Consider this puzzle: What is the square root of –1? We know that $1 \times 1 = 1$. We also know that $(-1) \times (-1) = 1$. There is no Real Number that, when multiplied by itself, results in a negative number. Thus, $\sqrt{-1}$ cannot exist on number line.

It is true that $\sqrt{-1}$ cannot exist on the real number line.

Reason :

By Brahmagupta's laws :

⇒ A positive number multiplied by a positive number gives a positive number : (+) × (+) = (+).

⇒ A negative number multiplied by a negative number gives a positive number : (-) × (-) = (+).

So, the square of any real number (positive, negative or zero) is always non-negative. There is no real number whose square is -1.

Hence, $\sqrt{-1}$ has no place on the real number line.

To handle expressions like $\sqrt{-1}$, mathematicians invented a new kind of number called the Imaginary Unit, denoted by i, where :

⇒ i² = -1

⇒ i = $\sqrt{-1}$.

This led to the development of Imaginary Numbers, which extend the real number line into a new dimension and form the basis of Complex Numbers (numbers of the form a + bi, where a and b are real numbers).

Imaginary numbers are essential in modern electrical engineering, quantum mechanics, signal processing and several other fields.

Hence, $\sqrt{-1}$ does not exist on the real number line; it is denoted by i, the imaginary unit, which forms the foundation of complex numbers.