Dave's Short Course in Complex Numbers

Mathematics reawakened in Western Europe in the 13th century. At that time works in mathematics weretranslated from the Arabic into Latin allowing Western European scholars to learn about the medieval Arabic-language mathematics and the older Greek mathematics, such as Euclid's Elements. In all this mathematics,only positive numbers were considered to be numbers. Negative numbers were not yet accepted as entities.(Some ancient cultures, including that of China and India, accepted negative numbers, but not the onesmentioned above.)

Solution of quadratics.

With negative numbers we understand that every quadratic equation in the variable x can be written in theform

ax2 + bx + c = 0,

where a, b, and c are constants. We also know that the general solution is given by the quadratic formula:

where there are two distinct real solutions if the discriminant b2 – 4ac is positive, one double real solution ifthe discriminant is 0, and no real solutions if the discriminant is negative.

Back in the 15th century, this was not understood. Instead, quadratic equations were classified into fourdifferent kinds depending on the signs of the coefficients a, b, and c. Since the leading coefficient a is notzero in a quadratic equation, you can always divide by it to get an equivalent quadratic equation where a

equals 1, that is, x2 + bx + c = 0. This one form gives rise to four forms when you move the negative terms tothe other side of the equation and when you drop zero terms from the equation:

x2 = c

x2 + bx = c

x2 + c = bx

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x2 = bx + c

There are other forms, but either they have no solutions among the positive numbers or else they can bereduced to linear equations. Each of these forms required a different form of a solution. With hindsight, wesee that the 15th century solutions are just special cases of the quadratic formula. One would think that theconsolidation of four cases into one might be enough justification for accepting negative numbers, butapparently it wasn't. It seems to take a lot of time before people will extend their concept of number toinclude new entities.

Solution of cubics.

Equations of the third degree are called cubicequations. The general form of a cubic is, afterdividing by the leading coefficient,

x3 + bx2 + cx + d = 0,

As with the quadratic equation, there are severalforms for the cubic when negative terms aremoved to the other side of the equation and zeroterms dropped.

Back in the 16th century it was a big deal to solvecubic equations. There was a great controversy inItaly between Cardano (1501–1576) and Tartaglia(1499–1557) about who should get credit forsolving the cubic equation. Any book on thehistory of mathematics will go into the details ofthis fascinating controversy. What’s interesting tous, though, is that negative numbers werebecoming legitimatized, a deeper insight intoequations was developed, and the first inkling of

a complex number appeared. Incidentally, at this time symbolic algebra had not been developed, so all theequations were written in words instead of symbols!

Cardano, in his Artiss Magnæ, or Great Art, found negative solutions to equations, and he called thesenumbers “fictitious”. He also noted an important fact connecting solutions of a cubic equation to its

coefficients, namely, the sum of the solutions is the negation of b, the coefficient of the x2 term. At one otherpoint, he mentions that the problem of dividing 10 into two parts so that their product is 40 would have to be5 + √(–15) and 5 – √(–15).

Bombelli’s investigations of complex numbers

Cardano did not go further into what later became to be called complex numbers thanthat observation, but a few years later Bombelli (1526–1572) gave several examplesinvolving these new beasts. Here’s one example.

One of Cardano’s cubic formulas gives the solution to the equation x3 = cx + d as


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where e = (d/2)2 – (c/3)3). Bombelli used this to solve the equation x3 = 15x + 4 to getthe solution

Now, the square root of –121 is not a real number; it’s neither positive, negative, norzero. Bombelli continued to work with this expression until he found equations thatlead him to the solution 4. He determined that

and, therefore, the solution x = 4. You can check those equations are correct by cubing 2 ± √–1 to get2 ± 11√–1.

This example is not given to show that Bombelli knew everything there is to know about complex numbers,only that he was starting to understand them.

On to The Fundamental Theorem of Algebra

Back to Introduction

Dave’s Short Course on Complex Numbers is locatedat http://www.clarku.edu/~djoyce/complex/

©1999, 2013David E. Joyce

Department of Mathematics and Computer ScienceClark University

Worcester, MA 01610


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As remarked before, in the 16th century Cardano noted that the sum of the three solutions to a cubic equation

x3 + bx2 + cx + d = 0

is –b, the negation of the coefficient of x2. By the 17th century the theory of equations had developed so faras to allow Girard (1595–1632) to state a principle of algebra, what we call now “the fundamental theorem ofalgebra.” His formulation, which he didn’t prove, also gives a general relation between the n solutions to an

nth degree equation and its n coefficients.

