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Physics (e.g., quantum mechanics, relativity, other subfields) makes use of imaginary numbers. "Complex analysis" (i.e., calculus that includes imaginary numbers) can also be used to evaluate difficult integrals and to perform other mathematical tricks.
Engineering, especially Electrical Engineering makes use of complex and imaginary numbers to simplify analysis of some circuits and waveforms.
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Where did complex and imaginary numbers come from?
The 16th century Italian mathematician, Gerolamo Cardano was the first to use imaginary and complex numbers in his work on cubic equations.
Why do you need imaginary numbers?
because somtimes there isn't an answer to every equation like what's the square root of -16.... there is no answer so we would just use an imaginary number which is i.It turns out that these are important in a practical sense. Imaginary numbers turn up all the time in quantum mechanics and certain types of electronic circuits as well.
Who are the mathematicians that discovered imaginary numbers?
The first person to write about them was Gerolamo Cardano in 1545, but he doesn't seem to have taken them seriously. See http://en.wikipedia.org/wiki/Gerolamo_Cardano . The first serious use of imaginary numbers (better, complex numbers) was by Rafael Bombelli, published in 1572. He used them as intermediate steps when solving cubic equations. See related link.
What are the real numbers that is not a whole number?
Real numbers are all numbers which do not contain "i", when "i" represents the square root of -1. All numbers which do contain "i" are "imaginary numbers" and are not real numbers. This means that all numbers you'd ordinarily use are real numbers - all the counting numbers (integers) and all decimals are real numbers. So in answer to your question, all the real numbers that are not whole numbers are all the decimal numbers - including irrational decimals such as pi.
What rule do you use to factor the sum of two squares?
It requires an understanding of imaginary numbers, specifically, the square root of negative one, which is abbreviated as i.a^2 - b^2 = (a - b)(a + b)a^2 + b^2 = (a - bi)(a + bi)
