-25} \,=\, \sqrt{-1} \times \sqrt{25}$

$\implies \sqrt{-25} \,=\, i \times 5$

$\implies \sqrt{-25} \,=\, i5$

$\,\,\, \therefore \,\,\,\,\,\, \sqrt{-25} \,=\, 5i$

Thus, the square root of any negative real number can be found in mathematics. Observe the following more examples to understand it much clear.

$(1)\,\,\,\,\,\,$ $\sqrt{-2} \,=\, i\sqrt{2}$

$(2)\,\,\,\,\,\,$ $\sqrt{-4} \,=\, 2i$

$(3)\,\,\,\,\,\,$ $\sqrt{-8} \,=\, i2\sqrt{2}$

$(4)\,\,\,\,\,\,$ $\sqrt{-49} \,=\, 7i$

$(5)\,\,\,\,\,\,$ $\sqrt{-100} \,=\, 10i$

$c$ is a real number and its negative form is $-c$. Find the square root of this negative quantity.

$\sqrt{-c} \,=\, i\sqrt{c}$

Take, the value of square root of $c$ is equal to $b$.

$\implies \sqrt{-c} \,=\, ib$

$\,\,\, \therefore \,\,\,\,\,\, \sqrt{-c} \,=\, bi$

The quantity $bi$ is called an imaginary number or imaginary quantity.

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