An nth degree equation can be written in modern notation as

xn + a1xn–1 + ... + an–2x2 + an–1x + an = 0

where the coefficients a1, ..., an–2, an–1, and an are all constants. Girard said that an nth degree equation

admits of n solutions, if you allow all roots and count roots with multiplicity. So, for example, the equation

x2 + 1 = 0 has the two solutions √–1 and –√–1, and the equation x2 – 2x + 1 = 0 has the two solutions 1 and 1.Girard wasn’t particularly clear what form his solutions were to have, just that there be n of them: x1, x2, ...,

xn–1, and xn.

Girard gave the relation between the n roots x1, x2, ..., xn, and xn and the n coefficients a1, ..., an–2, an–1, and

an that extends Cardano’s remark. First, the sum of the roots x1 + x2 + ..., + xn is –a1, the negation of the

coefficient of xn–1 (Cardano’s remark). Next, the sum of all products of pairs of solutions is a2. Next, the sum

of all products of triples of solutions is –a3. And so on until the product of all n solutions is either an (when n

is even) or –an (when n is odd).

Here’s an example. The 4th degree equation

x4 – 6x3 + 3x2 + 26x – 24 = 0

has the four solutions –2, 1, 3, and 4. The sum of the solutions equals 6, that is –2 + 1 + 3 + 4 = 6. The sum ofall products of pairs (six of them) is

(–2)(1) + (–2)(3) + (–2)(4) + (1)(3) + (1)(4) + (3)(4)

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which is 3. The sum of all products of triples (four of them) is

(–2)(1)(3) + (–2)(1)(4) + (–2)(3)(4) + (1)(3)(4)

which is 26. And the product of all four solutions is –24.

Descartes (1596–1650) also studied this relation between solutions and coefficients, and showed moreexplicitly why the relationship holds. Descartes called negative solutions “false” and treated other solutions(that is, complex numbers) “imaginary”.

Over the remainder of the 17th century, negative numbers rose in status to be full–fledged numbers. Butcomplex numbers remained in limbo through most of the 18th century. They weren’t considered to be realnumbers, but they were useful in the theory of equations. It wasn’t even clear what form the solutions toequations might take. Certainly complex numbers of the form a + b√–1 were sufficient to solve quadraticequations, but it wasn’t clear they were enough to solve cubic and higher-degree equations. Also, the part ofthe Fundamental Theorem of Algebra which stated there actually are n solutions of an nth degree equationwas yet to be proved, pending, of course, some description of the possible forms that the solutions mighttake.

On to The number i

Back to Quadratic and cubic equations


©1999David E. Joyce


Worcester, MA 01610

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Although the Fundamental Theorem of Algebra was still not proved in the 18th century, and complexnumbers were not fully understood, the square root of minus one was being used more and more.

Analysis, especially calculus and the theory of differential equations, was making great headway. Certainfunctions, including the trigonometric functions and exponential functions, appear in solutions to integralsand differential equations. Euler (1707–1783) made the observation, here written in modern notation, that

eix = cos x + i sin x

where i denotes √–1. This is an equation which allows you to interpret the exponentiation of an imaginarynumber ix as having a real part, cos x, and an imaginary part, i sin x. This was an especially usefulobservation in the solution of differential equations. Because of this and other uses of i, it became quiteacceptable for use in mathematics. Euler, a very influential mathematician, recommended the general use ofthese imaginary numbers in his Introduction to Algebra.

By the end of the 18th century numbers of the form x + yi were in fairly common use by researchmathematicians, and it became common to represent them as points in the plane. The standard conventionnow in use to display them is to place the real numbers, that is, those numbers of the form x + 0i, on thehorizontal x-axis, with positive numbers to the right and negative ones to the left. Also, imaginary numbers,that is, those numbers of the form 0 + yi, on the vertical y-axis, where positive values of y are up, andnegative ones down. Thus, i is located one unit above 0 (the origin, where the axes meet), and –i is locatedone unit below 0.

