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| Imaginary Numbers |
Imaginary Numbers
The complex numbers are an extension of the real numbers; they are the set of all numbers of the form a + bi where a and b are real numbers and i is a complex unit, typically taken to be the positive square root of -1. In other words, we let i = sqrt(-1) and then extend the rules of addition and multiplication with real numbers to this bigger set, given i^2 = (-i)^2 = -1.
For example, the polynomial x^2 + 1 does not factor over the real numbers, as the quadratic formula suggests the roots should be
[-0 ± sqrt(0^2 - 4*1*1)]/(2*1) = ± sqrt(-4)/2,
which does correspond to real numbers. However, in the complex numbers we see the roots immediately, i and -i, as sqrt(-4)/2 = 2*sqrt(-1)/2.
The imaginary numbers are a subset of the complex numbers, the set of complex numbers of the form bi, i.e., with real part 0. So, i, 2i and -3i are all imaginary numbers, while 1, 1 + i, 1 - i and -2 - 3i are not. As 0 = 0*i it is the only real and imaginary number.
One astounding connection between the set of imaginary numbers and the set of complex numbers is Euler's formula, which states that
e^(ix) = cos(x) + i*sin(x)
for any real number x. So every complex number can be represented as some real multiple of e to the power of an imaginary number.