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## On Inefficient Numeral Systems

A particular sentence in Luke’s post the other day made me chuckle – in a good way, of course. Re-reading it today produced the same effect, so I’ve decided to pursue it. The sentence is this:

“I think Hindu-Arabic Numerals make it too easy to write down numbers we don’t or can’t comprehend…”

Curse them. They’re so… efficient. Logarithmically so: let n be a positive integer we are going to transcribe. The number of digits required is D = 1 + floor( log10 n ). This is one way to represent the difficulty of writing a number out: let’s call it the representation cost function. That floor function just makes it even easier, so we can say that it’s at worst an O(log n) process. Any good numericist would rejoice at such compactness, and if the aim is to successfully notate abstract quantities that are beyond our experience, e.g. a billion, then it’s just the ticket. But just for today we are going to pretend that we have the opposite problem. We might believe that the ease with which overwhelmingly large numbers can be notated has led to complacency: AIDS has killed over 25 000 000 people since 1981. Doesn’t look like so many when you write it down, does it?

## Hans Bethe Lectures

I can’t recall from where I learned of these; thought it was CV but I couldn’t find the link there. Cornell has made available three lectures on quantum physics given by the 93-year old Bethe to folks at his retirement home.

They’re a wonderful overview of the whole of quantum theory, rather than just the quantum mechanics that undergraduates in Australia encounter in rigorous detail. The lack of mathematics means that one gains no facility with calculation whatsoever from watching them, but the concepts are lucid. Also noteworthy is Salpeter and Schweber’s introduction – the latter’s writing on the development of the quantum theory is particularly good.

## Getting your brain around the Universe

Given that I hope to make a career out of studying it, I feel like I should have a better intuitive understanding of the size of the universe. For example, how do you illustrate the size of the solar system? If I could hold the sun in my hand, could I walk to the other side of the galaxy?

The only way to do this is by a series of comparisons – simply stating that the galaxy is about 1,000,000,000,000,000,000,000 metres across doesn’t help anyone. We just don’t meet numbers that big in everyday life. (I think Hindu-Arabic Numerals make it too easy to write down numbers we don’t or can’t comprehend – but that’s a topic for another post).

So lets begin small … Everest. Suppose you make a model of Mount Everest that is as tall as a two storey house, entirely out of paper mache. Not just the top, either – all the way from sea level. On that scale, a scale model of yourself will stand 1.5 millimetres high. Now comes the hard bit. Have a picture in your mind of the two-storey house and the tiny model of yourself standing beside. Now stand back and imagine both objects growing in size until the you have a life sized model of yourself and an Everest-sized house. Alternatively, you might find it easier to imagine yourself shrinking until you’re 1.5mm high, and then looking up at the house. Try it a few times – its not easy. You might now have idea of the size of Everest. If you can do this after each step and keep track of all the steps, then you’re doing a lot better than I am.

Moving on … impressed by your paper mache skills, you decide to build a model of Earth, once again as high as a two-storey house. In order to create a scale model of Everest, mould a spare bit of glue into a bump half a centimeter high. (For those playing along at home, on this scale: the Earth’s crust is 2cm thick, the atmosphere is a bit under 6cm, and the moon is 2m across and 2 football fields away.)

Next, take out the yellow paint and you’ve got yourself a model Sun. On this scale, the Earth is about 7cm across and 800m away. Make sure you take the time to visualise – a big yellow ball two storeys high, next to a blue and green tennis ball.

Now let’s try for the solar system. The outermost planet is Neptune (forget Pluto – lousy freeloading dwarf.) Suppose Neptune’s orbit could be contained in the average backyard – 20 metres across. On this scale, the sun is 3mm across – about the size of a handwritten ‘o’. Incidentally, the nearest star to the sun is Proxima Centauri, and on this scale it is 90km away.

If you’re still with us, well done. It gets harder from here – the galaxy is really quite big. For our next trick, you’ll need a single human hair. The hair is essentially a long, thin cylinder – take a close look at the end. You will see a tiny circle, at most a tenth of a millimetre across. Now imagine taking a tiny pen and (steady now!), draw the orbits of the solar system on the end of your hair, with Neptune the outermost circle. On this scale, the galaxy is 20km across. Proxima Centauri is 60cm away. I still haven’t got my head around that one.

The Milky Way contains 100 Billion stars – but once again we really don’t have any intuitive grasp of that number. Try imagining cutting a piece of paper the size of a football field into pieces just large enough to print a full stop on. Or the grains in a cubic metre of sand. Or placing human hairs side by side for 7000km – start at Brisbane and visit Sydney, Melbourne and Adelaide on your way to Perth, laying down hairs as you go.

And finally, take a long hard look at this picture. It is a picture of a tiny part of the sky, taken by the Hubble space telescope. The picture shows an area of the sky the size of a couple of grains of sand held at arms length. Or a postage stamp, 20 metres away. Every dot and blob on that picture is a galaxy (unless it has “spikes” – they’re stars). Now, if you can blow up those grains of sand until they’re the size of the picture, a blob on the picture till its 20km across, then a human hair till its the size of a backyard, then an ‘o’ till its 2 stories high, then a tennis ball till its two stories high, then a 0.5cm bump till its 2 stories high, then a 1.5mm model of yourself until its life size, then you might have some idea of why I’m a cosmologist.

[Berian says: I’d started a post on a similar topic that I shouldn’t finish now, as Luke’s writing on the subject is splendid. The only thing missing is some surrealist comedy from back in the day:

Sorry to butt in on your post, Luke! I would have put this in the comments but I wanted to have a link to the video directly.]