This particular display of numbers of the form x + yi is attributed to various individuals including Wessel,Argand, and Gauss. It was easy to come by, since the usual (x,y)-coordinates for the plane had been used forover a century. Nonetheless, it is a very useful way to understand these numbers.

The Fundamental Theorem of Algebra–proved!

Still, at nearly the end of the 18th century, it wasn’t yet known what form all the solutions of a polynomialequation might take. Gauss published in 1799 his first proof that an nth degree equation has n roots each ofthe form a + bi, for some real numbers a and b. Once he had done that, it was known that complex numbers(in the sense of solutions to algebraic equations) were the numbers a + bi, and it was appropriate to call thexy-plane the “complex plane”.

Complex numbers: the number i http://www.clarku.edu/~djoyce/complex/numberi.html

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In some sense all the historical discussion before Gauss was prehistory of complex numbers. But that’s justthe history that is useful in understanding the need for complex numbers. Although there are other conceptsof numbers that either go beyond complex numbers or include something other than complex numbers, weknow that at least no other “numbers” are needed to solve polynomial equations. The use of complexnumbers pervades all of mathematics and its applications to science.

Next section: The complex plane, addition and subtraction

Previous section: The Fundamental Theorem of Algebra

On to The complex plane, addition and subtraction

Back to The Fundamental Theorem of Algebra




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Since Gauss proved the Fundamental Theorem of Algebra, we know that all complex numbers are of theform x + yi, where x and y are real numbers, real numbers being all those numbers which are positive,negative, or zero. Therefore, we can use the xy-plane to display complex numbers. We'll even call it thecomplex plane when we use the xy-plane that way. That gives us a second way to complex numbers, the firstway being algebraically as in the expression x + yi.

Notation.

The standard symbol for the set of all complex numbers is C, and we'll also refer to the complex plane as C.

We'll try to use x and y for real variables, and z and w for complex variables. For example, the equationz = x + yi is to be understood as saying that the complex number z is the sum of the real number x and the real

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number y times i. In general, the x part of a complex number z = x + yi is called the real part of z, while y iscalled the imaginary part of z. (Sometimes yi is called the imaginary part.)

When we use the xy-plane for the complex plane C, we'll call the x-axis by the name real axis, and the y-axiswe'll call the imaginary axis.

Real numbers are to be considered as special cases of complex numbers; they're just the numbers x + yi wheny is 0, that is, they're the numbers on the real axis. For instance, the real number 2 is 2 + 0i. The numbers onthe imaginary axis are sometimes called purely imaginary numbers.

Arithmetic operations on C

The operations of addition and subtraction are easily understood. To add or subtract two complex numbers,just add or subtract the corresponding real and imaginary parts. For instance, the sum of 5 + 3i and 4 + 2i is9 + 5i. For another, the sum of 3 + i and –1 + 2i is 2 + 3i.

Addition can be represented graphically on the complex plane C. Take the last example. The complexnumber z = 3 + i is located 3 units to the right of the imaginary axis and 1 unit above the real axis, whilew = –1 + 2i is located 1 unit left and 2 units up. So the sum z + w = 2 + 3i is 2 units right and 3 units up.

Parallelogram Rule.

Note in the last example that the four complex numbers 0, z = 3 + i, w = –1 + 2i, and z + w = 2 + 3i are thecorners of a parallelogram. This is generally true. To find where in the plane C the sum z + w of two complexnumbers z and w is located, plot z and w, draw lines from 0 to each of them, and complete the parallelogram.The fourth vertex will be z + w.


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Addition as translation.

Using the parallelogram rule can interpret addition by w as a transformation of the plane C. Adding w to 0gives w, of course, so 0 is moved to w in this transformation. Any other point z is moved to z + w, so z ismoved in the same direction the same distance. In other words, every point in C is moved the same directionand distance when w is added to it. We can say that addition by w gives a translation the plane C in thedirection and distance from 0 to w. The term "vector" is usually used in the description: "the plane istranslated along the vector 0w.

Negation and Subtraction.

There's a nice geometric interpretation ofnegation, too. Of course, the negation of x + yi is–x – yi, so the negation of a complex number willbe located just opposite 0 and the same distanceaway. For example, z = 2 + i is located 2 unitsright and one unit up, so its negation –z = –2 – iis located 2 units left and one unit down.

