How can a number be “imaginary”?
First of all, we know that the imaginary number, i, is defined as being the square root of negative 1.
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We know that the square root of a number is something that when squared, or multiplied by itself, gets that number. Say two; two multiplied by two is four, so the square root of four is two. This process is backward, as you might have noticed; you know the answer before you can look for it. But is that the whole answer? What else when squared gives you four?
Negative two.
A negative number multiplied or divided by another negative number gives you a positive number. Squaring any negative number will give you the same positive number as the square of the absolute value of that number.
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The square of a negative number can never be a negative number and the square of a positive number can never be a negative number. The square of zero, as anything multiplied by zero, is still zero.So what is the square root of a negative number, if it can’t be positive, negative or zero?
Imaginary.
i is represented by a letter because it can’t be represented by a number. It is an invented concept that we (the culmination of human civilization) have come up with to help us move forward.
There may be some measure of faith when it comes to mathematics. Mathematicians find themselves hitting brick walls, then “invent” ways through them only to find later that those inventions were more like discoveries and insights a little deeper into the workings of the universe.
Basic Math Using i
i+i = 1*i+1*i = (1+1)*i = 2*i
i-i = 1*i – 1*i = (1-1)*i = 0
i*i = sqrt(-1)*sqrt(-1) = -1
i/i= 1
Complex Numbers
Complex numbers are numbers with both real and imaginary components. The general form looks like this:
a*i+b
where both a and b represent real numbers.
a = Re(a*i+b) read “a is the real part [of a*i+b]“
b = Im(a*i+b) read “b is the imaginary part [of a*i+b]“
Note:
5 = 0*i + 5 (example cn.01)
i = 1*i + 0 (example cn.02)
2*i = 2*i+0 (example cn.03)
Any number can be written as a complex number, but where 5 can be written as a complex number, 2*i is explicitly a complex number.
Okay, so now you know how to write complex numbers. How do you use them?
Addition/Subtraction
The one thing that you always need to remember is that the general form of a complex number is the closest thing you can come to adding a real number to an imaginary number.
5+i = 1*i+5 (example ca.01)
5 + i + i = 2*i + 5 (example ca.02)
5+5+i = 1*i + 10 (example ca.03)
Notice a pattern? Associative Property to the rescue!
5+i+i (example ca.04)
=(5+i)+i
= (i+i) + 5
= 2*i+5
5+5+i (example ca.05)
=(5+i)+5
= i+ (5+5)
= 1*i+10
5+5+i+i (example ca.o6)
=(5+i)+(5+i)
=(i+i)+(5+5)
=2*i+10
(3*i+5)-(-2*i+4) (example ca.07)
=(3*i+2*i)+(5-4)
=1+5*i
Multiplication
This is basically just foiling, treating i as a variable even though it technically isn’t. Thought it would never come in handy? Wrong.
(5+i)*(5+i) (example cm.01)
=5*5 + i*5 + 5*i + i*i
=25+5*i+5*i+(-1)
=10*i + 24
(1+2*i)*(3+4*i) (example cm.02)
=1*3+ 1*(4*i) + (2*i)*3 +(2*i)*(4*i)
(2*i)*(4*i)
=(2*4)*(i*) thanks to the associative property of multiplication
=8*(-1) = -8
=3 +4*i + 6*i + (-8)
=(4+6)*i +(3-8)
=10*i -5
Note that on example cm.02 I took a part of the problem and solved it independently from the rest of the expression. If you ever have trouble doing all of your work on one page or all at once, it is useful to remember that you can use scrap paper and solve individual parts of a problem that is too complex to tackle in one go. Just remember to do it responsibly.
Division/Fractions
(10*i+5) / (5) (example cd.01)
=(10*i/5) + (5/5)
=(10/5)*i + (5/5)
=2*i+1
(10*i-5)/(2*i+1) (example cd.02)
This is not so simple without special tools. We need to factor. This is going to be broken up into three basic stages.
First, we’re going to assume that the solution is going to be a complex number that we’re going to need to solve for the real and imaginary parts of.
(2*i+1)*(a*i+b) = (10*i-5) cd.o2.1
(2*i)*(a*i) + (2*i)*(b) + (1)*(a*i) + (1)*(b) = 10*i -5
2*a*i*i + 2*i*b + a*i +b = 10*i-5
(2*b +a)*i +(b-2*i) =10*i -5
Second, we’re going to separate out the real part and the imaginary parts and solve them separately to get two equations in terms of “a” and “b”.
Re((2*b +a)*i +(b-2*i) )=Re(10*i -5)
2*i*a*i +1*b = -5
b-2*a = -5 cd.02.2
Im((2*b +a)*i +(b-2*i)) = Im(10*i -5)
2*i*b + 1*a*i =10*i
a+2*b =10 cd.02.3
Third, we’re going to treat the two equations we found in step 2 as a system of equations to solve for “a” and “b”.
b=2*a-5 is a way to re-write equation cd.02.2
a+2*(2*a-5)=10 rewrites cd.02.3 substituting 2*a-4 in for b
a+4*a-10 = 10
5*a = 10+10
a=20/5 = 4 cd.02.4
b=2*a – 4
b=2*4-5
b=10-4=3 cd.02.5
(10*i-5)/(2*i+1) = (4*i+3)
Do you want to know what would have made this process so much easier?
Complex Conjugates!
This is one of the most important parts of a complex number; the complex conjugate of a number, (generally written as that number with a line drawn over it, but written here as underlined), is the same magnitude but has an imaginary part of opposite sign. Complex conjugates are a special tool created to make your life easier!
a*i + b = b – a*i = (-a)*i +b (example cc.01)
Re(ai+b)=Re(ai+b)
Im(ai+b)=-Im(a*i+b) and Im(ai+b)=-Im(ai+b)
Note:
a*i+b * a*i+b (example cc.02)
= (a*i+b)*(b-a*i)
=(a*i)(b) + (a*i)(-a*i) +(b)(b) + (b)(-a*i)
=ab*i – ab*i + (b)(b) + (a)(a)(-1)(i)(i)
=b^2 + a^2
=Re(a*i+b)^2 + Im(a*i+b)^2
(a*i+b) / (c*i+d) (example cc.03)
=(a*i+b)(d-c*i) / (c^2 +d^2)
which is useful because it not only cuts down on how many steps it takes to solve something, but makes messier fractions more manageable.
(7i-5)/(2i+3) (example cc.04)
=(7i-5)(3-2i)/(4+9)
=(21*i-10*i+14-15) /13
=(11*i-1)/13
=(11/13)*i +(-1/13) <- (can you imagine doing that the other way?)
(10*i-5)/(2*i+1) (example cd.02 revisited)
=(10*i-5)(1-2*i)/(4+1)
=(10*i-5 -2*10*i*i +10*i)/5
=(20*i +15) /5
=(4*i + 3) <- (So much easier to find with specialized tools,no?)
The Complex Plane
The complex plane is very similar to the Cartesian Plane, except instead of the y axis you have the imaginary axis and instead of the x axis you have the real axis.
Generally, to plot (a*i+b), go up “a” and right “b” (down/left for respective negative values).
- Say a and b are both positive: go up “a” and right “b”, (or down “a” and right “b” for the complex conjugate)
- Say a is negative but b is positive: go down “a” and right “b”, ( or up “a” and right “b” for the complex conjugate)
- Say both a and b are negative: go down “a” and left “b”, ( or up “a” and left “b” for the complex conjugate)
