# Weird Arithmetic

Even in the age of the  computer and its old-fashioned relative, the calculator, some quirks of arithmetic are just too good to miss (to ludamus, at any rate).

Ask a friend to name their favourite digit between 1 and 9: they may say, for instance, “6”.

Now give them the number 12345679 (that’s right – with the digit 8 missing), and tell the to multiply it by 54 (which is just 6 times 9). They can do it on a calculator, or if they’re the kind of person who likes cold showers every morning and goes about wearing a hair shirt, they can try it by hand with pencil and paper.

Whatever digit they give you, just ask them to multiply 12345679 by that digit times 9: for instance, if they tell you that their favourite digit is 7, tell them to multiply by 63, which will give them 777777777, and so on.

It may be a pointless mathematical fact, or there may be a reason behind why it works which you might like to investigate. In any case, it’s fun and surprising.

# Richard Wiseman’s Incredible Vanishing Banana Illusion

This is quite astonishing. Without any CGI, or cutting/editing the video, Wiseman makes a banana apparently vanish, while its reflection in a mirror stays visible.

You can view this illusion at New Scientist’s site, and find out more about Richard Wiseman, and his site, here.

Do you have any suspicions as to how the trick is done?

# Stereogram Book

Ludamus has found, in a charity shop, a rather nice little book of stereograms. It was priced at £9.99 new, but the £1.50 I paid for it seemed more representative. There’s a collection of different types, some astonishingly three-dimensional, and others more mundane, and it doesn’t really go into the techniques for producing them, which would be really interesting.

No prizes are offered for seeeing what the hiden stereogram is, but you might like to have the glory of being the first to post a comment!

However, it strikes me that the more ordinary kind of stereo pair photos, at least, could be made with an ordinary camera and a block of wood. More on this teasing topic anon (maybe).

# Relativity is Real in GPS

We may think that relativity is abstract and far removed from everyday life and the “real” world. Time, for instance, is said to run differently depending on our speeds or environments, but is it so tiny as not to be noticeable in everyday life?

Philip Yam, in Scientific American, 2004, page 39, knows differently.

It is relevant, and it is noticeable. Global positioning systems, which we use in cars, planes and ships, depend on measuring the time radio waves take to reach us from a satellite of known position; this enables the receiver, which measures the travel time of the radio signal, to work out the distance from the satellite. Once four satellites have been measured, we can find our position.

Now if the satellite clocks were to run at a slightly wrong speed, say losing 2 nanoseconds a day (that’s 2 billionths of a second per day), then the satellites would appear to be 60cm further away each day, and we would soon have a build-up of error.

The times on the satellite clocks thus have to very accurate indeed,and any small effects (such as those due to relativity) can have a real effect on the navigation abilities of GPS systems.

There are in fact two relativistic effects: the speed of the satellite relative to the ground slows its clock by 7 microseconds per day (that’s 7000 nanoseconds per day); on the other hand, the weaker gravitational field up at the satellite orbit means that clocks run faster by 45 microseconds per day. The net effect is therefore that the clocks on the satellites run faster by (45-7), or 38 microseconds per day. That’s 38000 nanoseconds, or a cumulative error of over 11 kilometres per day! That’s a large error for such a non-real, airy-fairy theory that isn’t supposed to affect real life.

Cheer up. The people who run the systems know about this, and build in corrections so that you won’t try to drive the car into one of the craters on the Moon.

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