Negation can be interpreted as a transformationof the plane C, too. If you rotate the plane 180°around 0, then every point z is sent to its negation–z. Thus, negation gives a 180° rotation.

From addition and negation, you can determine what the geometric rule is for subtraction. To find wherez – w will be, first negate w by finding the point opposite 0, then use the parallelogram rule.

We can interpret subtraction of w as a transformation of C: the plane is translated along the vector from 0 to–w. Another way of saying that is that the plane is translated along the vector w0.


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An important concept for numbers, either real or complex is that of absolute value. Recall that the absolutevalue |x| of a real number x is itself, if it's positive or zero, but if x is negative, then its absolute value |x| is itsnegation –x, that is, the corresponding positive value. For example, |3| = 3, but |–4| = 4. The absolute valuefunction strips a real number of its sign.

For a complex number z = x + yi, we define theabsolute value |z| as being the distance from z to 0in the complex plane C. This will extend thedefinition of absolute value for real numbers, sincethe absolute value |x| of a real number x can beinterpreted as the distance from x to 0 on the realnumber line. We can find the distance |z| by usingthe Pythagorean theorem. Consider the righttriangle with one vertex at 0, another at z and thethird at x on the real axis directly below z (orabove z if z happens to be below the real axis). Thehorizontal side of the triangle has length |x|, thevertical side has length |y|, and the diagonal sidehas length |z|. Therefore,

|z|2 = x2 + y2.

(Note that for real numbers like x, we can drop

absolute value when squaring, since |x|2 = x2.) Thatgives us a formula for |z|, namely,

The unit circle.

Some complex numbers have absolute value 1. Of course, 1 is the absolute value of both 1 and –1, but it's

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also the absolute value of both i and –i since they're both one unit away from 0 on the imaginary axis. Theunit circle is the circle of radius 1 centered at 0. It include all complex numbers of absolute value 1, so it hasthe equation |z| = 1.

A complex number z = x + yi will lie on the unit circle when x2 + y2 = 1. Some examples, besides 1, –1, i, and–1 are ±√2/2 ± i√2/2, where the pluses and minuses can be taken in any order. They are the four points at theintersections of the diagonal lines y = x and y = x with the unit circle. We'll see them later as square roots of iand –i.

You can find other complex numbers on the unit circle from Pythagorean triples. A Pythagorean triple

consists of three whole numbers a, b, and c such that a2 + b2 = c2 If you divide this equation by c2, then you

find that (a/c)2 + (b/c)2 = 1. That means that a/c + i b/c is a complex number that lies on the unit circle. Thebest known Pythagorean triple is 3:4:5. That triple gives us the complex number 3/5 + i 4/5 on the unit circle.Some other Pythagorean triples are 5:12:13, 15:8:17, 7:24:25, 21:20:29, 9:40:41, 35:12:27, and 11:60:61. Asyou might expect, there are infinitely many of them. (For a little more on Pythagorean triples, see the end ofthe page at http://www.clarku.edu/~djoyce/trig/right.html.)

The triangle inequality.

There's an important property ofcomplex numbers relating addition toabsolute value called the triangle


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inequality. If z and w are any twocomplex numbers, then

You can see this from theparallelogram rule for addition.Consider the triangle whose verticesare 0, z, and z + w. One side of thetriangle, the one from 0 to z + w haslength |z + w|. A second side of thetriangle, the one from 0 to z, haslength |z|. And the third side of thetriangle, the one from z to z + w, isparallel and equal to the line from 0 tow, and therefore has length |w|. Now,in any triangle, any one side is less than or equal to the sum of the other two sides, and, therefore, we havethe triangle inequality displayed above.

On to Angles and polar coordinates

Back to The complex plane, addition and subtraction




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Multiplication done algebraically.

Complex multiplication is a more difficult operation to understand from either an algebraic or a geometricpoint of view. Let’s do it algebraically first, and let’s take specific complex numbers to multiply, say 3 + 2iand 1 + 4i. Each has two terms, so when we multiply them, we’ll get four terms:

(3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i2.

