complexmath

The Case for Learning Complex Math

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Resistance to complex math seems to never die out.  I see it frequently in PF posts.  Often it takes the form of challenges rather than questions.  First challenge: Complex is just a mathematical trick that has nothing to do with the physics.  Second challenge: Everything that complex does can be accomplished by ordinary real numbers.   My intention in this article is to address those challenges.

Can We Ignore Complex?

Suppose we have a simple rule that says, “The answer is the sum of the vectors.”  The sum of vectors is obtained by adding the tail of each vector to the head of the previous vector.  Adding them up including the different directions of A and B is right, and doing it ignoring the directions is wrong.

That seems obvious and trivial with just two vectors, but the same principle applies in the picture below that I extracted from Richard Feynman’s famous book Q.E.D. about quantum electrodynamics.  The picture examines the reflection of light by a piece of glass.   You don’t need to understand the physics to appreciate the truth of the following. Adding up those little vectors head-to-tail including their directions (as shown in the picture) gives a sum that agrees with experimentally observed statistics of reflected light.  Adding them up ignoring their directions gives the wrong answer.

slask

The point is this.  If you believe that you can simplify things in your mind and neglect the directions of those little vectors, you will come to the wrong answers.  Complex is more than a trick; it describes real physics.  You may not like that, but you must accept it to understand physics.

There are many examples where complex is used in science and engineering.  Following are two simple ones.

  1. An arrow can be considered a one dimensional object. It can point left or right, yet still be one dimensional.   But to flip an arrow from pointing left to pointing right, means that transiently it needs a minimum of two dimensions to describe it in the intermediate states.  Complex math is an elegant and simple way to do that.
  2. In AC electric circuit analysis, the phase angle of electric quantities is as important as their magnitudes. In fact, here is nearly everything you need to know about power grids in a single sentence. “Real power flow is proportional to the voltage phase angle difference between nodes, whereas imaginary power (A.k.a. VAR, or reactive) flow is proportional to the voltage magnitude difference between nodes.” Complex equations such as Ohm’s Law ##\bar V=\bar I\bar Z##, inherently express both magnitude and phase (the notation with the bar on top means the complex version of V or I or Z.)

Can We Achieve the Same Thing Without Complex?

The short answer to that is, “Yes you can, but it is more difficult.”  You can use coupled real equations, coupled real differential equations, trigonometry, or integral calculus to produce the same results.  However, a calculation that may take you seconds to complete using complex, could take hours or days by those alternate methods.  Your understanding of the concepts using the alternate methods may be impaired by the tedium of the expressions. Consider the experience of a historically renowned engineer, who trumped Nikola Tesla and the other geniuses of that era, when he leapfrogged their mathematical abilities by embracing complex:

Steinmetz’s work revolutionized AC circuit theory and analysis, which had been carried out using complicated, time-consuming calculus-based methods. In the groundbreaking paper, “Complex Quantities and Their Use in Electrical Engineering”, presented at a July 1893 meeting published in the American Institute of Electrical Engineers (AIEE), Steinmetz simplified these complicated methods to “a simple problem of algebra”. — from Wikipedia: Charles Proteus Steinmetz

Here is a famous example, the famous Schrödinger Equation, where ##\bar\psi## is complex.

$$i\hbar\frac{\partial{\bar\psi}}{\partial{t}}=\left[\frac{-\hbar^2}{2\mu}\nabla^2+V\right]\bar\psi$$

If we express  with A and B as real and imaginary components, then the following real simultaneous equations express the same thing.

$$\hbar\frac{\partial{A}}{\partial{t}}=\frac{-\hbar}{2\mu}\frac{\partial^2B}{\partial{x^2}}+VB$$

$$\hbar\frac{\partial{B}}{\partial{t}}=\frac{-\hbar}{2\mu}\frac{\partial^2A}{\partial{x^2}}+VA$$

If you think that the real form is simpler, good luck to you.   The bottom line is that all attempts to avoid complex by using equivalent real arithmetic are (in my opinion) harder, not easier.

Complex in History

I first learned of this history from the delightful book An Imaginary Tale, The Story of ##\sqrt{-1}## , by Paul J. Nahin.

All the way back to Archimedes, there was resistance to several mathematical concepts including infinity, irrational numbers, and even negative numbers.   To people who first learned arithmetic by the analogy of counting on their fingers, numbers are real positive things.   They could accept subtraction  5-4 as simply taking the real number 4 away from the real number 5.  But 4-5 was an absurdity.  After all, “Who ever heard of -1 goats?”

