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Numbers and Their Application - Lesson 15

Imagine More Complex Numbers

Lesson Overview

The Complex Numbers

It would seem that with so many real numbers, mathematicians would be satisfied. However, just as negative numbers allowed us to solve equations such as x + a = 0, so too do imaginary numbers, or more accurately complex numbers, allow us solutions to all quadratic and higher degree polynomial equations. The choice of the term imaginary has been somewhat unfortunate, but with exposure and practice, these numbers can become just as meaningful as the reals. Consider the following solution.
x2 + 1= 0
x2 = -1
x = ±[the square root of negative one]
Euler called these last quantities ±i. Euler also popularized the symbol e, f(x), and many others.

i = [the square root of negative one] is termed the unit imaginary—all imaginary numbers can be formed as multiples thereof.

For most students, the first exposure to complex numbers is in solving quadratic equations that have no real solutions, such as x2 + 3x + 3 = 0. Using the quadratic formula, we find that the discriminate (the part of the formula under the radical) is negative (-3)—but how do we take the square root of -3? Using this new symbol i = [the square root of negative one], and our rules for manipulating radicals, it becomes x=[square root of three] i, and the solutions to this equation are the complex numbers: (-3 ± [square root of three] i)/2.

Complex numbers are of the form a+bi, where a [caligraphic R] and b [caligraphic R].
a is called the real part, and b (not bi) is called the imaginary part.

Real and imaginary numbers are both small subsets of the complex numbers. Real numbers are represented by a, where b=0. Whereas, when a=0, a + bi is just bi—the imaginary numbers. The complex numbers are represented by the symbol [caligraphic C]. A common mistake is to refer to the complex numbers as the imaginary numbers. However, the imaginary numbers are only a very special subset of the complex numbers. The term non-real complex is often used, since all real numbers are complex numbers.

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The complex conjugate of a + bi is a - bi.

Complex numbers often appear in conjugate pairs—see the quadratic formula for why. i can be treated just like a variable, such as simplifying powers:
i0 = 1
i1 = i
i2 = -1
i3 = i2i = -1•i = -i
i4 = (i2)2 = (-1)2 = 1
in = in mod 4

Operations with Complex Numbers

Your TI-84+ graphing calculator will do extensive calculation with complex numbers. (Check your MODE and be sure you are in a+bi and not Real or re^0i.)
  1. Adding or subtracting complex numbers involves adding/subtracting like terms.

    (3 - 2i) + (1 + 3i) = 4 + 1i = 4 + i
    (4 + 5i) - (2 - 4i) = 2 + 9i
    (Don't forget subtracting a negative is adding!)

  2. Multiply: Treat complex numbers like binomials, use the FOIL method, but simplify i2.

    (3 + 2i)(2 - i) = (3 • 2) + (3 • -i) + (2i • 2) + (2i • -i)
    = 6 - 3i + 4i - 2i2
    = 6 + i - 2(-1)
    = 8 + i

  3. Divide: multiply numerator and denominator by the complex conjugate of the denominator.

    2 + 3i = (2 + 3i)(3 - i) = 6 + 7i - 3i2 = 6 + 7i + 3 = 9 + 7i
    3 + i (3 + i)(3 - i) 9 - i2 9 + 1 10

  4. If x > 0, then [square root] (-x) = i [square root] (x).

    [square root] (-9) • [square root] (-16) = i [square root] (9) • i [square root] (16) = i2•3•4 = -12. Notice how our order of operation is important (exponentiation before multiplication) as commonly, the incorrect answer [square root] (144) = 12 is obtained.

  5. Magnitude: |3 + 4i| = [square root] (32 + 42) = [square root] (9 + 16) = [square root] (25) = 5. Magnitude is often confusingly referred to as absolute value, since the same symbol is used. Notice how both are a measure of distance and the Pythagorean Theorem is used here. A common mistake is to include the i—avoid that error.

