## Archive for the ‘Ideas in Maths’ category

### The Imaginary

July 12, 2010

Our imagination is a wonderful thing. It allows us to conjure up wondrous, impossible events and entertain ourselves with an alternate, exciting reality. It allows us to dream. We can be whatever, whoever, we want to be in our own little world and all because of our amazing ability to create the unbelievable. This all sounds a bit poetic for maths though, doesn’t it? And yet, even in the world of absolute truths we find a bit of the imaginary.

Well, imaginary numbers to be precise. Imaginary numbers, as you may have guessed, aren’t real. This may be a step too far for some of you. I’ll be trying to convince you about invisible numbers next, won’t I? Or inventing the God number and explaining how it controls the rest of the numbers (I’d go with 2, incidentally. It’s always had an air of superiority about it). But I kid you not with this one, the imaginary numbers definitely exist (though sentences like that make my head hurt).

I do wish they had been named differently, though. There is nothing quite so confusing as explaining that a number is imaginary and exists but is not real. People begin to look at you funny. Saying that, people tend to accept negative numbers fairly easily. Obviously there is the analogy with debt, but have you ever seen -2 cars, or walked past a group of -5 people? Still, I’ll let you make your mind up about how ridiculous imaginary numbers are. Time for a bit of an explanation.

A very different kind of explanation

Now, you may have been told that you cannot square root a negative number, even if you try really hard. This is because no number multiplied by itself is negative, so you can’t take the root of a negative number. All sounds pretty straightforward so far. You can even try it on your calculator if you like. Done? It didn’t like that, did it (some calculators are capable of this, but they’re always a little too smug about it for my liking)? If your calculator can’t find an answer then what hope do you have?

Ok, let’s not panic. We can do this. We just need to think about the problem a bit differently, use a little imagination. Imagination! That’s it! If there’s no real answer to square rooting a negative number, why don’t we create an imaginary one? All we have to do is define $\sqrt{-1}$ as $i$ and we’re done, imaginary numbers to the rescue. Simple, eh? All of our problems solved with one little trick. Not seeing it?

Consider $\sqrt{-64}$

$\sqrt{-64}=\sqrt{64*-1}$

$\sqrt{64*-1}=\sqrt{64}*\sqrt{-1}$

Therefore $\sqrt{-64}=8i$

Gotta love math humour

How easy was that? All it took was the addition of one letter and we have our answer. What we see is that the root of negative numbers produces an imaginary number. This is, of course, a great lesson for life: if you can’t work out the answer, just make it up.

What happens if we want to add some real numbers to our imaginary mix? Here things get a little complex. Literally. Complex numbers are numbers consisting of a real and imaginary part (in the same way Christmas has Santa and awkward family reunions) , such as $4+8i$. However, it’s difficult to visualise that number, isn’t it? It’s like trying to imagine what love looks like, or a social life.

This is where Argand diagrams come in (in visualising complex numbers, not a social life). An Argand diagram is a graph with one real axis and one imaginary axis. Our complex number then becomes a point on the graph, making things much easier.

Not so complex now

There you have it, the wonderfully weird world of imaginary numbers. Throw out your calculator, play with the imaginary and question your very perception of reality. What else are you going to do on a Monday?

### Happiness

July 8, 2010

Happiness is a state of mind or feeling characterized by contentment, love, satisfaction, pleasure, or joy. Whether that be from hanging out with friends, reading a humble maths blog or finally, at two o’clock in the morning after many, many attempts (and several deep periods of self-reflection), successfully completing a Rubik’s cube for the first time, happiness is the most sought after of human emotions. Its elusiveness (that Rubik’s cube took me to many dark places) makes it all the more satisfying.

But is happiness limited to humans? Many animals seem to experience happiness. The joy my dog takes in rolling in whatever decaying mess she can find is only matched by her guilt and confusion when I tear out of the house shouting and gesticulating wildly. Cats, I believe, purr contentedly when fed, before returning to their quiet, judgemental state.

He does not approve of your life choices

Can a number be happy? OK, I know. Obviously a number cannot experience happiness (though I always thought 3 looked a bit smug). To say that a number is happy is similar to remarking that your front door has a particularly cheery disposition today, or that your fridge has been looking a bit down lately. People will stop making polite conversation and instead nervously edge away.

