Posted tagged ‘odious’

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?

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