How To Put Imaginary Number In Matrices

Imaginary numbers always confused me. Like agreement e, most explanations fell into one of two categories:

It's a mathematical brainchild, and the equations work out. Deal with information technology.
Information technology's used in avant-garde physics, trust us. But wait until higher.

Gee, what a great way to encourage math in kids! Today we'll set on this topic with our favorite tools:

Focusing on relationships, not mechanical formulas.
Seeing complex numbers as an upgrade to our number arrangement, just like zip, decimals and negatives were.
Using visual diagrams, not just text, to empathise the idea.

And our hugger-mugger weapon: learning by analogy. We'll arroyo imaginary numbers by observing its ancestor, the negatives. Here's your guidebook:

imaginary number properties

Information technology doesn't make sense yet, but hang in there. By the stop nosotros'll chase down i and put it in a headlock, instead of the reverse.

Video Walkthrough:

Actually Agreement Negative Numbers

Negative numbers aren't easy. Imagine you're a European mathematician in the 1700s. You have 3 and 4, and know you lot can write four – 3 = one. Unproblematic.

But what almost 3-four? What, exactly, does that hateful? How can you take 4 cows from 3? How could yous have less than aught?

Negatives were considered absurd, something that "darkened the very whole doctrines of the equations" (Francis Maseres, 1759). Yet today, it'd be cool to think negatives aren't logical or useful. Try request your instructor whether negatives corrupt the very foundations of math.

What happened? Nosotros invented a theoretical number that had useful properties. Negatives aren't something we can touch or concord, just they depict certain relationships well (like debt). It was a useful fiction.

Rather than saying "I owe you 30" and reading words to meet if I'm up or down, I can write "-thirty" and know information technology means I'm in the hole. If I earn money and pay my debts (-30 + 100 = 70), I tin record the transaction easily. I have +70 afterwards, which ways I'chiliad in the clear.

The positive and negative signs automatically continue track of the direction — you don't need a sentence to draw the bear on of each transaction. Math became easier, more than elegant. Information technology didn't matter if negatives were "tangible" — they had useful properties, and we used them until they became everyday items. Today you'd call someone obscene names if they didn't "go" negatives.

But let'southward not exist smug nigh the struggle: negative numbers were a huge mental shift. Fifty-fifty Euler, the genius who discovered due east and much more, didn't understand negatives every bit we do today. They were considered "meaningless" results (he later on made up for this in style).

It'southward a attestation to our mental potential that today's children are expected to understand ideas that once confounded ancient mathematicians.

Enter Imaginary Numbers

Imaginary numbers have a similar story. We can solve equations like this all twenty-four hours long:

$\displaystyle{x^2 = 9}$

The answers are three and -3. But suppose some wiseguy puts in a teensy, tiny minus sign:

$\displaystyle{x^2 = -9}$

Uh oh. This question makes most people cringe the first time they run across information technology. You want the square root of a number less than zippo? That's absurd! (Historically, there were real questions to reply, merely I like to imagine a wiseguy.)

It seems crazy, just like negatives, zero, and irrationals (non-repeating numbers) must take seemed crazy at first. There'southward no "real" meaning to this question, correct?

Wrong. So-called "imaginary numbers" are as normal as every other number (or just as simulated): they're a tool to describe the world. In the same spirit of assuming -1, .3, and 0 "be", let'southward assume some number i exists where:

$\displaystyle{i^2 = -1}$

That is, you lot multiply i past itself to get -1. What happens at present?

Well, beginning we go a headache. Simply playing the "Let's pretend i exists" game actually makes math easier and more elegant. New relationships sally that we can describe with ease.

You may not believe in i, but like those fuddy former mathematicians didn't believe in -1. New, brain-twisting concepts are hard and they don't make sense immediately, even for Euler. But equally the negatives showed us, foreign concepts can still be useful.

I dislike the term "imaginary number" — information technology was considered an insult, a slur, designed to hurt i's feelings. The number i is just as normal as other numbers, just the name "imaginary" stuck and so we'll use it.

Visual Understanding of Negative and Complex Numbers

Equally we saw terminal time, the equation $x^2 = 9$ really ways:

$\displaystyle{1 \cdot x^2 = 9}$

$\displaystyle{1 \cdot x \cdot x = 9}$

What transformation x, when applied twice, turns 1 to nine?

The two answers are "10 = 3" and "x = -iii": That is, you can "scale by" 3 or "scale past three and flip" (flipping or taking the opposite is ane estimation of multiplying by a negative).

At present let's think virtually $10^2 = -1$, which is really

$\displaystyle{1 \cdot x \cdot x = -1}$

What transformation x, when applied twice, turns 1 into -ane? Hrm.

Nosotros can't multiply by a positive twice, because the result stays positive
We tin't multiply by a negative twice, because the result will flip dorsum to positive on the 2nd multiplication

Simply what about… a rotation! It sounds crazy, but if we imagine 10 existence a "rotation of 90 degrees", then applying x twice volition be a 180 degree rotation, or a flip from 1 to -1!

