# Reflections on a number...

• 06 September, 2007 22:00

…93560524018599910506210816359772643138060546701029356997 10424251057817495310572559349844511269227803449135066375687 47760283162829605532422426957534529028 … etc.

“What the heck is this, Greg?”

Okay, so my editor didn’t quite put it like that but I know that’s what he was thinking. Actually, I’d half expected his call to quiz me about what I’d submitted. It’s not like a number makes it as a newspaper column every day of the week. [Reseller News, August 24]

I hope you saw it. 1.41421356237309 … etc.

It is an irrational number, which in mathematics is one that can’t be expressed as a fraction a/b, where ‘a’ and ‘b’ are whole numbers, and b is not zero. Perhaps the best known irrational number is pi (or π), which as you know, is the ratio of a circle’s circumference to its diameter in Euclidean geometry, approximately 3.14159.

And my number? Well, some of you knew it was the square root of two (√2) and also known as Pythagoras’ constant.

√2 is the number which when multiplied by itself gives two. Physically, if you walk forward one metre, turn right and walk another metre, you will be √2 metres from where you started. For the fractionalists among you, its closest approximation is 99 over 70. And it has had quite a history. According to one legend, Pythagoras (he of the square on the hypotenuse and right-angled triangle fame) could not accept the existence of irrational numbers, despite not being able to disprove their existence through logic. So, he sentenced his maths rival, and irrational number supporter Hippasus to death by drowning, or drowned him himself, or merely kicked him out of their maths club.

√2 is also a number with many uses. Take paper. Not the newspaper you’re holding (because that’s a bit different) but the stuff you put into printers and copiers. The humble piece of A4 (or maybe even A3) has the ratio 1 to √2. A0 is a metre square in area. A1 is A0 folded in half (and turned the other way up). A2 is A1 folded in half, and so on. All sheets are the same shape. This will only work if they all have their sides in proportion 1 to √2.

Then there’s electricity. Peak voltage of AC electricity is √2 times the quoted (RMS) value – for example, 240 volts RMS peaks at 339 volts.

Yep, the square root of two is a fascinating number. You can even watch it being calculated on YouTube.

Perhaps the most relevant to the world of ICT, however, relates to a paper written by Intel founder Gordon Moore, in which he made the now famous observation that the number of transistors on an integrated circuit for minimum component cost doubles every 24 months. This was back in 1965 – today, it’s still commonly used to refer to the rapidly continuing advance in computing power.

If doubling occurs every 24 months, guess the amount of increase each year? That’s right by a factor of 1.4142135623730950488 …