diff --git a/guide/english/mathematics/imaginary-numbers/index.md b/guide/english/mathematics/imaginary-numbers/index.md index f7f646b2b7..a59385b254 100644 --- a/guide/english/mathematics/imaginary-numbers/index.md +++ b/guide/english/mathematics/imaginary-numbers/index.md @@ -3,12 +3,19 @@ title: Imaginary Numbers --- ## Imaginary Numbers -An Imaginary Number "i" is defined simply as the square root of -1. In other words i2 = -1. One of the reasons you would use Imaginary Numbers (or Complex Numbers as some others call them) is so that you can manipulate equations or expressions that contain the square root of -1. +The [complex numbers](https://en.wikipedia.org/wiki/Complex_number) are an extension of the [real numbers](https://en.wikipedia.org/wiki/Real_number); 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`. -Imaginary numbers are used to represent two-dimensional numbers. Much like radio waves, there is an electric part and a magnetic part that orthogonal to each other. Imaginary numbers are used to represent such a number - think of it as a mathematical construct. The number, 1+i represents a 45 degree angle (the Tan-1(1) = 45 degrees. The imaginary part you can think of as the y-axis and the real part as the x-axis. a+bi is a general representation of the number and the inverse tangent (b/a) represents the phase angle while the sqrt (a^2 + b^2) represents the magnitude of the number. - -#### More Information: - -- [Better Explained: great intuitive explanations for all things mathematics ](https://betterexplained.com/) +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](https://en.wikipedia.org/wiki/Euler%27s_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. \ No newline at end of file