Imagine you're a complex number, which is just a type of arrow.

Exponentiation is to do with growth.

Growth at a speed which is a multiple of how big you are already.

i is the multiplication which turns you through ninety degrees.

If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude.

Your head moves at the speed which is your length. So pi is how long it takes you to turn through a half circle.

So if you grow at right angles to yourself for time pi, you are pointing the opposite way.

That is the meaning of the famous identity e^(i*pi) = -1.

If we know what sin and cos are, we can see equally easily that e^it = cos t + i*sin t, which is known as Euler's formula. Sin and cos are just the coordinates of a point on a circle that has moved a distance t anticlockwise round a circle of radius 1.

In more detail:

Complex numbers are arrows. They have direction and length.

When you exponentiate something, you are growing it by a multiple of itself.

e^t is the number which 1 becomes after t seconds of smooth exponential growth.

e^2t is the number which 1 becomes after t seconds of growing at twice that rate.

e^(-t) is t seconds of growth in the opposite direction (a big thing gets smaller at a rate proportional to itself). (Otherwise known as exponential decay)

But multiplication by a complex number can also involve turning.

Consider i*9 -> 9i

The complex number 9 is an arrow pointing east length 9

The complex number i is an arrow pointing north length 1

Multiply them together and you get an arrow length 9 pointing north.

Multiplication by i is the same as rotation through 90 degrees anticlockwise.

What does it mean to grow at right angles to yourself?

It means to turn. And because the direction you're growing in is always at right angles to the direction you're pointing in, you never get longer or shorter.

So e^(it) is what happens after t seconds of turning anticlockwise.

How fast are we turning? Well, our length is always 1, so the speed of turning is always 1. After t seconds, the tip of the turning arrow will have moved along an arc of length t.

Once it's moved along an arc of length pi, the arrow is pointing in the opposite direction to where it started.

An arrow length 1 pointing west is also known as -1.

So e^(i pi) = -1.

In fact e^(it) is where the turning arrow is after t seconds. It will have traced out an arc of length t.

So e^(i pi/2) is an arrow pointing north. e^(i pi/2) = i

More generally, if you resolve the turning arrow into a north arrow and an east arrow, the east component will be one side of a triangle with an angle of t radians and the north component will be the other.

e^(i t) = cos t + i sin t.

## Friday, October 30, 2009

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Nice explanation, but I'm still not sure I get the "grow at right angles to yourself" and the triangle part at the end, could you make some diagrams? :)

ReplyDeleteBy the way, your way of explaining reminds me a lot of http://betterexplained.com.

Take an arrow, length 1, pointing east, and draw another arrow length 0.0001, pointing north, starting at the tip of the first one.

ReplyDeleteDraw another arrow to make up the third side of the triangle, then you've got an arrow that's very nearly length 1 (~1.000000005) but turned through a small angle. (~0.0001 radians)

If you repeat this process ten thousand times, always making the tiny new arrow at right angles to the arrow you've constructed, then it will turn a whole radian whilst still staying about length 1.

In the limit of lots and lots of very small steps, then it will turn at speed 1, but not lengthen at all.

Your explanation is one of the best I have seen. I will try to give a similar one in my efforts to better understand it.

ReplyDeletemultiplying by -1 is the transformation you do once to turn a number negative, and thus is a flip along one dimension.

multiplying by i is the transformation you do twice to turn a number negative, and thus is a rotation by 90 degrees (into a second dimension.)

unless we want a new way to get from 1 to negative 1 (ideas?) we don't need another kind of number.

e^1t is the number 1 turns into after growing at a rate equal to its size per second continuously for t seconds.

e^it is the number 1 turns into after moving perpendicular to itself (not growing in size) at a speed equal to its size per second continuously, which has to always be 1 length per second. after about 3.14 ish seconds, we find that 1 has turned into -1, and thus e^_i*pi = -1

not saying your explaination is inadequate in any way, just trying to figure out eulers formula and helping others.

actually replace for e^1t "growing at a rate" with "moving along the regular numbers"

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