Ask Quille The Pony anything.

[Quill The Pony. 175]

AMA pretty much.

[Gekoncze 327]

1) Which school subject do you like most?

2) What is the square root of -1?

3) Why did you choose your avatar?

^.^

[SasQ 1,356]

I let myself answer the 2nd question above:

 

Square root of -1 is a new type of number, called "imaginary number", which cannot be reduced to any other number you might be familiar with. Therefore the only thing you can do with it is leave it the way it is, and use the symbol root(-1) in whole as a name of that new number, or replace it with some shorter symbol (usually the small letter i). You probably feel upset that you cannot "simplify" it any further, but you don't have to feel that way, because this is how we discovered (and named) all the other numbers. Let me explain it further...

 

When playing with numbers, from time to time you will encounter an operation you cannot perform, or a number you cannot reduce any further. For example your first encounter of this kind might be an equation like this:

   3 - 5 = ?

What number does this reduce to? You cannot subtract more than you have, right? If you have 3 sheep, and you take 3, you have 0 sheep, but this equation requires you to still take 2 sheep you don't have. How could it be even possible?

But imagine you owe someone 5 sheep, having only 3. If you give him your 3 sheep to pay your debt, you still have 2 sheep to pay, so you're in hole. Your creditor still expects 2 sheep from you, and whenever you get a sheep, you will have to give it to him immediately. So it is as you had "less than you have".

In arithmetics, you cannot subtract more than you have, but you can break the 5 into 3 and 2, and reduce it this way:

  3 - 5  =  3 - (3+2)  =  3 - 3 - 2  =  0 - 2

since you know that 3 - 3 = 0. And you're left with 0 - 2, which is still "impossible", but here comes the trick:

You can take this whole symbol 0 - 2 and assume this is a name of a new number, which is somehow less than zero by 2. To further simplify your notation, you can drop the leading 0, since it will always be there for "less than zero" numbers, right? And this way you'll be left with just -2 as a name for such a "number demanding 2" (or "still hungry of 2").

And that's how "negative numbers" were invented. The name "negative" is from Latin "negare", which means "to deny", in opposition to "positive", which is also from Latin "positio" and it means "to put", "to place", "to take place". That's because positive numbers are considered "real" and "existing" (those which actually take some space), and "negative" numbers are considered non-existent, and lots of people denied its reality as long as up to 1700's. (Some of them do to this day )

You can be doubtful about the existence of such a "demanding" number, but this shouldn't stop you from using it in equations to convince yourself that it can be helpful. It can allow you to solve problems you couldn't solve earlier. For example, you can calculate debts and loans, and this little minus sign will automatically keep track of your balance for you, whether you're in the hole or in the clear. This patch is an upgrade to your number system  (BTW "minus" is also from Latin and it means "less". So 0 - 2 could be read "0, less by 2").

 

But soon you will encounter another kind of "irreducible" problem:

When you have 6 loafs of bread and 3 hungry men, you can distribute the loafs evenly among them:

  6/3 = 2

But suppose you have only 1 loaf of bread and 2 hungry men, and you are required to serve them equally:

  1/2 = ?

When you give your only loaf to one of these guys, you'll run out of bread but you still have one man hungry! How can you distribute one loaf of bread evenly between 2 men? This is impossible!

Unless... you split the loaf into 2 equal parts, and give half a loaf to each men, and this way everyone is happy

In arithmetics terms, you cannot divide 1 into 2, because 2 is bigger than 1. But you can assume that this whole symbol 1/2 is a name for some new type of number, which represents half, or a second part of a whole (that is, you tear the whole into 2 equal parts, and take the second one of them). This number, a fraction, is also useful, because it can help you solve new problems, such as distributing bread equally among men.

 

With time, you will encounter another problem:

When you have 9 square tiles, you can make a 3x3 square out of them. When you have 16 tiles, you can make a 4x4 square. You can easily answer how many tiles do you need to build a square with a given number of tiles as its side: just repeat this side as many times, as the number of tiles it consists of. This is called "squaring", or raising to the second power:

  3² = 9 ,  4² = 16  etc.

But suppose you want to know the opposite: You have a bunch of square tiles and you want to make a square out of them. How many tiles do you need to put as its side, or base, or "root" (as a root of a tree)? Finding this answer is called "taking the square root", and it uses a fancy symbol similar to a small letter r (from "root", or "radix" in Latin) before and above the number you're taking the square root of. From technical reasons, I will denote it as root(number). So you have:

  r(9)   = 3

  r(16) = 4

But what if you have 7 tiles? Or 5? Or just 2? There's no way to arrange them into a square, so you cannot tell either what would its "root" be. But if you know a bit of Geometry and the Pythagorean Theorem, you can observe that in every square you can draw a diagonal (a straight line going across the square and connecting its most opposite corners). And you can always use Pythagorean Formula to calculate its length: a square built on this diagonal is a sum of two squares built on the sides of the original square. So you just need to build squares on both sides of your original square, add them up (or double, since those will be squares of the same size), and take the square root of the result to find the length of the diagonal.

Suppose you have a 1x1 square, and you want to know the length of its diagonal. So you calculate: 1² + 1² = 1 + 1 = 2, and here comes the trouble: Somehow you now need to take a square root of 2! But this is impossible! When you have just 2 square tiles, you cannot build a perfect square out of them, no matter how much you split them into smaller parts! (In fact, you would need to split them infinitely many times to do that.) So how can we find the answer to this problem?:

  root(2) = ?

We cannot. 2 is not a perfect square, so we cannot take its square root to reduce it any further.

