Enough with this Square Root of -1 Business!

Like so many other people, I was kept from appreciating the beauty and utility of mathematics because of the way it was taught to me.

The majority of introductions to complex numbers start with the elusive and mysterious square root of -1, denoted by i.

A number that has an i stuck on to it is called “imaginary” (a convenient differentiator to “real”). Being asked to learn something that is called “imaginary” is not very motivating to young learners who work best starting with concrete metaphors.

51tPEpSto+L._SX322_BO1,204,203,200_

The imaginary number is counterintuitive and confusing. And it’s not the coolest part. Sure, i was an important invention at a critical stage in the history of math when there was no good way to express z2 = -1. And yes, it makes a good ending to a long story (which happens to be true): math has advanced through several expansions of the concept of “number” … from the counting numbers to the wholes – to the negatives – to the fractions – to the irrationals – and finally to complex numbers – where i came along and saved the day.

But…does this mean that invoking i is the best way to explain complex numbers to novices – to everyday people? I join many others in saying that there is a better way to learn about the wonderful world of two-dimensional numbers. One voice among those is Kalid Azad.

kalidphoto-color-homeHe speaks in metaphors and freely engages the visual mind to help us grasp math concepts using our whole brain. In his explanation on complex numbers, Azad says this about i: “It doesn’t make sense yet, but hang in there. By the end we’ll hunt down i and put it in a headlock, instead of the reverse.”

…..

When you get an intuitive, aesthetic feeling for why certain mathematical ideas are being taught, you become more motivated to learn the notation. The corollary: learning math notation without understanding why is like learning musical notation before ever being allowed to listen to or play music.

Paul-Lockhart2Paul Lockhart, in A Mathematician’s Lament, compares the way math is taught to a nightmare scenario in which music is taught to students using sheet music notation only (no actual music is played or heard) – until the student is advanced enough to start “using” it.

 

What is a Two-Dimensional Number?

When I read that complex numbers are really no more “imaginary” than real numbers, I decided that I would start dismantling my old worldview. Why should I assume that numbers have to be one-dimensional? Over time, I became more accustomed to the notion that a number can occupy a plane (the complex plane) and not just a line (the number line). Learning how to make images of the Mandelbrot Set helped a lot.

Think of Multiplication as Rotation

Screen Shot 2015-12-18 at 9.49.07 AMInstead of trying to wrap your mind around i, and how it magically makes equations come out right, let’s start with geometry. Think of multiplication as rotation and expansion. In the blog Girls Angle, Ken Fan introduces complex number multiplication in a nice visual way… here.

Here’s a video explaining complex numbers in terms of physical metaphors, and eventually explaining why the square root of -1 becomes a necessary part of the notation.

screen-shot-2015-12-17-at-4-03-15-pm

Squaring

Consider the following diagram showing what happens when you square certain complex numbers that lie on the unit circle:

complex plane gravitation

The dot on the right represents the complex number (1+0i). When you square it, it stays the same (no surprise: 1×1=1). The number at the left is (-1+0i). When you square that, it becomes (1+0i). But when you square the number at top (0+1i) it “rotates” by 90 degrees to (-1+0i). Finally, at the bottom, the number (0-1i) rotates…but would it be correct to say that it rotates by 90 degrees clockwise to (-1,0i)? Depends on how you look at it. Rotating by 270 degrees counter-clockwise has the same result. This is the nature of rotation and angular reality: it is periodic – it cycles…it repeats.

What an awesome idea. Multiplication is like doing a whirling dervish jig.

Animated Squaring

Here’s an interactive tool I made that allows you to play with 200 dots (complex numbers) randomly scattered on the complex plane. You can experience what happens when complex numbers are squared. It also allows you to multiply the dots (using a complex number dot that you can drag along the screen).

http://ventrella.com/ComplexSquaring/

Screen Shot 2015-12-17 at 3.57.18 PM

This interactive tool might make you feel as if the dots on the screen are obeying some sort of gravitational law of physics. Well, in a way, yes, that’s what’s happening. When you add, multiply, or exponentiate numbers, you get a new number. In the complex plane, the space where that change takes place is two-dimensional. That’s cool! We like images.

Here’s another visual tool: when we multiply two complex numbers, such as (a+bi) and (c+di), we can visualize the operation in this way:

mult

In pseudocode:

realPart      = (a*c) - (b*d);
imaginaryPart = (a*d) + (b*c);

This explanation of multiplication does not require i.

To this day, I STILL do not feel very much music when I think about the square root of -1.

On the other hand, the more I play around with visualizing and animating complex numbers, the more intuitive they become, and the deeper my sense that these numbers are as real as any old one-dimensional number.

They are not imaginary at all.

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5 thoughts on “Enough with this Square Root of -1 Business!

  1. Conversely, all numbers are imaginary since math is entirely invented. We can certainly agree that these particular numbers are very poorly named. I’ve never really gotten my head around why complex numbers are so natural and useful because I’ve never needed them, but I do find them fascinating. Here’s a YouTube playlist on the topic that I found particularly helpful: https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF

  2. Did you try any other interpolation functions? I was thinking it would be fun to try and to the interpolation as a rotation.

    • I assume you are referring to the interactive tool I made that animates squaring. Yes, this is a question I have been mulling over. I recently learned about De Moivre’s formula for raising a complex number by an exponent (in this case, a non-integer that interpolates from 1 to 2). The idea is to use that to interpolate the animation. But I think there may be a problem in that the angles can be expressed in different ways causing a conflict between clockwise and counterclockwise rotation. In other words, there are two directions to interpolate an exponent from 1 to 2 to arrive at the squared value. There seems to be a periodic nature of complex squaring.

  3. What I think I see here is binary logic introduced to a relational 3rd dimensional (or imaginary) space to illustrate numerical values in visual terms. Fine. Notice how the quadrant style box above resembles the “decision box” of political science: yes and no on the x ( so, plus and minus), and no and yes (minus and plus) on the y. It leaves you with regions (dimensional) of yes/no, yes/yes, no/no and no/yes. It also gives you positive NOs and negative YESes. Notice how difficult it become to discuss when you introduce numerical concepts into language.

    • I’m not familiar with the 4-quadrant decision box of political science, so I have no idea what the four combinations of ‘yes’ and ‘no’ actually mean. But there is something rather general about taking two binary regimes and “crossing” them. I wouldn’t call complex numbers ‘binary’ in any sense…unless you are only considering the positive and negative sides of the two dimensions. At any rate, it’s fun to explore number theory in relation to things like language.

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