Gradually, the metaphor for numbers shifted.  Instead of counting on fingers, numbers could be considered as distances that could be measured with a ruler or a string. Obviously, the ruler could measure to the left or the right, so negative numbers seemed OK.  But given acceptance of -1, ##\sqrt{-1}## or the square root of any negative number was still considered absurd.  There is no positive or negative number, that yields -1 when squared.

The ancients also knew how to solve for the roots of cubic equations.  They knew that there must be three roots, but in some cases they found one real root and two “nonsense” roots that required the square root of negative numbers.  Their reaction of the time was to throw up their hands in disgust and to declare those equations unsolvable.

It was not until 1797 when Caspar Wessel first discovered the validity and enormous simplifications possible by complex numbers (see below). Wessel was ignored, but in 1831 the more famous Carl Friedrich Gauss published his views that permitted complex to be accepted in academic circles.

But even in modern times, we still encounter resistance.  Indeed, the repeated challenges to complex on PF are the inspiration for this Insights article.   I believe that the problem is related to language.  ##\sqrt{-1}## is associated with the words “real” “imaginary” and “complex”.  In natural language, real is associated with reality, whereas imaginary is associated with fantasy, and complex with complicated.  It shouldn’t be surprising that anyone with the slightest touch of math phobia would be warned away by use of those words.   I compare it with the hugely unfortunate choice of words “big bang” in cosmology.  PF mentors are doomed to endlessly answer and re-answer misconceptions rooted in that phrase.  We can blame Leibniz for the word imaginary and Gauss for the word complex.  But Gauss himself recognized the language problem when he said,

If this subject has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed.  Had +1 -1 and ##\sqrt{-1}##  instead of being called positive, negative, and imaginary (or worse still impossible) unity, been given the names, say, of direct, inverse, and lateral unity, there would hardly have been any scope for such obscurity.” — Gauss

So, just as cosmologists are stuck with The Big Bang, we are stuck with the word imaginary.  We can regret it but we can’t change it.

Wessel’s Great Insight

The great leap of insight that led Einstein to special relativity where others failed, had to do with the definition of simultaneity.  In an analogous fashion, Wessel was the first to think of ##\sqrt{-1}## as a rotation operator. Including Wessel’s innovation, we can think of the conceptual evolution regarding elementary numbers as (1) beginning with counting digits, then (2) advancing to arrows and rulers, and finally to (3) the concept that arrows can be rotated.

Let us focus on multiplication, setting aside the scaling part for the moment.  The number 1 can be considered as “no rotation”.  The number -1 can be considered as “rotate 180 degrees”.   ##\sqrt{-1}## rotates 90 degrees.  In that context, ##\sqrt{-1}## can be said to be the geometric mean of +1 and -1. Indeed, Ernst Mach said exactly that in his 1906 book Space and Geometry.  Now think back on what I said before about rotating an arrow.  Start at 0 degrees, direction +1.  Rotate 90 degrees, direction  ##\sqrt{-1}##.  Rotate that another 90 degrees, and we have flipped 180 degrees. That is equivalent to multiplication by ##\sqrt{-1}^2##  which is, of course ##-1##.

For rotations of intermediate angles, we can use the general expression ##a+ib## where a is the real part and b is the imaginary part.   It can also be written in polar form: magnitude c at angle ##\theta## using the notation ##c\angle\theta##.  Complex addition and subtraction are most easily visualized in rectangular form while multiplication and division are most easily visualized in polar form. I am also fond of thinking of repeated rotations where ##\theta## is a small angle, such as one degree.  Then rotation by any integer angle N degrees is equivalent to multiplication by ##(1\angle1)^N##.

In general, multiplication by ##c\angle\theta## simply changes the scale by c and rotates by angle ##\theta##. That was Wessel’s breakthrough.  If only we were taught in elementary school that multiplication is scaling plus rotation, then later told about the special case of real numbers where b=0, or ##\theta=0##.   The best inventions always seem obvious in hindsight. Such was the case here.  A rotation operator seems so simple and obvious to us, but those who preceded Wessel failed to see it.

Just think of i as the fifth basic arithmetic operator, right after, ##+-*/##.  I’ll say it again:  think of the basic arithmetic operators as addition, subtraction, multiplication, division, and rotation.  School children could be taught that from the beginning.  Hopefully, they might have a much easier time accepting complex when they get older.  Then, they could also learn that complex multiplication incorporates both scaling and rotation, so that the list of basic operators could be reduced to four once again.

Nahin’s book can show you numerous examples of how exceedingly difficult problems become simple using complex.  Let me show one where the freedom to choose rectangular or polar notation is valuable.  Expand the expression ##(a+ib)^9##.  After much work, and with no errors, you should arrive at $$a^9+9ia^8b-36a^7b^2-84ia^6b^3+126a^5b^4+126ia^4b^5-84a^3b^6-36ia^2b^7+9ab^8+9ib^9$$

But if we put it in polar form ##c\angle\theta## instead of rectangular ##a+ib## and consider each successive multiplication as a repeated rotation the answer is trivially ##(c\angle\theta)^9=c^9\angle(9\theta)##.