Now the 12i + 2i simplifies to 14i, of course. What about the 8i2? Remember we introduced i as an

abbreviation for √–1, the square root of –1. In other words, i is something whose square is –1. Thus, 8i2

equals –8. Therefore, the product (3 + 2i)(1 + 4i) equals –5 + 14i.

If you generalize this example, you’ll get the general rule for multiplication

Remember that (xu – yv), the real part of the product, is the product of the real parts minus the product of theimaginary parts, but (xv + yu), the imaginary part of the product, is the sum of the two products of one realpart and the other imaginary part.

Let’s look at some special cases of multiplication.

Multiplying a complex number by a real number

In the above formula for multiplication, if v is zero, then you get a formula for multiplying a complex numberx + yi and a real number u together:

(x + yi) u = xu + yu i.

In other words, you just multiply both parts of the complex number by the real number. For example, 2 times3 + i is just 6 + 2i. Geometrically, when you double a complex number, just double the distance from theorigin, 0. Similarly, when you multiply a complex number z by 1/2, the result will be half way between 0 andz. You can think of multiplication by 2 as a transformation which stretches the complex plane C by a factor of

Complex numbers: multiplication http://www.clarku.edu/~djoyce/complex/mult.html

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2 away from 0; and multiplication by 1/2 as a transformation which squeezes C toward 0.

Multiplication and absolute value.

Even though we’ve only done one case for multiplication, it’s enough to suggest that the absolute value of zw(i.e., distance from 0 to zw) might be the absolute value of z times the absolute value of w. It was when w wasthe real number u just above. In fact, this is true in general:

The verification of this identity is an exercise in algebra. In order to prove it, we’ll prove it’s true for the

squares so we don’t have to deal with square roots. We’ll show |zw|2 = |z|2|w|2. Let z be x + yi, and let w beu + vi. Then, according to the formula for multiplication, zw equals (xu – yv) + (xv + yu)i. Recall from thesection on absolute values that

|z|2 = x2 + y2

Similarly, we have

|w|2 = u2 + v2

and, since zw = (xu – yv) + (xv + yu)i,

|wz|2 = (xu – yv)2 + (xv + yu)2

So, in order to show |zw|2 = |z|2|w|2, all you have to do is show that

(xu – yv)2 + (xv + yu)2 = (x2 + y2) (u2 + v2)

and that’s a straightforward exercize in algebra.

Powers of i.


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For our next special case of multiplication, consider the various powers of the imaginary unit i. We started

with the assumption that i2 = –1. What about i3? It’s just i2 times i, and that’s –1 times i. Therefore, i3 = –i.

That’s interesting: the cube of i is its own negation. Next consider i4. That’s

the square of i2, that is, the square of –1. So i4 = 1. In other words, i is afourth root of 1. You can show that –i is another fourth root of 1. And sinceboth –1 and 1 are square roots of 1, we now know all four fourth roots of 1,namely, 1, i, –1, and –i. This observation connects to the Fundamental

Theorem of Algebra since the equation z4 = 1 is a fourth-degree equation somust have exactly four roots.

Higher powers of i are easy to find now that we know i4 = 1. For example, i5

is i times i4, and that’s just i. You can reduce the power of i by 4 and not

change the result. For another example, i11 = i7 = i3 = –i.

How about negative powers of i? What is the reciprocal of i, that is, i–1? For the same reason that you can

subtract 4 from a power of i and not change the result, you can also add 4 to the power of i. That means i–1 =

i3 = –i. Thus, the reciprocal of i is –i. Imagine–a number whose reciprocal is its own negation! Of course, it’seasy to check that i times –i is 1, so, of course, i and –i are reciprocals.

Roots of unity.

The various roots of 1 are called roots of unity. In general, by the FundamentalTheorem of Algebra, the number of n-th roots of unity is n, since there are n

roots of the n-th degree equation zu – 1 = 0. The square roots of unity are 1and –1. The fourth roots are ±1, ±i, as noted earlier in the section on absolutevalue. Also, in that section, it was mentioned that ±√2/2 ± i√2/2 were squareroots of i and –i, and now with the formula for multiplication, that’s easy toverify. Therefore, the eight eight-roots of unity are ±1, ±i, and ±√2/2 ± i√2/2.Notice how these eight roots of unity are equally spaced around the unit circle.