But wait: The diagonal of the square is there! It is real! It exists! It has a definite length! So it is some number, right? We just don't know how to write its name, because it's not any of the numbers we met so far. It's a new type of number, so called "irrational number," which literally means "unthinkable".

(See she pattern yet? Every time we encountered a new type of number we couldn't express in terms of the numbers we knew, we called it some offensive name describing our inability to grasp it.)

Being taught by our experience, we can again use the whole symbol root(2) as a name for the number just discovered, and remember that this symbol, root(2), is a name for the number representing the length of the diagonal in the unit square. Or a side ("root") of a square made of 2 unit square tiles (if you somehow were able to "melt" them and mold into a form of another square, its side would measure root(2) in length). So your number system is again upgraded: you can express numbers you couldn't previously, and this allows you to solve problems you couldn't solve in the past.

 

OK, so what about that intriguing square root of -1? (or any other negative number)

Well, if you think of it, making a square is just multiplying a number with itself: to make a 2x2 square, you need 4 (=2x2) tiles. So what number do you need to multiply with itself to get -1?

Two positive ones won't do, because positive times positive is still positive, and we need a negative one.

So maybe two negative ones? No, this won't work either, because negative times negative is also positive.

And you cannot mix positive and negative (which could give you negative), because both numbers multiplied have to be the same.

Positives are out, negatives are out... What about 0? 0*0 is still 0, so we cannot get -1 from any numbers we know! We tried positives, negatives, and zero, but none of them worked. So we give up? The square root of -1 seems to be irreducible.

Unless... we do the same trick we did so far: We pretend that there is some number which is neither positive, nor negative, nor zero, and we use the whole symbol root(-1) as its name? We can also call it i in short, from "imaginary". It is supposed to mean a number which is "half-multiplication by -1" (because that's what square roots really are: they allow you to split one multiplication operation into two equal "half-multiplications"), so that you need to multiply twice by it to finally get -1. You can also use this number to construct new numbers, e.g. root(-9)  =  3 root(-1)  =  3, or even combine them with the numbers you know: 3 + root(-4)  =  3 + 2 root(-1)  =  3 + 2 i. So it seems that this number can be useful.

 

OK, but leaving its usefulness alone, you'd probably want to know what does this new number mean? What it represents? We can understand negative numbers as debts, or temperatures below zero degrees centigrade. We can understand fractions as parts of a whole. We can even picture irrational numbers, such as root(2), as the lengths of some geometrical lines. But how can we picture root(-1) in our minds? It is neither positive, nor negative, nor zero, so we cannot put it anywhere in our number line. It doesn't seem to exist at all! This is absurd!

Well, there's so many people out there which still think that way: that imaginary numbers are unreal, or "just mathematical notation which can simplify some calculations". But if you understand it the proper way, imaginary numbers are as real (or as unreal) as the "real" numbers we know. Here's the secret:

 

Imaginary numbers are neither positive, nor negative, nor zero, because they don't lie on the number line at all! They lie beside it! In the second dimension. To understand it, let's make some exercise:

Stand up and make a step forward. This is +1 (one step forward from 0). Subtraction "undoes" the addition. If I told you now to "undo" this step, which direction would you go? Yes, you will make one step back, and you're back where you started, that is, at 0. Can you repeat this operation once again? I hope you won't have any trouble with it: you can make another step back, and now you're at -1. And that's the meaning of the number -1: it moves backwards, in the opposite direction to your default "forward".

Now make a step forward to return to 0.

So far you have moved along just one line (one dimension), which is equivalent to the line where all real numbers lie upon; positive, negative and zero. But we need a number which is neither positive, nor negative, nor zero, remember? So we need to go in some totally new direction we haven't traveled yet. We need to step out of our 1D line world and into the 2D plane world. If I told you to make a step in a totally new direction, which is the most different from all directions you've traveled so far, where would you go?

You've made a step sideways, perpendicularly to your "forward-backward" line, haven't you? ;>

And that's exactly where the root(-1) lies! So it isn't unreal at all! If someone tells you imaginary numbers don't exist, please tell him that if this is true, then north doesn't exist either, only south and west

 

You might notice that you could actually go in two different directions: You could go left, or you could go right. Both of them are correct "square roots of 2", in the same way you could get 4 from multiplying 2x2 or (-2)x(-2). Every number has always two square roots which lie opposite to each other. But we need some way to distinguish them. So we can reuse our "minus" sign for that, by agreeing upon that whenever we go left, this is the "positive imaginary" direction, and whenever we go right, this is the "negative imaginary" direction. This decision is arbitrary: we could have chosen it the other way around, too. But whatever we choose, we need to stick to it for any future calculations. So mathematicians agreed upon that "left" is positive and "right" is negative imaginary.

 

So now you can understand all the mechanics of root(-1) and why does it lie precisely one step to the left (or right):

If you start at 0, facing in the "forward" (positive real) direction, and you make a step forward, so you land at +1. But now you want to transform yourself into -1, but in two separate "steps", what do you need to do? Well, you can rotate yourself 90 degrees to the left, around 0, so that you'e at root(-1). If you repeat this action once again, you will rotate yourself another 90 degrees, and now you're at -1. See? You repeated multiplication of your unit (1) by root(-1) twice to get to -1, so you have squared the root(-1) to get -1. So our new "sideways number" really is the square root of -1, since you really have had to multiply your unit twice by that number to get -1.

 

Does that answer your question? 

 

Edit: As a bonus, you can check out how I tried to solve another similar problem: of division by zero, in the same lines as described here: by supposing that there is a number with such properties (which I called "creative number", since it can create something out of nothing, when multiplied back with 0), and its name is the whole irreducible part of the equation, 1/0. Neat, eh?

Edited February 19, 2014 by SasQ