Graphing Complex Numbers

Complex numbers are graphed on the complex plane—the cartesian product of the reals and the imaginaries. As such, it is very similar to the xy-plane. The familiar x-axis is still the familiar real number line and the y-axis is replaced with a number line containing the imaginary numbers. This is often termed an argand diagram. Cantor showed it was possible to construct a one-to-one correspondence between every point in the plane and the real number line. He did this by mapping the ordered pair with decimal expansion (a1a2a3...,b1b2b3...) to the point (a1b1a2b2a3b3... thus interlacing the decimal expansion. Thus, it would seem, the complex numbers have the same cardinality as the reals.

Polar Form

Complex numbers are also often located on the complex plane by their distance from the origin and angle from the positive x-axis. The angle might be given in either degrees or radians. Your TI-84+ calculator uses degrees but may be set in either polar or a+bi format and will interconvert for you. The following relationship due to Euler
Kei0=K(cos0 + isin0),
 where sin and cos are the trigonometry relationships, is often used. Thus if K = 1 and 0 = [pi]/2 = 90°, the complex number located one unit directly above the origin is obtained. This is i, because sin(90°)=1 and cos(90°)=0. r is a much more common choice of variable to represent magnitude, but the author feels the choice of K will be much more meaningful and memorable for his students! The five most important numbers in mathematics can be interrelated this way: e-i[pi]+1=0

Finding Roots

Abraham de Moivre was born in France in 1667 but moved to England while a teenager for political refuge. There he chanced to met Newton's Principia Mathematica and supported himself by lecturing. He soon established himself as a respected first-rate mathematician and was eventually asked to decide between Newton and Leibnitz regarding the invention of the calculus. His name lives on in de Moivre's Theorem which states that zn = kncis(n0), where
cis0 is an abbreviation for cos0 + isin0.
n may be fractional thus  z1/n = k1/ncis([0 + 360j]/n)° where j is an integer ranging from 0 to n-1.

We can apply this to the multiplicative identity (1) which also has a magnitude of 1. It is clear 1 has two square roots: ±1. Since -1 has two square roots, it should now be clear that 1 has four fourth roots: ±1 and ±i. We can apply de Moivre's Theorem to obtain the eight eighth roots as follows.

The Eight Eighth Roots of Unity are ±1, ±i, ±[square root of two]/2±i• [square root of two]/2. (This last expression is generally considered ambiguous as to how many points it represents, but here represents four distinct points.) Note how they are very symmetrically arranged (on a circle) on the complex plane. Note also how the radical relates to sin(45°+90°n) and cos(45°+90°n). The table of Greek letters (above) with names and phonetic English equivalents should be committed to memory by the grade A math and science student.

Errata

With this eighth printing, fewer typographical errors hopefully exist. However, some interesting ones keep appearing, so please bring them to my attention. Although no printer, operating system, nor browser changes occurred since last year, there were pagination differences on pages I did not change! However, fewer image issues appeared this time. Please note that these pages are optimized for Netscape 4.76, one of the legal, less buggy, and less virus-prone browsers.

Students should start organizing their booklets for stapling soon after homework 15 has been graded. Check to be sure you have all your pages. I prefer your homework to be interleaved—see table of contents instructions. Most lectures at least spill onto a 5th page and added material makes that excessive at times. The exceptions are 2, 10, and 14. All 16 homeworks are 1 sheet, however. Be sure you have activity 11, perhaps others, and 2 quizzes. Do not have your test or test key stapled within the booklet.

A few developmental errors or omissions are also list here since they are still fresh in my mind.

Epilogue

This series of lectures on numbers are in many ways a dream come true. Some work remains to smooth out the level of effort required and make the homework do what I want it to. Links to biographies (math projects) are still needed as well. This material (especially the 1998-99 junior stuff) helps expand many topics discussed rather concisely at times here. Links to (Dr. Math, Fibonacci, etc.) were just added here. It is planned for Center students to take some responsibility to clarify the less clear and extend the more interesting aspects. Continued feedback is appreciated.

We have created the list of mathematicians and Latin/Greek words, abbreviations, and phrases in lesson 1 this year. For distribution, however, we will print them here.

List of Mathematicians

Several different mathematicians are referenced in this series of lectures and some have asked for a list. Here is a start.

 

 

 

List of Greek/Latin Terms

Several different Greek and Latin terms are purposefully used in this series of lectures and some have asked for a list. Here is a start.

 

 

 

 

 

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