We can use geometry to find some other roots of unity, in particular the cube roots and sixth roots of unity.But let’s wait a little bit for them.

Multiplying a complex number by i.

In our goal toward finding a geometric interpretation of complex multiplication, let's consider nextmultiplying an arbitrary complex number z = x + yi by i.

z i = (x + yi) i = –y + xi.

Let's interpret this statement geometrically. The pointz in C is located x units to the right of the imaginaryaxis and y units above the real axis. The point z i islocated y units to the left, and x units above. What hashappened is that multiplying by i has rotated to point z 90° counterclockwise around the origin to the pointz i. Stated more briefly, multiplication by i gives a 90°


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counterclockwise rotation about 0.

You can analyze what multiplication by –i does in thesame way. You'll find that multiplication by –i gives a90° clockwise rotation about 0. When we don'tspecify counterclockwise or clockwise when referringto rotations or angles, we'll follow the standardconvention that counterclockwise is intended. Thenwe can say that multiplication by –i gives a –90°rotation about 0, or if you prefer, a 270° rotation about0.

A geometric interpretation of multiplication.

To completely justify what we're about to see, trigonometry is needed, and that is done in an optional section.For now, we'll see the results without the justification. We've seen two special cases of multiplication, one byreals which leads to scaling, the other by i which leads to rotation. The general case is a combination ofscaling and rotation.

Let z and w be points in the complex plane C. Draw the lines from0 to z, and 0 to w. The lengths of these lines are the absolute values|z| and |w|, respectively. We already know the length of the linefrom 0 to zw is going to be the absolute value |zw| which equals|z| |w|. (In the diagram, |z| is about 1.6, and |w| is about 2.1, so |zw|should be about 3.4. Note that the unit circle is shaded in.) Whatwe don't know is the direction of the line from 0 to zw.

The answer is that “angles add”. We'll determine the direction ofthe line from 0 to z by a certain angle, called the argument of z,sometimes denoted arg(z). This is the angle whose vertex is 0, thefirst side is the positive real axis, and the second side is the linefrom 0 to z. The other point w has angle arg(w). Then the productzw will have an angle which is the sum of the anglesarg(z) + arg(w). (In the diagram, arg(z) is about 20°, and arg(w) isabout 45°, so arg(zw) should be about 65°.)

In summary, we have two equations which determine where zw is located in C:

On to Reciprocals, conjugation, and division


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This section assumes a knowledge of trigonometry. For information on trigonometry, see Dave’sShort Trig Course at

http://www.clarku.edu/~djoyce/trig/

Polar coordinates will help us understand complex numbersgeometrically. On the one hand, the usual rectangular coordinatesx and y specify a complex number z = x + yi by giving thedistance x right and the distance y up. On the other hand, polarcoordinates specify the same point z by saying how far r awayfrom the origin 0, and the angle for the line from the origin tothe point. We’ve already called the distance r the absolute value|z| of z, and we saw how the Pythagorean theorem gave relationbetween it and x and y:

Next, we need to deal with the angle . We’ll follow the standardconvention for specifying the angle . This convention takes thepositive x-axis (our real axis) to be at angle 0°, the positive y-axis(our imaginary axis) at angle 90°, the negative x-axis angle 180°, and the negative y-axis at angle 270°. Also,360° can be added or subtracted from any angle and the direction is not changes. So, 0°, 360°, 720°, and–360° all refer to the positive x-axis. Similarly, 270° and –90° both refer to the negative y-axis. A 45° angleruns along the line y = x, up to the right. And so forth.

Complex numbers: angles and polar coordinates http://www.clarku.edu/~djoyce/complex/polar.html

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A point z can be specified by either pair, the pair of rectangular coordinates, x and y, or the pair of polarcoordinates, r, which is |z|, and , which is arg (z). Since either pair determines the point, each pair should

determine the other pair. There should be four equations,connecting them, and so there are. The Pythagorean identity wasmentioned above, but the others require trigonometry. From thesame triangle we used for the Pythagorean theorem, we find thefollowing three relations:

tan = y/x, x = r cos , and y = r sin .

Now, if we apply these relations to our complex numberz = x + yi, then we get an alternate description for z

z = x + iy

= r cos + i r sin

= r (cos + i sin )

= |z| (cos + i sin )

Note that the complex number cos + i sin has absolute value 1 since cos2 + sin2 equals 1 for any angle. Thus, every complex number z is the product of a real number |z| and a complex number cos + i sin .

