Complex Numbers Made Intuitive This essay explains why complex numbers behave the way they do. I could follow the conventions of a typical high school text and start by writing down the rules for complex arithmetic. But that would only confuse the intelligent reader, who would rightly demand to know why he or she should accept these arbitrary rules as sensible. So instead I'm going to begin by examining a series of simpler number systems, starting with the natural numbers, so that when we do finally arrive at complex numbers, the reader will be able to appreciate the logic behind their definition. number systems: mathematical constructs with pleasing regular structure that describe aspects of the world intro rant: why i = sqrt(-1) is a terrible way to introduce complex numbers example: natural numbers - application: counting beans - operations (+ *), identities (0 1), properties (commutative, associative) - rings, fields, groups, etc. missing additive inverse - so add negative numbers: application: credit/debit ==== sidebar: a textual approach to arithmetic. Mathematically, -6 and 6 are both perfectly good numbers; 6 could be written +6 to emphasize that it appears to the right of 0 on the number line. But textually, "6" is a symbol for a natural number, and we have no symbols for non-natural integers. Instead they are named by construction, e.g., appending a minus sign to get the phrase "-" "6". What this phrase means is: "starting at zero, proceed six clicks out on the number line, but in the negative direction instead of the usual positive direction." We can develop linguistic rules for arithmetic, as follows: to negate x, write "-" x adjacent minus signs cancel: "-" "-" --> "" rule for subtraction: x-y if x >= y then answer is z such that y+z = x [only defined if x>=y] else answer is "-" (y-x) These linguistic rules are how many people actually do math in their heads. Continuuing this idea, we can develop linguistic rules for fractional notation: a/b + c/d = (ad+bc)/bd, plus simplification rules like n/1 --> n, and p/-q --> -p/q. This gets interesting when you add rules for visually nested fractions like: a ___ b ad ------- --> ------ c bc --- d ==== circular arithmetic - application: telling time non-numeric groups: no magnitude ordering of members example: flavor table in quantum chromodynamics example: the group defined by 90 degree rotations about x, y, or z axes missing multiplicative inverse - so go to rationals application: titrating a recipe need for the continuum - so go to reals gives powers with negative exponents; irrationals like pi and e modular arithmetic with reals - application: angles vector arithmetic - addition and scalar multiplication make sense - element-wise multiplication can be done but isn't very interesting what about 2d vectors? - dual representations: rectangular and polar What if we chose a different rule for multiplication of 2d vectors? - u * v = - intuitively: rectangular vector describes a linear displacement polar vector describes a scale and twist - application: wave function amplitudes in quantum mechanics - we can do both addition and multiplication in either form; it's just more natural to do addition in rectangular and multiplication in polar - show additive and multiplicative inverse in each form - show additive and multiplicative identity in each form - show how multiplication in rectangular form is the familiar outer product rule - show how complex conjugate is a useful hack, not fundamental; useful for computing magnitude, and for complex division What about complex exponentials? What about exp(i pi) ? ================ The Bloch or Poincare sphere is a way of encoding a constant-magnitude complex mixture of two objects. See diagram in Science 318:1567 (7 Dec 2007). Given the alternative objects H and V representing horizontal and vertical polarization states of a photon, the north pole of the sphere is H, the south pole is V, and the equator is a mixture of alpha*H + beta*i*V where |alpha|^2+|beta|^2 = 1. This formalism is used to represent mixtures of photon polarization states alpha|H> + beta|V>. ================================================================ 6/28/2012 complex numbers outline: number systems: naturals, integers, reals, etc. operator +, commutative, identity 0: closed under + wnat an inverse? add negative numbers operator *, commutative, identity 1: closed under * want an inverse? switch to reals (and don't count 0) relationship of + and *: n copies of x combined using + gives x*nx n copies of x combined using * gives x^n can't keep going because ^ isn't commutative * distributes over + but ^ does not distribute over * ==== How to define math over 2-vectors? Element-wise addition makes sense. What about multiplication? Element multiplication works but is uninteresting; it produces no interactions between the two components, so it's no different than arithmetic over two scalars done in parallel. Also, there is no good physical analog for this definition of multiplication. What we want is a definition for multiplication that provides for some interaction between the elements, that still has nice properties like commutativity, associativity, and distributivity, and ideally that corresponds to some physical phenomenon. Complex numbers meet this definition, as you will see. ==== why starting from i is bad: multiplying two imaginaries gives a real, but not vice versa sqrt(1) is 1, sqrt(-1) is i, but sqrt(i) is ugly: (1+i)/sqrt(2) real/imaginary coordinate axes look symmetric but can't interchange them the way you can swap x and y axes, due to asymmetry in multiplication complex conjugate treats real and imaginary parts differently. why? There is one little bit of symmetry: the FOIL rule for multiplying two complex numbers: a+bi * c+di = (ac-bd) + (bc+ad)i is just the distributivity of multiplication over addition, combined with the fact that i*i = -1. There is a more profound symmetry in complex numbers, but to see it you have to express them differently. Here is a hint about what that symmetry will look like: Draw the value 1 as a vector on the complex plane. Multiply by i and draw that vector, i. Multiply by i again and you get -1. Multiply by i a third time and you get -i. Do it a fourth time and you get back to 1. These points are 90 degree rotations of each other. So multiplying by i produces a 90 degree rotation. You can easily verify that multiplying any complex number a+bi by i produces a 90 degree rotation of that vector, -b+ai. ==== Now discuss converting between Cartesian and polar representations of 2-vectors. Addition is easy in cartesian; multiplication has a definition in polar that is easy and that turns out to convey the nice properties we desire for math over 2-vectors. ==== Complex powers: see http://www.clarku.edu/~djoyce/complex/powers.html