Imaginary Number Rotation

Yowza! And if we recollect most it more, nosotros could rotate twice in the other direction (clockwise) to turn i into -ane. This is "negative" rotation or a multiplication by -i:

imaginary number Negative Rotation

If we multiply by -i twice, the first multiplication would plough 1 into -i, and the second turns -i into -1. So at that place'due south really ii square roots of -ane: i and -i.

This is pretty absurd. We have some sort of answer, merely what does it mean?

i is a "new imaginary dimension" to mensurate a number
i (or -i) is what numbers "become" when rotated
Multiplying i is a rotation by xc degrees counter-clockwise
Multiplying by -i is a rotation of 90 degrees clockwise
Two rotations in either direction is -1: it brings united states of america dorsum into the "regular" dimensions of positive and negative numbers.

Numbers are 2-dimensional. Yep, it's mind angle, merely like decimals or long division would exist heed-bending to an ancient Roman. (What do yous mean there's a number betwixt 1 and 2?). It'south a strange, new fashion to think near math.

Nosotros asked "How practice we plow one into -1 in two steps?" and found an answer: rotate it 90 degrees. Information technology'south a strange, new way to think virtually math. But it's useful. (By the fashion, this geometric interpretation of complex numbers didn't arrive until decades afterward i was discovered).

Also, proceed in mind that having counter-clockwise be positive is a human convention — information technology easily could have been the other fashion.

Finding Patterns

Let'southward dive into the details a bit. When multiplying negative numbers (like -1), you become a design:

i, -1, 1, -1, one, -1, i, -i

Since -1 doesn't change the size of a number, just the sign, you flip back and forth. For some number "10", you'd get:

10, -10, x, -x, 10, -ten…

This thought is useful. The number "x" can correspond a good or bad hair week. Suppose weeks alternate between good and bad; this is a good week; what will information technology be like in 47 weeks?

$\displaystyle{x \cdot (-1)^{47} = x \cdot -1 = -x}$

And then -10 means a bad hair week. Notice how negative numbers "proceed track of the sign": nosotros can throw $(-1)^{47}$ into a figurer without having to count ("Week ane is good, week ii is bad… calendar week 3 is skilful…"). Things that flip back and forth can be modeled well with negative numbers.

Ok. At present what happens if we proceed multiplying past $i$?

$\displaystyle{1, i, i^2, i^3, i^4, i^5}$

Very funny. Let'due south reduce this a flake:

$1 = 1$ (No questions here)
$i = i$ (Can't do much)
$i^ii = -1$ (That's what i is all about)
$i^3 = (i \cdot i) \cdot i = -1 \cdot i = -i$ (Ah, iii rotations counter-clockwise = 1 rotation clockwise. Neat.)
$i^iv = (i \cdot i) \cdot (i \cdot i) = -1 \cdot -1 = i$ (4 rotations bring u.s. "full circle")
$i^5 = i^4 \cdot i = 1 \cdot i = i$ (Here we go again…)

Represented visually:

imaginary number cycle

We cycle every 4th rotation. This makes sense, right? Whatever kid can tell yous that iv left turns is the same as no turns at all. Now rather than focusing on imaginary numbers ($i$, $i^2$), look at the full general pattern:

X, Y, -X, -Y, X, Y, -X, -Y…

Like negative numbers modeling flipping, imaginary numbers tin can model anything that rotates between two dimensions "X" and "Y". Or anything with a circadian, circular human relationship — accept anything in listen?

'Cos it'd be a sin if you didn't. In that location'll de Moivre be more in future manufactures. [Editor'south note: Kalid is in electroshock therapy to treat his pun addiction.]

Understanding Circuitous Numbers

There's another detail to cover: tin a number be both "existent" and "imaginary"?

Y'all bet. Who says nosotros have to rotate the entire 90 degrees? If we continue 1 foot in the "real" dimension and another in the imaginary ane, it looks like this:

imaginary number i plus i

We're at a 45 degree angle, with equal parts in the real and imaginary (1 + i). It'south like a hotdog with both mustard and ketchup — who says you need to choose?

In fact, we tin can option any combination of real and imaginary numbers and make a triangle. The angle becomes the "angle of rotation". A complex number is the fancy name for numbers with both real and imaginary parts. They're written a + bi, where

a is the real part
b is the imaginary function

imaginary number a plus bi

Non too bad. Merely in that location's ane last question: how "big" is a circuitous number? We tin can't mensurate the real office or imaginary parts in isolation, considering that would miss the big movie.

Let's step back. The size of a negative number is not whether you can count it — it'due south the distance from zero. In the case of negatives this is:

$\displaystyle{\text{Size of } \ -x = \sqrt{(-x)^2} = |x|}$

Which is another mode to find the absolute value. But for complex numbers, how practice we measure out two components at xc degree angles?

It's a bird… it'southward a plane… it's Pythagoras!

