The Definition
As I’ve already stated, I am assuming that you have seen complex numbers to th
you’re aware that i = √−1 and so i2 = −1. This is an idea that most people fir
class (or wherever they first saw complex numbers) and i = √−1 is defined so
with square roots of negative numbers as follows,
√−100 = √(100) (−1) = √100 √−1 = √100 i = 10
What I’d like to do is give a more mathematical definition of a complex numbers a
i2 = −1 (and hence \i = \sqrt { - 1} \) can be thought of as a consequence of this
take a look at how we define arithmetic for complex numbers.
What we’re going to do here is going to seem a little backwards from what you’ve
seen but is in fact a more accurate and mathematical definition of complex numbe
this section is not really required to understand the remaining portions of this doc
solely to show you a different way to define complex numbers.
So, let’s give the definition of a complex number.
Given two real numbers a and b we will define the complex number z as,
z = a + bi
Note that at this point we’ve not actually defined just what i is at this point. The nu
the real part of z and the number b is called the imaginary part of z and are ofte
Rez = a Imz = b
There are a couple of special cases that we need to look at before proceeding. Fi
at a complex number that has a zero real part,
z = 0 + bi = bi
In these cases, we call the complex number a pure imaginary number.
Next, let’s take a look at a complex number that has a zero imaginary part,
z = a + 0i = a
In this case we can see that the complex number is in fact a real number. Becaus
think of the real numbers as being a subset of the complex numbers.
We next need to define how we do addition and multiplication with complex numb
As I’ve already stated, I am assuming that you have seen complex numbers to th
you’re aware that i = √−1 and so i2 = −1. This is an idea that most people fir
class (or wherever they first saw complex numbers) and i = √−1 is defined so
with square roots of negative numbers as follows,
√−100 = √(100) (−1) = √100 √−1 = √100 i = 10
What I’d like to do is give a more mathematical definition of a complex numbers a
i2 = −1 (and hence \i = \sqrt { - 1} \) can be thought of as a consequence of this
take a look at how we define arithmetic for complex numbers.
What we’re going to do here is going to seem a little backwards from what you’ve
seen but is in fact a more accurate and mathematical definition of complex numbe
this section is not really required to understand the remaining portions of this doc
solely to show you a different way to define complex numbers.
So, let’s give the definition of a complex number.
Given two real numbers a and b we will define the complex number z as,
z = a + bi
Note that at this point we’ve not actually defined just what i is at this point. The nu
the real part of z and the number b is called the imaginary part of z and are ofte
Rez = a Imz = b
There are a couple of special cases that we need to look at before proceeding. Fi
at a complex number that has a zero real part,
z = 0 + bi = bi
In these cases, we call the complex number a pure imaginary number.
Next, let’s take a look at a complex number that has a zero imaginary part,
z = a + 0i = a
In this case we can see that the complex number is in fact a real number. Becaus
think of the real numbers as being a subset of the complex numbers.
We next need to define how we do addition and multiplication with complex numb
