Happiness

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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7 Comments on “Happiness”

  1. Deborah Says:

    I am going to be so cheeky now. It’s just that I love the idea of this blog and you are one of the minute number of people (perhaps you could do a post on that number) who can actually write, with proper spelling and everything. (I’m impressed – seriously). Also, I love all this quirky numbers stuff. Health warning, cheeky remarks about to start: I think you should make your posts shorter and easier to digest, and – 64 million dollar question (what’s that all about – more post material for you?) – who are you?? Where’s your ‘About’ page? Truly, I really, really like this blog. Don’t stop, will you?x

    • mostlymath Says:

      Thank you very much for the comment! A healthy dose of praise and criticism is always refreshing. With regards to the blog, I aim (not very successfully, admittedly) for around 500 words. There’s so much potential ground to cover in each topic I often find myself getting a bit carried away, but I will keep your suggestion in mind. I try to limit the tricky maths in order to keep it light and readable. I have a rather brief ‘About’ page on the blog, but it will need some padding out. I’ll get round to that soon, still quite new to all of this. I’m very pleased you enjoyed my writing; I have no plans to quit any time soon. Thanks again for the remarks.


    • I totally agree with you Deborah. I have been writing for quite w while in http://math4allages.wordpress.com but reading this guy’s post really made me realized how boring of a writer I am.

      Anyway, I am glad, that there are people out there who are really good at math and who can write well.

      Keep up the good work James. And continue writing. The world needs math writers like you. 🙂

      • mostlymath Says:

        Thanks again for the support! It’s very uplifting to get such positive comments. I think you do yourself a disservice to call yourself boring though, you have so many interesting and helpful posts.


  2. […] Happiness « A mathematician’s guide to… […]


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