Download complex numbers

Pre-calculus: Complex numbers
What does the number system look like?
To understand what defines a complex number, we have to study the
number system. Here’s a simplified complex number system (source:
tpub.com):
Every complex numbers can be represented as a + b i, where a and b are
real numbers and i is defined as the basis of imaginary part.
definition of i: i 2 = −1
Examples:
3 = 3 + 0 i (real number)
2 i = 0 + 2 i (pure imaginary)
What are the properties of complex numbers?
Commutative property
If x and y are complex numbers, x+y=y+x and xy=yx
Pre-calculus: Complex numbers
Associative property (Both leftwards and rightwards)
If x,y and z are complex numbers, x(y+z)=xy+xz
If x,y and z are complex numbers, (x+y)z=xz+yz
Linear independence of real part and imaginary part
If x and y are complex numbers where x = a + b i and y = c + d i,
x+y = (a+c) + (b+d) i (The real and imaginary part add separately)
Uniqueness of the representation of a + b i
If x is a complex number, where x = a + b i, the numbers a and b must be
unique.
Mathematically, it means if we can represent x as c + d i, then c must equal
to a and b must equal to b.
What can we do with complex numbers?
There are tons of operations that can be done using complex numbers. The
following two are examples of mostly used operations.
Complex conjugate ( a or a* )
If a is a complex number where a = b + c i, then the complex conjugate of
a, a is a = b - c i.
To go from a to its conjugate, we simply replace i in a by -i.
If a is the complex conjugate of a, a is also the complex conjugate of a .
The product (a* a )
If a is a complex number where a = b + c i, the product a* a is b2 + c2 and is
always positive.
Proof:
a = b + ci
a = b − ci
!
!
!
by definitions
a ⋅ a = (b + ci) ⋅ (b − ci)
= b ⋅ b − bci + cib − ci ⋅ ci !
expanding (b + c i )*( b - c i)
=b +c !
using commutative and associative property, and the definition
2
2
!
!
i 2 = −1 .