What are imaginary numbers?
For starters: despite the given name, imaginary numbers are NOT imaginary. They exist, and can be useful in some cases.
Normally, when we use our calculators to try and find the negative square root of a number, we receive this error message:
=
Although, thanks to imaginary numbers, we don't have to see this error message anymore. There is a way to find the square root of a negative number, and that way is called 'i' (the unit for imaginary numbers).
Imaginary numbers can be written as real numbers multiplied by the unit i (imaginary number). For example, the square root of a negative number could be an imaginary number.
The square root of -16 = 4i (four times the imaginary number)
An imaginary number could also be defined as the negative result of any number squared.
4^2 = -16
Because normally, 4^2 would equal +16, but because we are using the unit i, (4)(4) = -16i. We could also use this method for negative squares. For example, (-4)(-4) = -16i.
When using imaginary numbers, we can look past these mathematical laws:
(+)(+) = (+)
(-)(-) = (+)
(+)(-) = (-)
(-)(+) = (-)
And we can make these equations possible:
(+)(+) = (-)
(-)(-) = (-)
(+)(-) = (+)
(-)(+) = (+)
[source]
"Algebra" (Novel):
"Algebra" is a book written by Italian mathematician Rafael Bombelli. In this book, he had explained the rules for multiplying real numbers with positives and negatives. Here is how he explained it:
"Plus times plus makes plus
Minus times minus makes plus
Plus times minus makes minus
Minus times plus makes minus
Plus 8 times plus 8 makes plus 64
Minus 5 times minus 6 makes plus 30
Minus 4 times plus 5 makes minus 20
Plus 5 times minus 4 makes minus 20"
He also explained the laws of multiplying complex numbers:
"Plus by plus of minus, makes plus of minus.
Minus by plus of minus, makes minus of minus.
Plus by minus of minus, makes minus of minus.
Minus by minus of minus, makes plus of minus.
Plus of minus by plus of minus, makes minus.
Plus of minus by minus of minus, makes plus.
Minus of minus by plus of minus, makes plus.
Minus of minus by minus of minus makes minus."
[source] (Note: These laws were directly copied from Wikipedia as the original text was taken directly from the book "Algebra.")