How did imaginary numbers come into existence? What was the first problem that required use of imaginary number?
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To solve a quadratic (something that looks like 3x^2 + 2x + 9 = 0) you can use the quadratic formula. The [quadratic formula](https://www.google.com/search?client=firefox-b-d&q=quadratic+formula) is basically just a way of [completing the square](https://www.mathsisfun.com/algebra/completing-square.html) but using ax^2 + bx + c = 0 as the quadratic.
The problem with this though is if you look at the quadratic formula a cannot be 0 (assumption #1). If it were 0 we would be dividing by 2*0 and that’s a no-no. So a > 0. But we can also see b^2 – 4ac is under the square root. In order for it to make sense this must also be greater than or equal to 0 because its impossible to multiply any two numbers together to get a negative square number (assumption #2).
So b^2 -4ac >= 0 or b >= sqrt(4ac). 4ac is under the square root here too we know a > 0 so it must also be true that c >= 0 in order to prevent negative numbers under the root. So the quadratic formula only works for values of a > 0, b >= sqrt(4ac) and c >= 0. But we have a problem… we know that for (a = 1, b = 0, c = -9) or x^2 – 9 = 0 has a solution. Look, add 9 to both sides x^2 = 9, square root both sides x= +/-3. So c must be able to be negative, even though we have just shown it must be c >= 0
So one of our assumptions must have been wrong. It can’t be 1, which leaves 2… the only wrong assumption here is that the value under a square root must be a positive number. It therfor is possible to multiply two of the same numbers together and get a negative square. (here 3i * 3i = -9, or in otherwords sqrt(-9) = 3i)
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