I'll see what I can do without images...
Complex NumbersComplex numbers is the combination of real and imaginary numbers. Instead of being purely "real" or being purely "imaginary," they are considered "complex."
If you don't already know, complex numbers are numbers that are multiples of the square root of negative one. This, in the real number system, is prohibited. Why?
| Operation | Explanation |
| y=x2 | y is x squared |
| y=xx | x2=xx |
When you square a number, you multiply a number by itself. If x is positive, then y is positive: a positive times a positive is positive. If x is negative, then y is still positive: a negative times a negative is positive. It is impossible in the real number system to have a number that can become negative after being squared.
The imaginary number system solves this dilemma by defining a number by an operation that would otherwise be considered problematic in the real number system.
i=(-1)
1/2Technically, you can't take the square root of a negative because there is no number that is negative after squaring; however, the imaginary number system defines this number and ignores the consequences, substituting all problems with multiples of
i. The main purpose of
i is to create a number that is negative after squaring.
2
i=x
-4=x
2A real number is composed of several subsets of numbers. There are counting numbers, whole numbers, integers, rational numbers, and irrational numbers.
The following spoiler is the set of these subsets.
Counting numbers: any number from one to infinity with a "step" of one (i.e. the next number minus the previous number equals one). E.g. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...
Whole numbers: All counting numbers AND the number zero... E.g. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...
Integers: All counting numbers, their opposites, and zero (which does not have an opposite). E.g. ...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...
Rational numbers: All possible quotients of any two integers that result in a number which can be written with a finite number of decimal places. E.g. ...-75/10, -7, -6, 0, 13/2 ...
Irrational numbers: This subset is part of real numbers, but is not comprised of the previous subsets. This subset represents all numbers that cannot be written out with a finite number of decimal places, such as radicals, certain constants achieved with infinite summing, etc. E.g. e=sum(1/n!,n,0,infinity), pi (ratio of circle's circumference to its diameter), 51/2 (square root of 5, which isn't itself a square), 81/4 (fourth root of eight, which isn't itself a number achieved by raising an integer to the power 4), etc.
The imaginary numbers are any real numbers multiplied by
i (also known as "the imaginary unit").
Complex numbers are the sum of real and complex numbers. They follow the basic form of a+b
i, where a and b are real numbers.
Addition and Subtraction of Complex Numbers with Real, Imaginary, and Other Complex NumbersOperations with complex numbers are easy enough, really. When adding or subtracting, just think of a complex number as a polynomial with two terms, a real term and an imaginary term. When you add a real number to a complex number, combine the real terms for the new complex number. When you add an imaginary number to a complex number, combine the imaginary terms for the new complex number. When you add a complex number to another complex number, just combine the real terms with each other and combine the imaginary terms with each other to get the new complex number. Let's make an example out of 2+5
i.
2+5
i+3
2+3+5
i5+5
i2+5
i-7
2-7+5
i-5+5
i2+5
i+5
i2+10
i2+5
i-5
i2+0
i2
(2+5
i)+(8-32
i)
2+5
i+8-32
i2+8+5
i-32
i10-27
iMultiplication of Complex numbers with Real or Imaginary numbersMultiplication of a complex number with a real number or an imaginary number is easy enough: just use the Distributive Property of Equality.
7(2+5
i)
(7*2)+(7*5
i)
14+35
i6
i(2+5
i)
(6
i*2)+(6
i*5
i)
12
i+(-35)
-35+12
iMultiplication of Complex numbers with Other Complex numbersMultiplication of two complex numbers will be just like multiplication with binomials (polynomials with two terms): just employ the operation FOIL (First, Outer, Inner, Last).
