Complex Numbers: Kalid's Page (2024)

Summary

Complexnumbers can be represented in cartesian form (a + bi) or in polarform (r*e^(i * theta) ). The magnitude of a complex numberis found by multiplying by its complex conjugate (a-bi) and thentaking the square root of the product. In polar form, r isthe magnitude.

Imaginary Numbers: What are they?

Easyanswer: The square root of -1 is represented by the number"i". "i" looks like a variable but itis not; it is the number such that its sqaure equals negative one.The square roots of other negative numbers can be represented interms of i. For example, the square root of -9 is 3i.

Trickystuff: How a negative number can even have a square bogglesthe mind. That's why "i" is imaginary, I suppose,but the fact that we defined and use it is funky. Anotherpoint: don't all numbers have two square roots?The square root of 9 is 3 and -3. What about -1? Itshould have two roots as well (since we said that it is allowedto have roots). Maybe -i will work as well (just like -3 worksfor 9). Well, if we square -i we get

(-i) * (-i) = (-1)i * (-1)i = i * i = -1.

Anotherpoint: imaginary numbers have real squares. Why? When you squarea number b*i (like 3i, or -6i), you get:

(bi)*(bi) =(b * b)(i * i) = -(b^2).

Sono matter what number b you choose, you get a real result.Fourth powers are another matter, but sticking to squares we aresafe.

Whoa.So +i and -i have the same square. Why do we choose one overthe other? I shall return to this shortly, but for now I willlet the excitement build.

Complex Numbers

Wenow have two types of numbers:real, which are all the regularnumbers you know and love, and imaginary numbers, the numbers thatare square roots of negative numbers. It is easy to tell themapart: imaginary numbers have an "i". Real numbersdon't. Now, complex numbers are numbers whichhave two parts: a real part and an imaginary part. One wayto write them is like this: a + bi. "a"is the real part (no i) and "bi" is the imaginary part(has an i). An example of some complex numbers are:
3 + 4i a=3, b=4
3a=3, b=0
-6i a=0,b=-6

As you can see, every number can be written as a complex number.Some numbers, like 5 or -9 don't have imaginary parts, and othernumbers, like 3i, don't have real parts. Complex numbers are commonlywritten in the form

z= a +bi (z is complex, a and b are real).

Noticethat z is a single number. It has two components, but it isstill one number. Think of it in terms of fractions: a fractionis a single number that has two parts: a numerator and a denominator.In general, each component of the fraction is different from thefraction itself. The same goes for complex numbers:zhas two components (1 real and 1 imaginary), and each componentalone is (generally) different from z.The components combineto create the complex number z.

GraphingComplex Numbers

Using a + bi notation, we can even drawcomplex numbers on the complex plane.We are used to x andy axis: we plotted points as (x,y) pairs. Now, instead ofhaving x and y coordinates, we have real and imaginary coordinates.Notice how the complex numbers can be broken down into (a,b) pairs.3+4i becomes (3,4) on the complex plane.

Wedraw a vector from the origin (0,0) to the point (a, b) that representsthe complex number. 3 +3i looks like this, with imaginarynumbers on the vertical and real numbers on the horizontal:

Thisis just like a normal graph, except we have changed the labelson the axes...

Youprobably know that a point can be represented in cartesian or polarcoordiantes.Cartesian coordiantes are in the form (x,y) andgive the two (or more) components of a point.Polar coordinatesuse a direction and magnitude, and have points in the form (r, theta).

Forexample, the point (1,1) in cartesian is (2^.5, 45) in polar.45 represents the direction (45 degrees above the horizontal) and2^.5 represents the amount of distance to go (thank you Pythagoras).The angles start at zero and go counter-clockwise. To go 1 unitdownward:

cartesian: (0,-1)

Polar:( r, theta )
Cartesian:a = r*cos(theta) b= r*sin(theta)

You don't have to memorize these by any means. Draw a triangle andyou can figure it out (link). It's better to learn the intuitionbehind a concept and derive it when you need it. Intuition is hardto forget; formulas are easy.

