The complex numbers are all the numbers that you get by putting together real
numbers and the square root of -1, which is called i. Anything you can get by
adding, subtracting, multiplying, and dividing real numbers and i is called a
complex number. As it turns out, this means that they are anything of the form
a+bi, where a and b are real numbers. This article is about how to do the four
basic arithmetic operations, adding, subtracting, multiplying, and dividing with
complex numbers. To master this arithmetic all you really need to know is that i
is the square root of -1 and that anything you normally do in algebra with
variables, you can do with i.
Addition
To add complex numbers, just treat the i like a variable and
combine like terms.
Example:
(3+2i)+(1-5i)=3+2i+1-5i=4-3i
Subtraction
To subtract to the same thing, but just like with variables
you have to make sure to change the signs of both terms in the second one.
Example:
(5-i)-(3-2i)=5-i-3+2i=2+i
Multiplication
Here just like with variables, we use FOIL, but one
different thing happens, we will get an i squared, which we have to turn into -1
because i is the square root of -1.
Example:
(4+3i)(3-2i)=12-8i+9i-6i
2=12-8i+9i+6=18+i
Division
Division is just a little bit harder. To get it into its
standard form of a+bi we need to do something sort of like rationalizing the
denominator. We need to multiply top and bottom by what is called the complex
conjugate. The complex conjugate is what you get if you change the sign of the
imaginary part, but keep the real part as is.
Example:
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Now you know all about complex number arithmetic.
