Q1.
Consider this puzzle: What is the square root of –1? We know that 1 × 1 = 1. We also know that (–1) × (–1) = 1. There is no Real Number that, when multiplied by itself, results in a negative number. Thus, √(–1) cannot exist on number line.
Answer
There is no real number whose square is −1. The reason is Brahmagupta's own sign rule.
if x > 0 then x × x > 0 (fortune × fortune = fortune)
if x < 0 then x × x > 0 (debt × debt = fortune)
if x = 0 then x × x = 0
if x < 0 then x × x > 0 (debt × debt = fortune)
if x = 0 then x × x = 0
Every real number falls into one of these three cases, so x² ≥ 0 for every real x. Since −1 < 0, the equation x² = −1 has no solution anywhere on the number line.
Why it happens: the number line was built by filling gaps — fractions filled the gaps between integers, irrationals filled the gaps the fractions left. But √(−1) is not sitting in a gap; it is excluded by the arithmetic of signs itself. No amount of further filling can produce it, so mathematicians had to leave the line altogether and add a new dimension, writing i for a quantity with i² = −1.
Did you know? The pattern of this chapter repeats once more here. Every time an operation could not be carried out — subtraction inside ℕ, division inside ℤ, square roots inside ℚ — the number system was enlarged. Imaginary numbers are the next enlargement, and they are what make alternating-current circuits and quantum mechanics calculable.