[Anaside:To express a point in two dimensions you need two peicesof data.We are used to the data coming in an (x,y) pair.Now we see it can also be represented as an (r, theta) pair.Are there any more ways to represent a point on a plane? To representa point in three dimensions, we need three pieces of data. Thereare a few ways to do this (link).]

You will probably seen theta written in terms of radians. Polarcoordinates may seem like a hassle: we have to take our complexnumber and figure out the magnitude and direction. With cartesiancoordinates, it is simply (a, b). The next section will justifywhy we use polar.

IncredibleMath Relation

Ok,I'll admit that very few things in math can be called "exciting".Intersting, maybe (don't roll your eyes) but exciting? This,my friends, is one of those rare moments. I was in hystericswhen I first learned of it. The relation is:

Thisformula is just... amazing. It relates e, which is an irrational(infinite decimal places) and funky number to begin with, to i,an imaginary numbers, and also to sine and cosine, which are justregular functions that have rational values . Whoa.To see why it is true, click here. For example, e^(i*pi) =-1. That equation has two irrantional numbers, and somehowthe exponential e pops out a negative number. Ok, that's enoughblathering about the beauty of that equation, let's see what itcan do.

Suppose we multiply both sides by some number r. Then we get:

Let's look at this for a bit. It isstrikingly similar to some of the equations for converting betweencartesian and polar coordinates. Indeed, (rcos(theta),rsin(theta)) is the (x,y) pair for a point originally expressedin (r, theta) form. But the sin has an i term, so the numberis complex. Now we have an (a,b) term, with a = rcos(theta)and b = rsin(theta).

We have found the polar form for complex numbers. Insteadof being an (r, theta) pair we can write any complex number z as:

z= a +bi or z = re^(i*theta)

Therules for converting between the two are the same. r = sqrt(a^2+ b^2) and theta = arctan(b/a)
If we choose the right numbers for r and theta, then z = a +bi =re^(i * theta). This is all thanks to the beauty of the aboveformula.

ComplexConjugates and Magnitudes

Remember how you were at the edge of yourseat wondering why we choose +i instead of -i as the square rootof -1? Now we can see where it comes in.

The normal method of finding a magnitudeis to square a number and then take its square root. For positivereal numbers this just gives us the original number, and for negativereal numbers (like -9) it will give the absolute value (its magnitude). Thus, both 9 and -9 have the same magnitude of 9. Theyare the same distance from the origin, just in different directions.

Complex numbers aren't quite so simple.Taking 1 + i as an example, if we try and square this and take thesquare root we get: magnitude(?) = sqrt((1+i)^2) = sqrt(1 +2i -1) = sqrt(2i). But we showed earlier that imaginarynumbers can't have square roots. (LINK WHY). Uh oh.

Complex conjugates save the day. Becauseour decision to define i as positive was arbitrary, we can't excludethe possibility of a negative i. We define the complexconjugate of (a + bi) as (a - bi). If z is a complexnumber, its complex conjugate is usually written as z with a barover it. Now, instead of squaring a complex number then takingits square root, we multiply it by its complex conjugate then takethe square root. For any number (a + bi) we get

Magnitude= sqrt( (a+bi) * (a -bi) ) = sqrt(a^2 +abi -abi+ b^2) = sqrt(a^2 + b^2).

Itlooks just like the formula for regular cartesian coordinates!(Pythagorean theorem to find lengths). Thus, the magnitudeof (3 + 4i) is sqrt(9 + 16) = 5. On a last note, if you wantto find the complex conjugate of any complex number, justswitch all the i's to "-i". It doesn't matter ifthey are in exponentials or denominators or inside square roots:just switch them all. For complex numbers in polar form (re^(i*theta)),the magnitude is just r.

The polar form of imaginary numbers is usefulbecause multiplication becomes addition when you are dealing withexponentials. This is much, much easier than expanding outloads of cosine and sine terms. Also, you don't have to rememberthe sine and cosine angle addition formulas; the exponentitals cando it for you. This is very useful when you are analyzingcircuits.