Geez, his theorem shows up everywhere, fifty-fifty in numbers invented 2000 years after his time. Yes, we are making a triangle of sorts, and the hypotenuse is the distance from aught:

$\displaystyle{\text{Size of } \ a + bi = \sqrt{a^2 + b^2}}$

Neat. While measuring the size isn't as easy every bit "dropping the negative sign", complex numbers do have their uses. Let's accept a look.

A Real Example: Rotations

Nosotros're not going to wait until college physics to utilise imaginary numbers. Let's try them out today. At that place's much more to say virtually complex multiplication, but keep this in mind:

Multiplying by a circuitous number rotates by its angle

Let's take a expect. Suppose I'yard on a gunkhole, with a heading of iii units East for every 4 units North. I desire to alter my heading 45 degrees counter-clockwise. What's the new heading?

imaginary number example

Some hotshot volition say "That's simple! But take the sine, cosine, gobbledegook by the tangent… fluxsom the foobar… and…". Crevice . Sorry, did I intermission your calculator? Care to answer that question again?

Let'south endeavour a simpler arroyo: we're on a heading of iii + 4i (whatever that angle is; we don't really intendance), and want to rotate past 45 degrees. Well, 45 degrees is 1 + i (perfect diagonal), and then nosotros tin multiply by that amount!

imaginary number rotation example

Here's the idea:

Original heading: three units East, 4 units Due north = 3 + 4i
Rotate counter-clockwise by 45 degrees = multiply by 1 + i. (Hither's why multiplication, not addition, performs the rotation.)

If we multiply them together we get:

$\begin{aligned} (3 + 4i) \cdot (1 + i) &= 3 + 3i + 4i + 4i^2 \\ &= 3 + 7i \hspace{8mm} + 4(-1) \\ &= -1 + 7i \end{aligned}$

So our new orientation is 1 unit Westward (-1 East), and 7 units Northward, which you lot could draw out and follow.

Only yowza! We constitute that out in 10 seconds, without touching sine or cosine. At that place were no vectors, matrices, or keeping track what quadrant we are in. Information technology was just arithmetic with a touch of algebra to cross-multiply. Imaginary numbers have the rotation rules baked in: it just works.

Even meliorate, the result is useful. We have a heading (-1, 7) instead of an angle (atan(7/-one) = 98.13, keeping in mind we're in quadrant ii). How, exactly, were you planning on drawing and following that angle? With the protractor you keep around?

No, you'd convert information technology into cosine and sine (-.14 and .99), find a reasonable ratio between them (well-nigh 1 to 7), and sketch out the triangle. Circuitous numbers vanquish you to information technology, instantly, accurately, and without a estimator.

If you lot're similar me, you'll find this use mind-blowing. And if y'all don't, well, I'm afraid math doesn't toot your horn. Sorry.

Trigonometry is swell, only complex numbers can brand ugly calculations simple (like calculating cosine(a+b) ). This is but a preview; later manufactures will requite yous the full repast.

Bated: Some people remember "Hey, it's non useful to have Northward/East headings instead of a degree angle to follow!"

Really? Ok, expect at your right paw. What's the angle from the bottom of your pinky to the top of your alphabetize finger? Skilful luck figuring that out on your own.

With a heading, y'all can at least say "Oh, information technology's 10 inches across and Y inches up" and accept some take a chance of working with that begetting.

Complex Numbers Aren't

That was a cyclone tour of my basic insights. Accept a look at the kickoff nautical chart — information technology should make sense at present.

There'southward and then much more to these beautiful, zany numbers, but my brain is tired. My goals were simple:

Convince you that complex numbers were considered "crazy" but tin can be useful (but like negative numbers were)
Show how complex numbers tin brand certain bug easier, like rotations

If I seem hot and bothered about this topic, in that location'south a reason. Imaginary numbers take been a bee in my bonnet for years — the lack of an intuitive insight frustrated me.

Now that I've finally had insights, I'thousand bursting to share them. But it frustrates me that yous're reading this on the web log of a wild-eyed lunatic, and not in a classroom. We suffocate our questions and "chug through" — because we don't search for and share clean, intuitive insights. Egad.

Simply better to low-cal a candle than expletive the darkness: here'due south my thoughts, and one of you lot will smoothen a spotlight. Thinking nosotros've "figured out" a topic like numbers is what keeps the states in Roman Numeral land.

There's much more than complex numbers: check out the details of circuitous arithmetic. Happy math.

Epilogue: But they're still foreign!

I know, they're notwithstanding strange to me too. I try to put myself in the heed of the kickoff person to discover goose egg.

Zero is such a weird idea, having "something" correspond "nothing", and it eluded the Romans. Complex numbers are like — information technology'southward a new way of thinking. Just both zero and circuitous numbers make math much easier. If we never adopted strange, new number systems, we'd still be counting on our fingers.

I repeat this analogy because information technology'south so easy to start thinking that complex numbers aren't "normal". Permit's keep our mind open: in the future they'll chuckle that complex numbers were one time distrusted, even until the 2000'south.

If you want more than nitty-gritty, check out wikipedia, the Dr. Math give-and-take, or another argument on why imaginary numbers exist.