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Free Homework Math Help: Complex Numbers

Complex Numbers

When solving quadratic equations in algebra, sometimes the answer ends up in a form that is not a real number. For example, when solving for x in the equation x2+1 = 0, the next step would be x2 = -1. The after taking the square root of both sides, the answer would be x = ±√-1. The √-1 is not a real number. Instead, it is called an imaginary number and the symbol is i. In equation form, it is written as i = √-1. Furthermore, i2 = -1, i3 = -√-1, i4 = (-1)(-1) = 1 while i5 = i4*i = i.

Complex numbers are formed when a real number is added to an imaginary number: 2+3i. The 2 is the real part while the 3i is the imaginary part. The rectangular form of complex numbers is a+bi. Each complex number has another one related to it called a conjugate. The complex conjugate of a+bi is a-bi. When a complex number is multiplied by its conjugate the result is a real number. For example, (a+bi)(a-bi) = a2 + b2. The following table lists some of the rules and properties of complex numbers in rectangular or standard form.

i is also called the complex unit. The symbol for a conjugate is a line above the complex variable z. If z and w are complex variables or numbers, then z and w are conjugates.


Properties of Complex Numbers

NameMathematical Expression
Definition of i i = √-1
Definition of √-a √-a = i√a
Complex Number a+bi, a,b ∈ R
Real Number a+0i, a ∈ R
Imaginary Number 0+bi, b ∈ R
Equality a+bi = c+di; a=c, b=d
Addition a+bi + c+di = (a+c)+(b+d)i
Subtraction a+bi - c+di = (a-c)+(b-d)i
Multiplication (a+bi)(c+di) = (ac-bd)+(ad+bc)i
Division
(a+bi)
(c+di)		   =   (ac+bd)
(c2+d2)		   +   (bc-ad)
(c2+d2)		i
Powers of i in = ir where r is the remainder of n/4, n > 0
Conjugate of a+bi a-bi
Conjugate Sum z+z is a real number
Conjugate Product z*z is a real number
Conjugate Equality z = z if and only if z is a real number
Conjugate Equivalent Sums (z+w) = z+w
Conjugate Equivalent Products (z*w) = z*w
Double Conjugate
z		    =    z
Conjugate Power (zn) = zn, integer n > 0
Reciprocal
1
a+bi		   =   a
(a2+b2)		   -   b
a2+b2		i

Complex numbers can also be represented in polar form. The rules and formulas for this form are usually introduced in higher math courses such as precalculus or calculus.

Powers of i table

i = √-1 i5 = √-1 i9 = √-1
i2 = -1 i6 = -1 i10 = -1
i3 = -√-1 i7 = -√-1 i11 = -√-1
i4 = 1 i8 = 1 i12 = 1

Note that the cycle repeats after every fourth power.