We’re almost to the point where we can provethe last unproved statement of the previoussection on multiplication, namely, thatarg(zw) = arg(z) + arg(w). As above, we takearg(z) to be , and now let arg(w) be . Then,


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z = |z| (cos + i sin )and

w = |w| (cos + i sin )

We need to show that arg(zw) is + . In otherwords

zw = |zw| (cos ( + ) + i sin ( + ))

If we use the addition formulas for cosine andsine at one crucial point, we’ll have it. Recallfrom trigonometry these addition formulas:

cos ( + ) = cos cos – sin sin

sin ( + ) = cos sin + sin cos .

Now we’re ready to show arguments add in the product zw.

zw = |z| (cos + i sin ) |w| (cos + i sin )

= |zw| (cos + i sin ) (cos + i sin )

= |zw| ((cos cos – sin sin ) + i(cos sin + sin cos ))

= |zw| (cos ( + ) + i sin ( + ))

Thus, arg(zw) is + , as claimed.

On to Multiplication

Back to Absolute value




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We’ve studied addition, subtraction, and multiplication. Now it’s time for division. Just as subtraction can becompounded from addition and negation, division can be compounded from multiplication and reciprocation.So we set ourselves the problem of finding 1/z given z. In other words, given a complex number z = x + yi,find another complex number w = u + vi such that zw = 1. By now, we can do that both algebraically andgeometrically. First, algebraically. We’ll use the product formula we developed in the section onmultiplication. It said

(x + yi)(u + vi) = (xu – yv) + (xv + yu)i.

Now, if two complex numbers are equal, then their real parts have to be equal and their imaginary parts haveto be equal. In order that zw = 1, we’ll need

(xu – yv) + (xv + yu)i = 1.

That gives us two equations. The first says that the real parts are equal:

xu – yv = 1,

and the second says that the imaginary parts are equal:

xv + yu = 0.

Now, in our case, z was given and w was unknown, so in these two equations x and y are given, and u and vare the unknowns to solve for. You can fairly easily solve for u and v in this pair of simultaneous linearequations. When you do, you’ll find

So, the reciprocal of z = x + yi is the number w = u + vi where u and v have the values just found. Insummary, we have the following reciprocation formula:

Complex numbers: reciprocals, conjugates, and division http://www.clarku.edu/~djoyce/complex/div.html

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Reciprocals done geometrically, and complex conjugates.

From what we know about the geometry of multiplication, we can determine reciprocals geometrically. If zand w are reciprocals, then zw = 1, so the product of theirabsolute values is 1, and the sum of their arguments (angles) is0.

This means the length of 1/z is the reciprocal of the length of z.For example, if |z| = 2, as in the diagram, then |1/z| = 1/2. It alsomeans the argument for 1/z is the negation of that for z. In thediagram, arg(z) is about 65° while arg(1/z) is about –65°.

You can see in the diagram another point labelled with a barover z. That is called the complex conjugate of z. It has thesame real component x, but the imaginary component isnegated. Complex conjugation negates the imaginarycomponent, so as a transformation of the plane C all points arereflected in the real axis (that is, points above and below thereal axis are exchanged). Of course, points on the real axisdon’t change because the complex conjugate of a real number isitself.

Complex conjugates give us another way to interpretreciprocals. You can easily check that a complex numberz = x + yi times its conjugate x – yi is the square of its absolute

value |z|2.

Therefore, 1/z is the conjugate of z divided by the square of its absolute value |z|2.

In the figure, you can see that 1/|z| and the conjugate of z lie on the same ray from 0, but 1/|z| is only

one-fourth the length of the conjugate of z (and |z|2 is 4).

Incidentally, complex conjugation is an amazingly “transparent” operation. It commutes with all thearithmetic operations: the conjugate of the sum, difference, product, or quotient is the sum, difference,product, or quotient, respectively, of the conjugates. Such an operation is called a field isomorphism.

Division.


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Putting together our information about products and reciprocals, we can find formulas for the quotient of onecomplex number divided by another. First, we have a strictly algebraic formula in terms of real and imaginaryparts.