- Multiply the first terms of each complex number
- Multiply the first term of the first complex number with the last term of the second complex number
- Multiply the last term of the first complex number with the first term of the second complex number
- Multiply the last terms of both complex numbers
- Combine like terms
(2+5
i)(9+
i)
(2*9)+(2*
i)+(5
i*9)+(5
i*
i)
18+2
i+45
i+(-5)
(18-5)+(2
i+45
i)
13+47
iDivision of Complex numbers with Real numbersDivision of a complex number by a real number should also be relatively easy, especially if you are allowed to keep your answers in fractional form.
(2+5
i)/5
(2/5)+(5
i/5)
(2/5)+
ior if you must...
0.4+
iDivision of Complex numbers with Imaginary numbersDivision of a complex number by an imaginary number will be slightly more challenging (I was going to say "complex," but then you might get confused): since imaginary terms technically ought not to be in the denominator, you will have to multiply by an expression that equals one in order to cancel out the imaginary units on the bottom.
(2+5
i)/8
i((2+5
i)/8
i)*(
i/
i) *
i/i=1, so the expression is left unaffected since anything times one is itself*
(2+5
i)
i/8
i*
i(2
i+(-5))/(-8)
(-5+2
i)/8
(-5/8)+((1/4)*
i)
(-5/8)+(
i/4)
or if you must...
-0.625+0.25
iDivision of Complex numbers with Other Complex numbersDivision of a complex number by a complex number will require that you multiply both the numerator and the denominator by a special form equivalent to (a-b
i), assuming that the denominator is of the form (a+b
i) (since you do it to both the numerator and the denominator, it is like multiplying by one, which has no effect on the answer). The reason that you multiply by the factor (a-b
i) is so that you get a real number in the denominator.
(a+b
i)(a-b
i)=a
2-ab
i+ab
i-(b
i)
2=a
2+b
2(2+5
i)/(1-3
i)
((2+5
i)(1+3
i))/((1-3
i)(1+3
i))
((2*1)+(2*3
i)+(5
i*1)+(5
i*3
i))/((1*1)+(1*3
i)+(-3
i*1)+(-3
i*3
i))
(2+6
i+5
i-15)/(1+3
i-3
i+9)
(-13+11
i)/10
(-13/10)+(11/10)
ior if you must...
-1.3+1.1
iAbsolute Values of Complex numbersThis is the fun part: absolute values of complex numbers! You must imagine the complex number as a point on the xy-plane, the coordinate grid, for this to make sense. This is where an image would come in handy.
Imagine the standare coordinate grid with the x-axis and the y-axis. Instead of an x-axis, rename it the real axis; instead of a y-axis, rename it the imaginary axis. Rewrite the complex number as an ordinate pair, with a being the real coordinate and b being the imaginary coordinate. Plot the point (a,b) on the real-imaginary coordinate grid. For example, if you have the number 2+5
i, then rewrite it as (2,5) and plot the point on your grid.
You need to know the Distance Formula for this.
d
2=(x
2-x
1)
2+(y
2-y
1)
2d=((x
2-x
1)
2+(y
2-y
1)
2)
1/2The distance from the complex number to the origin is the absolute value of the complex number. This definition works for real and pure imaginary numbers, too! Using 2+5
i as an example...
d=((2-0)
2+(5-0)
2)
1/2d=((2)
2+(5)
2)
1/2d=(4+25)
1/2d=29
1/2The absolute value of 2+5
i is the square root of 29.
Let's use 5 as our number to see if the distance formula really can be used for a definition of the absolute value. The point form of 5 would be (5,0) in the complex coordinate grid.
d=((5-0)
2+(0-0)
2)
1/2d=(5
2+0
2)
1/2d=(5
2)
1/2d=5
That works, since the absolute value of 5 is indeed five.
Now let's use -5, whose absolute value we know already to be 5...
d=((-5-0)
2+(0-0)
2)
1/2d=((-5)
2+0
2)
1/2d=((-5)
2)
1/2d=25
1/2d=5
That also works.
The absolute value of a complex number is its distance from the origin. The formula for finding the absolute value of any number can be written as
|z|=(a
2+b
2)
1/2...when z=(a+b
i).
[M:7000]
