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Complex number forms review

Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms.

What are the different complex number forms?

Rectangulara, plus, b, i
Polarr, left parenthesis, cosine, left parenthesis, theta, right parenthesis, plus, i, sine, left parenthesis, theta, right parenthesis, right parenthesis
Exponentialr, dot, e, start superscript, i, theta, end superscript

Rectangular form

start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, i
The rectangular form of a complex number is a sum of two terms: the number's start color #11accd, start text, r, e, a, l, end text, end color #11accd part and the number's start color #1fab54, start text, i, m, a, g, i, n, a, r, y, end text, end color #1fab54 part multiplied by i.
As such, it is really useful for adding and subtracting complex numbers.
We can also plot a complex number given in rectangular form in the complex plane. The real and imaginary parts determine the real and imaginary coordinates of the number.
Want to learn more about complex number rectangular form? Check out this video about the complex plane and this video about adding and subtracting complex numbers.

Polar form

start color #e07d10, r, end color #e07d10, left parenthesis, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, plus, i, dot, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, right parenthesis
Polar form emphasizes the graphical attributes of complex numbers: start color #e07d10, start text, a, b, s, o, l, u, t, e, space, v, a, l, u, e, end text, end color #e07d10 (the distance of the number from the origin in the complex plane) and start color #aa87ff, start text, a, n, g, l, e, end text, end color #aa87ff (the angle that the number forms with the positive Real axis). These are also called start color #e07d10, start text, m, o, d, u, l, u, s, end text, end color #e07d10 and start color #aa87ff, start text, a, r, g, u, m, e, n, t, end text, end color #aa87ff.
Note that if we expand the parentheses in the polar representation, we get the number's rectangular form:
start color #e07d10, r, end color #e07d10, left parenthesis, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, plus, i, dot, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, right parenthesis, equals, start overbrace, start color #e07d10, r, end color #e07d10, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, end overbrace, start superscript, start color #11accd, a, end color #11accd, end superscript, plus, start overbrace, start color #e07d10, r, end color #e07d10, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, end overbrace, start superscript, start color #1fab54, b, end color #1fab54, end superscript, dot, i
This form is really useful for multiplying and dividing complex numbers, because of their special behavior: the product of two numbers with absolute values start color #e07d10, r, start subscript, 1, end subscript, end color #e07d10 and start color #e07d10, r, start subscript, 2, end subscript, end color #e07d10 and angles start color #aa87ff, theta, start subscript, 1, end subscript, end color #aa87ff and start color #aa87ff, theta, start subscript, 2, end subscript, end color #aa87ff will have an absolute value start color #e07d10, r, start subscript, 1, end subscript, r, start subscript, 2, end subscript, end color #e07d10 and angle start color #aa87ff, theta, start subscript, 1, end subscript, plus, theta, start subscript, 2, end subscript, end color #aa87ff.
Want to learn more about complex number polar form? Check out this video.

Exponential form

start color #e07d10, r, end color #e07d10, dot, e, start superscript, i, start color #aa87ff, theta, end color #aa87ff, end superscript
Exponential form uses the same attributes as polar form, start color #e07d10, start text, a, b, s, o, l, u, t, e, space, v, a, l, u, e, end text, end color #e07d10 and start color #aa87ff, start text, a, n, g, l, e, end text, end color #aa87ff. It only displays them in a different way that is more compact. For example, the multiplicative property can now be written as follows:
left parenthesis, start color #e07d10, r, start subscript, 1, end subscript, end color #e07d10, dot, e, start superscript, i, start color #9d38bd, theta, start subscript, 1, end subscript, end color #9d38bd, end superscript, right parenthesis, dot, left parenthesis, start color #e07d10, r, start subscript, 2, end subscript, end color #e07d10, dot, e, start superscript, i, start color #9d38bd, theta, start subscript, 2, end subscript, end color #9d38bd, end superscript, right parenthesis, equals, start color #e07d10, r, start subscript, 1, end subscript, end color #e07d10, start color #e07d10, r, start subscript, 2, end subscript, end color #e07d10, dot, e, start superscript, i, left parenthesis, start color #9d38bd, theta, start subscript, 1, end subscript, plus, theta, start subscript, 2, end subscript, end color #9d38bd, right parenthesis, end superscript
This form stems from Euler's expansion of the exponential function e, start superscript, z, end superscript to any complex number z. The reasoning behind it is quite advanced, but its meaning is simple: for any real number x, we define e, start superscript, i, x, end superscript to be cosine, left parenthesis, x, right parenthesis, plus, i, sine, left parenthesis, x, right parenthesis.
Using this definition, we obtain the equivalence of exponential and polar forms:
start color #e07d10, r, end color #e07d10, dot, e, start superscript, i, start color #aa87ff, theta, end color #aa87ff, end superscript, equals, start color #e07d10, r, end color #e07d10, left parenthesis, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, plus, i, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, right parenthesis

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