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.
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