Complex math is most commonly associated with two dimensions.  But it can be used with any number of dimensions (but it is easiest with 2, 3, 4, 6 or 8 dimensions).  For example, three dimensional space coordinates can be expressed as ##x+iy+_jz## where ##i## and ##j## are mutually orthogonal unit vectors used as rotation operators.

Complex Conjugates

What are the roots of the cubic equation ##x^3=-1##?  The answer is simple and obvious to us, but it is one of the problems that bedeviled ancient mathematicians.  The roots are at ##\sqrt[3]{-1}##, but what does that mean?  It means there are three roots, ##-1##, ##1\angle 60##, and ##1\angle -60##.  Once again, Complex math makes its simple.  It becomes simpler still when we think of circles and rotation operators and show the results graphically. To take a cube root, simply divide the circle into three equal parts.

 

The two complex roots above, are complex conjugates.  Conjugates have the same magnitude of rotation, but the opposite direction of rotation.  In rectangular coordinates, the conjugate of ##a+ib##  is ##a-ib##. In physics and engineering, complex quantities very often appear as complex conjugate pairs.  Now you know what that means.

Mathematicians and scientists use the symbol i for ##\sqrt{-1}##. Electrical engineers (EEs) use the symbol ##I## to mean current.  For example, ##P=EI##.  But that could lead to the awkward complex expression ##iI##.  To avoid that, EEs use ##j## instead of ##i## as the symbol for ##\sqrt{-1}##.  The term ##j\omega t## appears frequently in AC analysis.

Conclusion

So if you are one of those who strives to understand physics and engineering, but who has been resisting learning about complex math, cut it out.  You need to learn the basics of complex arithmetic. Even a very basic tutorial that will take you less than 30 minutes to master may be sufficient.   Here is one that I like.

http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/

Afterward

Any student of quantum mechanics will encounter the famous Pauli matrices.  In the context of the above discussion about flipping and rotating, how would you characterize them?  How would you express them without complex numbers?

##\sigma_1 = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}## ##\sigma_2 = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}## ##\sigma_3 = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}##

Acknowledgements

PF regular Hesch and DeanC, both contributed suggestions for this article.

 

Dick Mills is a retired analytical power engineer. Power plant training simulators, power system analysis software, fault-tree analysis, nuclear fuel management, process optimization, power grid operations, and the integration of energy markets into operation software, were his fields. All those things were analytical. None of them were hands-on.

During the years 2005-2017. Dick lived and cruised full-time aboard the sailing vessel Tarwathie (see my avatar picture).  That was very hands on.  During that time, Dick became a student of Leonard Susskind and a physics buff. Dick’s blog is at dickandlibby.blogspot.com

15 replies
  1. Ralph Dratman
    Ralph Dratman says:

    I agree about using ordered pairs. If one considers a vector space of two dimensions equipped with a specific multiplication operation (vector * vector -> vector) corresponding with complex multiplication, the concept of an "imaginary" something goes away, while leaving notation and computation basically unchanged.

  2. mfig
    mfig says:

    Great article. I do have to disagree with the contention that the resistance of many to complex numbers is due to the term imaginary. People don’t have a problem accepting all kinds of imaginary things when viewing Star Wars, so I don’t think it bogs them down that much in math class. Rather, I think the aversion is quite natural. As was pointed out in the article, the reaction to ## n = 4 – 5 ## was, “Who ever heard of -1 goats?!” That is not an irrational reaction. It is perfectly natural because, well, nobody has ever been able to spot their -1 goat out browsing on the mountain. It is only when -1 is seen as an accounting tool that we can understand what -1 goats means.

    I think a similar reaction is provoked when one encounters ## i = sqrt{-1} ## for the first time. It is not because it is called the imaginary unit, but because people naturally read ## A = sqrt{B} ## as, “Some number A, which when multiplied by itself, equals B.” They then make the immediate deduction that there is no such number A, which when multiplied by itself, equals -1 or any other negative number! This is one of the reasons I personally think complex numbers should first be taught as ordered pairs with specific operations simply defined. Then the fact that ## (0,1)*(0,1) = (-1,0) ##, does not arouse natural suspicion at all. Later, the convenient notation can be introduced. IMO, this would go a long way towards eliminating the aversion to complex numbers.

