September: season of mists and mellow whatsisname… And surds revision for students beginning A level maths.
GCSE pupils hate surds, and are horrified to discover that once you are in the sixth form you are expected to use the blasted things all the time. How ridiculous is that? I mean, who cares that √12 = 2√3 ? Who would ever use something that stupid in real life? They’re worse than fractions!
It’s a fair argument. We have phones that work as calculators these days, and tablets that work as calculators, and computers…. oh, and calculators as well. If I need the value of √12, a few keystrokes give me 3.464101615, which is more than enough precision for any practical purpose. So what’s the point of surds?
The point is, of course, that there is no point. No decimal point. √12 is an irrational number, which means that it can’t be written as a fraction and its decimal expansion goes on forever without repeating. So 3.464101615 is only an approximate value for √12, good enough for government work (to quote the great uncle of a friend who surveyed railroads in the Amercan west) but WRONG! If you square it on your calculator, the answer will be 12 (it is on all mine), but this is because the calculator is rounding 11.999999999045608225 to 10 significant figures.
(If you want to try full precision multiplication like this, a good resource is at https://www.mathsisfun.com/calculator-precision.html Or you could try it with a pencil and paper…)
Using surds saves us from unnecessary rounding errors. Suppose we approximate √12 as 3.464 to save space.
Then 3√12 x 7√12 = 3( 3.464) x 7( 3.464) = 251.985216
But √12 x √12 = 12, so the correct answer is 3 x 7 x 12 = 252….
AND it’s actually easier to work this out than to get the approximate answer with a calculator. And less error prone – the more buttons you press, the more mistakes you make.
√12 is the number which when squared gives us 12.
It doesn’t matter that we don’t know exactly what the value is, because whenever we square it the surd vanishes and we get a proper number, with NO errors.
We also get better answers by rationalising the denominator of a surd fraction.
We have a square root which we can only evaluate approximately and, worse, we have to divide by it. That’s two potential sources of error.
(If you don’t know how to rationalise the denominator, there are lots of good videos on You Tube, such as https://www.youtube.com/watch?v=T_cZqeLNCHM to show you how it works.)
The same principle applies to ‘exact values’ calculations involving π. If you ask most children what π is, they are likely to give the approximate value 3.14. In fact:
π is the number you get when you divide the circumference of a circle by its diameter.
If we leave the symbol for π in our calculations, and we get lucky, it will cancel out. At worst, we carry out one computation with 3.14 (or whatever) at the end. Either way, we simplify the calculation hugely. Again, fewer keystrokes = fewer errors.
So learn to love surds, and be right more often.
(PS I’m new to WordPress. I’m sure there has to be a better way of incorporating fractions that bringing them over from Libre Office as images. I’ll look into it…)