Next, we have an expression in complex variables that uses complex conjugation and division by a realnumber.

Both formulations are useful and well worth knowing and understanding.

On to Powers and roots

Back to Multiplication




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Powers of complex numbers are just special cases of products when the power is a positive whole number.We have already studied the powers of the imaginary unit i and found they cycle in a period of length 4.

and so forth. The reasons were that (1) the absolute value |i| of i was one, so all its powers also have absolutevalue 1 and, therefore, lie on the unit circle, and (2) the argument arg(i) of i was 90°, so its nth power willhave argument n90°, and those angles will repeat in a period of length 4 since 4·90° = 360°, a full circle.

More generally, you can find zn as the complex number (1)

whose absolute value is |z|n, the nth power of the absolutevalue of z, and (2) whose argument is n times the argumentof z.

In the figure you see a complex number z whose absolutevalue is about the sixth root of 1/2, that is, |z| = 0.89, andwhose argument is 30°. Here, the unit circle is shaded blackwhile outside the unit circle is gray, so z is in the blackregion. Since |z| is less than one, it’s square is at 60° andcloser to 0. Each higher power is 30° further along and evencloser to 0. The first six powers are displayed, as you cansee, as points on a spiral. This spiral is called a geometric orexponential sprial.

Roots.

Note that in the last example, z6 is on the negative real axis at about -1/2. That means that z is just about equalto one of the sixth roots of -1/2.

Complex numbers: powers and roots http://www.clarku.edu/~djoyce/complex/powers.html

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There are, in fact, six sixth roots of any complex number. Let w be a complex number, and z any of its sixth

roots. Since z6 = w, it follows that

the absolute value of w, |w| is |z|6, so |z| = |w|1/6, and1. 6 arg(z) is arg(w), so arg(z)=arg(w)/6.2.

Actually, the second statement isn’t quite right since 6 arg(z) could be any multiple of 360° more than arg(w),so you can add multiples of 60° to arg(w) to get the other five roots.

For example, take w to be -1/2, the green dot in the figure to the right. Then

|w| is 1/2, and arg(w) is 180°. Let z be a sixth root of w. Then (1) |z| is |w|1/6

which is about 0.89. Also, (2) the argument of w is arg(w) = 180°. But thesame angle could be named by any of

180°, 540°, 900°, 1260°, 1610°, or 1970°.

If we take 1/6 of each of these angles, then we’ll have the possible argumentsfor z:

30°, 90°, 150°, 210°, 270°, or 330°.

Since each of the angles for z differs by 360°, therefore each of the possible angles for z will differ by 60°.These six sixth roots of -1/2 are displayed in the figure as blue dots.

More roots of unity.

Recall that an “nth root of unity” is just another name for an nth root of one. The fourth roots are ±1, ±i, asnoted earlier in the section on absolute value. We also saw that the eight 8th roots of unity when we looked atmultiplication were ±1, ±i, and ±√2/2 ± i√2/2.

Let’s consider now the sixth roots of unity. They will be placed aroundthe circle at 60° intervals. Two of them, of course, are ±1. Let w be theone with argument 60°. The triangle with vertices at 0, 1, and w is anequilateral triangle, so it is easy to determine the coordinates of w. Thex-coordinate is 1/2, and the y-coordinate is √3/2. Therefore, w is(1 + i√3)/2. The remaining sixth roots are reflections of w in the real andimaginary axes. In summary, the six sixth roots of unity are ±1, and(±1 ± i√3)/2 (where + and – can be taken in any order).

Now some of these sixth roots are lower roots of unity as well. Thenumber –1 is a square root of unity, (–1 ± i√3)/2 are cube roots of unity,and 1 itself counts as a cube root, a square root, and a “first” root

(anything is a first root of itself). But the remaining two sixth roots, namely, (1 ± i√3)/2, are sixth roots, butnot any lower roots of unity. Such roots are called primitive, so (1 ± i√3)/2 are the two primitive sixth roots ofunity.

It’s fun to find roots of unity, but we’ve found most of the easy ones already.

Complex numbers: powers and roots http://www.clarku.edu/~djoyce/complex/powers.html

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Dave's Short Course in Complex Numbers

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Transcript of Dave's Short Course in Complex Numbers