Paul P. Mealing

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Saturday, 14 April 2012

i, the magic number that transformed mathematics and physics

You might wonder why I bother to beleaguer people with such esoteric topics like complex algebra and Schrodinger’s equation (May 2011, refer link below). The reason is that I’ve struggled with these mathematical milestones myself, but, having found some limited understanding, I attempt to pass on my revelations.

Firstly, I contend that calling i an imaginary number is a misnomer; it’s really an imaginary dimension. And if it was called such it would dispel much of the confusion that surrounds it. We define i as:

i = √-1

But it’s more intuitive to give the inverse relationship:

i2 = -1

Because, when we square an imaginary number, we transfer it from the imaginary plane to the Real plane. Graphically, i rotates a complex number by 900 in the anti-clockwise direction on the complex plane (or Argand diagram). Or, to be more precise, multiplying any complex number (which has both an imaginary and a Real component) by i will rotate its entire graphical representation through 900. In fact, complex algebra is a lot easier to comprehend when it is demonstrated graphically via an Argand diagram. An Argand diagram is similar to a Cartesian diagram only the x axis represents the Real numbers and the y axis is replaced by the i axis, hence representing the i dimension, not the number i.

It’s not unusual to have mathematical dimensions that are not intuitively perceived. Any dimension above 3 is impossible for us to visualise. And we even have fractional dimensions that are called fractals (Davies, The Cosmic Blueprint, 1987). So an imaginary dimension is not such a leap of imagination (excuse the pun) in this context. Whereas calling i an imaginary number is nonsensical since it quantifies nothing.

In an equation, i appears to be a number, and to all intents and purposes is treated like one, but it’s more appropriate to treat it as an operator. It converts numbers from Real to imaginary and back to Real again.

In quantum mechanics, Schrodinger’s wave function is a differential complex equation, which of itself tells us nothing about the particle it’s describing in the physical world. It’s only by squaring the modulus of the wave function (actually multiplying it by its conjugate to be technically correct) that we get a Real number, which gives a probability of finding the particle in the physical world.

Without complex algebra (therefore i ) we would not have a mathematical representation of quantum mechanics at all, which is a sobering thought. We have long passed the point in our epistemology of the physical universe whereby our comprehension is limited by our mathematical abilities and knowledge.

There are 2 ways to represent a complex number, and we need to thank Leonhard Euler for pointing this out. In 1748 he discovered the mathematical relationship that bears his name, and it has arguably become the most famous equation in mathematics.

Exponential and trigonometric functions can be expressed as infinite power series. In fact, the exponential function is defined by the power series:

ex = 1 + x + x2/2! + x3/3! + x4/4! + ….

Where n! (called n factorial) is defined as: n! = n x (n-1) x (n-2) x …. 2 x 1

But the common trig functions, sin x and cos x, can also be expressed as infinite power series (Taylor’s theorem):

sin x = x – x3/3! + x5/5! – x7/7! + ….

cos x = 1 – x2/2! + x4/4! – x6/6! + ….

Euler’s simple manipulation of these series by invoking i was a stroke of genius.

eix = I +ix – x2/2! – ix3/3! + x4/4! + ix5/5! – x6/6! – ix7/7! + …

i sin x = ix – ix3/3! + ix5/5! – ix7/7! + …

I’ll let the reader demonstrate for themselves that if they add the power series for cos x and isin x they’ll get the power series for eix .

Therefore:   eix = cos x + i sin x

But there is more: x in this equation is obviously an angle, and if you make x = π, which is the same as 1800, you get:

sin 1800 = sin 0 = 0

cos 1800 = - cos 0 = -1

Therefore:  eiπ = -1

This is more commonly expressed thus:

eiπ  + 1 = 0

And is known as Euler’s identity. Richard Feynman, who discovered it for himself just before his 15th birthday, called it “The most remarkable formula in math”.

It brings together the 2 most fundamental integers, 1 and 0 (the only digits you need for binary arithmetic), the 2 most commonly known transcendental numbers, e and π, and the operator i.

What I find remarkable is that by adding 2 infinite power series we get one of the simplest and most profound relationships in mathematics.

But Euler’s equation (Euler’s identity is a special case): eiθ = cos θ + i sin θ
gives us 2 ways of expressing a complex number, one in polar co-ordinates and one in Cartesian co-ordinates.

We use z by convention to express a complex number, as opposed to x or y.

So  z = x + iy (Cartesian co-ordinates)

And z = reiθ  (polar co-ordinates)

Where r is called the modulus (radius) and θ is the argument (angle).

If one looks at an Argand diagram, one can see from Pythagoras’s theorem that:

r2 = x2 + y2

But the same can be derived by multiplying the complex number by its conjugate, x – iy

So  (x + iy)(x – iy) = x2 + y2 = r2 

(I’ll let the reader expand the equation for themselves to demonstrate the result)

But also from the Argand diagram, using basic trigonometry, we can see:

x = r cos θ  and y = r sin θ (from cos θ = x/r and sin θ = y/r)

So  x + iy  becomes  r cos θ + i r sin θ

There is an advantage in using the polar co-ordinate version of complex numbers when it comes to multiplication, because you multiply the moduli and add the arguments.

So, if:    z1 = r1eiθ1   and   z2 = r2eiθ2

Then:   z1 x z2 = r1eiθ1 x r2eiθ2 = r1r2ei(θ1 + θ2)

And, obviously, you can do this graphically on an Argand diagram (complex plane), by multiplying the moduli (radii) and adding the arguments (angles).

Addendum 1: Given its role in quantum mechanics, I think i should be called the 'invisible dimension'.

Addendum 2: I've been re-reading Paul J. Nahin's very comprehensive book on this subject, An Imaginary Tale: The Story of √-1, and he reminds me of something pretty basic, even obvious once you've seen it.

tan θ = sin θ/cos θ or y/x (refer the Argand Diagram)

So θ = tan-1(y/x) where this represents the inverse function of tan (you can calculate the angle from the ratio of y over x, or the imaginary component over the Real component).

You can find this function on any scientific calculator usually by pressing an 'inverse' button and then the 'tan' button.

The point is that you can go from Cartesian co-ordinates to polar co-ordinates without using e. According to Nahin, Caspar Wessel discovered this without knowing about Euler's earlier discovery. But Wessel, apparently, was the first to appreciate that you sum angles when multiplying complex numbers and invented the imaginary axis when he realised that multiplying by i rotated everything by 900 anticlockwise.

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