I suppose, then, if we are to call a number happy we shall need a new definition of happiness. Try taking the sum of the square of the digits of a number, iterate this process and see what happens. If this process eventually leads to 1 then we have ourselves a happy number. If not then we’ve unfortunately found ourselves an unhappy number. I’ll show you what I mean.

Take 7.

$7^2=49$
$4^2+9^2=97$
$9^2+7^2=130$
$1^2+3^2=10$
$1^2+0^2=1$

And there we have a happy number!

Unhappy numbers eventually end up in the cycle

$4,16,37,58,89,145,42,20,4,\ldots$

Perhaps it’s unsurprising that 7 is the second happy number (after 1). All those years of being regarded as exceptionally lucky have obviously done a lot for its self-esteem. While I don’t quite comprehend why 7 is deemed so lucky (or why rabbits’ feet are considered lucky, I’m fairly confident the rabbit felt it was luckier attached to its leg) but it’s obviously quite pleased with the state of affairs. Interestingly 13 is also happy, not a twinge of guilt for all the bad luck it has bestowed on people.

Damn you 13!

Which would almost lead you to believe that 13 is a bit evil. That would be silly though, wouldn’t it? We’ve just about got our heads around it being happy, evil is a step too far. And 13 is most definitely not an evil number. It’s odious.

You see, a number is evil if it has an even number of 1s in its binary expansion. That means that if, when you write the number in base 2, there is an even number of 1s in the number you have yourself a very evil number indeed. If there are an odd number of 1s, you merely have an odious number on your hands.

For instance, 13 is, as I said before, an odious number.

$13=1*2^3+1*2^2+0*2^1+1*2^0=(1101)_{base2}$

Thereby making it odious.

Why you would call a group of numbers odious is beyond me. You would think that if one group of numbers is evil its opposing group would be good, or angelic. Obviously the person naming them had experienced many traumas at the hands of rogue integers and decided they should all be detestable in one way or another. Rather depressingly, there are far more unhappy numbers than happy ones and they’re evenly split between being evil or odious. It really makes you worry about the sad state of affairs our numbers are in, doesn’t it?

### Irrationality

July 4, 2010

﻿﻿
When I talk of irrationality, I mean, of course, irrational numbers. Now these numbers are not irrational in the same way your girlfriend is, these numbers go on and on and on forever and ever and ever. Ok, maybe there are some similarities, but unlike your girlfriend these numbers never repeat themselves.

Perhaps I should clarify (the extended girlfriend comparison made things a little complicated). An irrational number is one that cannot be expressed as the ratio of two integers, a fraction if you will. So while numbers like a third (0.333…) and a ninety-ninth (0.0101…) extend infinitely, they can be written as 1/3 and 1/99 respectively.

Pi (π = 3.1415…) is an example of an irrational number. “Hang on”, I hear you cry. “You’ve only quoted 5 numbers there, how do we know there’s no repeating sequence?” Well, you can either trust me or check for yourself. No, go on, I’ll wait. Nothing? Still, this doesn’t really prove anything, does it? It could repeat after the millionth digit, or the billionth. Proving it never repeats by checking every digit is like trying to prove you can hold your breath for an hour. It’s a remarkable effort, but you’ll definitely die before you reach the end.

No, George, that wasn't a challenge.

To add to the confusion, many of you may have used the approximation pi ≈ 22/7 (333/106 if you were really clever), which seems to show pi as the ratio of two integers. However, this doesn’t quite cut it after a certain number of decimal places. There is a proof of the irrationality of pi, but it’s a bit complicated for this blog.

A much simpler proof involves √2.
Start by assuming √2 is rational.
Then √2 = p/q where p and q are integers and p/q is in its simplest form.
Squaring both sides gives 2 = p²/q²
Multiply both sides by q² to give 2q² = p²q²/q² then cancel
So 2q² = p² meaning p² is even
Therefore p is even (it cannot be odd, as this would make  p² odd) and can be written as p = 2m
Then 2q² = (2m)² = 4m²
This simplifies to  q² = 2m²
By the same argument q is even and can be written as q = 2n
Going back to the start then, √2 = 2m/2n = m/n
But we specified at the start that p/q was the simplest form, so we’ve found a contradiction. Therefore √2 is irrational.

Pretty neat, huh? This is a beautiful example of proof by contradiction, a method whereby you assume the opposite to be true and then show that it is logically inconsistent. It’s like assuming your friend isn’t stealing from you, finding all of your bank statements in his house and concluding that you are an extremely poor judge of character.