  3. kith
    kith says:

    There are a number of little errors with your Schrödinger equation: the [itex]hbar[/itex] needs to be squared, the derivative with respect to [itex]x[/itex] should be of second order and neither the [itex]bar psi[/itex] nor the [itex]nabla[/itex] should be there in the bracket.

    Nice insight!

  4. anorlunda
    anorlunda says:

    There are a number of little errors with your Schrödinger equation: the [itex]hbar[/itex] needs to be squared, the derivative with respect to [itex]x[/itex] should be of second order and neither the [itex]bar psi[/itex] nor the [itex]nabla[/itex] should be there in the bracket.

    Nice insight!

    Thanks for your sharp eye. Corrected.

  5. QuantumQuest
    QuantumQuest says:

    A really good insight. What everyone resisting to the very idea of complex numbers has to be aware of, is that the concept is not something ad-hoc which after some work, made its way and conquered the world of math, but rather it was somewhere hidden and finally came into light. That explains its use as rotation as described in the article and it definitely has its important place in Physics.

  6. anorlunda
    anorlunda says:

    When writing that Insights article, I came upon a curiosity.

    Rather than (add,subtract,multiply,divide) as the basic arithmetic operations, students could be taught (sum,negate,multiply,invert). Where sum adds signed numbers, while add implies unsigned positives such as “3 goats plus 2 goats”. Subtraction could be defined as negate, then sum. Division is invert, then multiply.

    But then someone could note that negation is just multiplication by -1, so that the operators could be reduced to three (sum, multiply, invert.)

    If we introduce complex numbers, then we could have (sum, negate, multiply, invert, rotate). But negate and rotate are both special cases of multiply, so we are back to three once again (sum, multiply, invert).

    But with complex, there is an additional basic operation so we are back to four (sum, multiply, invert, conjugate)

    I’m curious. Has anyone else been down this path before of redefining the basic arithmetic operators?

  7. jbriggs444
    jbriggs444 says:

    But then someone could note that negation is just multiplication by -1, so that the operators could be reduced to three (sum, multiply, invert.)

    From a abstract algebra perspective, I am not comfortable making multiplication part of the definition of a group under addition.

  8. anorlunda
    anorlunda says:

    making multiplication part of the definition of a group under addition.

    I don’t know what you mean by that.

    Start at the beginning. Are you comfortable with (add, subtract, multiply, divide) being the four basic operators taught in grade school?

  9. jbriggs444
    jbriggs444 says:

    I don’t know what you mean by that.

    Start at the beginning. Are you comfortable with (add, subtract, multiply, divide) being the four basic operators taught in grade school?

    Yes. That is fine. And I would also have no problem with (add, negate), (multiply, invert).
    I have a problem with using multiplication by -1 as part of the definition of the additive inverse. It’s superfluous and introduces a multiplication operation that may not even exist.

  10. anorlunda
    anorlunda says:

    Yes. That is fine. And I would also have no problem with (add, negate), (multiply, invert).
    I have a problem with using multiplication by -1 as part of the definition of the additive inverse. It’s superfluous and introduces a multiplication operation that may not even exist.

    OK, then I agree. I hadn’t thought so deeply into the implications.

    But would you consider negate different than multiply by -1? It is hard for me to think of negate except as a special case of multiply.

  11. jbriggs444
    jbriggs444 says:

    But would you consider negate different than multiply by -1? It is hard for me to think of negate except as a special case of multiply.

    It is hard for me as well. All that training starting from elementary school dealing with integers and rational numbers makes it difficult to get away from that mode of thought and to start thinking of addition and multiplication as abstract operations on other sets.

    Yes, putting my mathematician’s hat on, negation is different from multiplication by -1 since the latter operation may not exist.

    That said, if multiplication exists and additive inverses exist and if the distributive law for multiplication over addition holds, then I am pretty sure that one can demonstrate that negation and multiplication by -1 must be equivalent.

  12. mnb96
    mnb96 says:

    "Can We Achieve the Same Thing Without Complex? Yes you can, but it is more difficult."I have the feeling that this is a somewhat arguable statement and that the "mistery" behind complex numbers is often overemphasized. In my opinion all these "misteries" about imaginary numbers and the square root of -1 disappear when you simply see complex numbers as the even-grade Clifford algebra of R^2. The "misterious" imaginary unit is just the bivector e1*e2 (or a unit pseudoscalar of R^2, in general). Multiplication of complex numbers is equivalent to the Clifford product of scalar+bivector quantities in R^2, with the difference that such quantities can even be multiplied by ordinary vectors in R^2, and they have the effect of rotating (and scaling) the vector. I think Clifford algebras, Geometric Algebra, Clifford products and rotors should have been mentioned in this article. Reference https://www.physicsforums.com/insights/case-learning-complex-math/

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