Wednesday, July 26, 2017

Of philistines, maths and log tables

One of the things which most disappoints me which stops people achieving their full potential is a disdain for key educational subjects, and there is no worse example than people convincing themselves that they cannot do maths, or that this essential subject is more difficult than it actually is.

There was an example on BBC Radio Cumbria yesterday, with two presenters who referred to a sportsman with a degree in maths. I think they were trying to suggest that he was more clever than people often assume athletes are, but it came over as "maths is boring and difficult" and this is not an opinion which people in positions of influence in the media or anywhere else should be encouraged to promote, even accidentally.

Examples of the comments made on Radio Cumbria included one about algebra being difficult and another which was disparaging about logarithmic tables "what are they and what use are they?"

Oh dear, oh dear. Sigh.

I suppose it doesn't help that dictionaries are often written by people who are good at English and not necessarily good at maths - I've just checked the definition of a logarithm in an otherwise quite good dictionary - one of the Oxford range - and if I didn't already know what they are I would have found that definition given completely incomprehensible. It's so poor I'm not going to quote it.

I find logarithms quite useful in the statistical aspects of my job as an economic analyst, though I usually get a computer to work them out rather than looking them up in a table.

My late mother, who was a teacher, once jokingly described log tables as "a means of multiplying and dividing for people who only know how to add and subtract."

That is an example of a true word spoken in jest because you can indeed use them that way, though it is not really what logarithms are most helpful for.

Let's see if I can explain why logarithms can be useful.

A logarithmic table is a table which expresses the value of the numbers in the table as powers of a base number; most often that base number is ten.

(Occasionally instead mathematicians use a number called "e" which is about 2.718 - it is used because certain formulas incorporating this number have useful properties in differential and integral calculus.)

Any number to power Zero is One.

Any base number to the power One, is the base number itself.

Any base number to the power Two is the base number squared.

Any base number to the power Three is the base number cubed/

In a table with the base of ten,

One expressed as a log 10 value is zero (because ten to the power zero is one)
Ten expressed as a log 10 value is one (as ten to the power one is ten)
100 expressed as a log 10 value is two (as ten to the power two, or ten squared, is 100)
1,000 expressed as a log 10 value is three (as ten to the power 3, or ten cubed, is 1000)

The point where most people's eyes start to glaze over when even a very good teacher (which I do not claim to be) tries to explain logarithms is when you start on the two components of a logarithm, which are called the characteristic and the mantissa, and yet understanding these is really helpful if you want to be easily able to manipulate numbers.

Any "real number" (the definition of a real number is an issue for another day) can be expressed as a number between One and Ten multiplied by a power of ten. This is known as "standard form" (or sometimes as scientific notation.)

So for example. 100 in standard form is equal to One times ten squared (ten to the power two)

e.g. 100 = 1 * 10(2)

2,000 is two times ten cubed (ten to the power three)

e.g. 2,000 = 2 * 10(3)

31,623 is 31.623 times ten to the fourth power

e.g. 31,623 = 3.1623 * 10(4)

If you convert any of these numbers to a logarithm using base ten, the whole integer part of that logarithm - the bit to the left of the decimal point - is equal to the power of ten in the standard form of the number. This is called the characteristic and it tells you what order of magnitude the number is for which this is the logged value.

The left-over part, the fractional value to the right of the decimal point, is called the mantissa
and it tells you what the first (and subsequent) digits of the number are, e.g. the first term of the standard form will always correspond to the same mantissa.

So in the above examples,  the log of 100 is 2.0, with the characteristic of 2 representing ten to the power two and a mantissa of zero representing the number One in  100 = 1 * 10(2)

The log of 2,000 is approximately 3.301, with the characteristic of 3 representing ten to the power three and a mantissa of 0.301 representing the number 2 in the standard form 2,000 = 2 * 10(3) and whenever you see a logarithm with the fractional component or mantissa of 0.301 or thereabouts it means that the number for which this is the log is approximately equal to two multiplied by ten to the power of the part of the number to the left of the decimal point.

So if you see any logged figure of around 6.301 it's in the ball park of the log of two times ten to the power six, which is about two million.

The log of 31,623 (or thereabouts) is 4.5. In this case the characteristic of 4 representings ten to the power four so the number is in the tens of thousands, and a mantissa of 0.5 representing the number 3.1623 in the standard form 31,623 = 3.1623 * 10(4) and whenever you see a logarithm with the fractional component or mantissa of 0.5 or thereabouts it means that the number for which this is the log has a first integer of three (and if the mantissa is exactly 0.5 the next four digits will be one, six, two and three.)

OK, you may or may not find this interesting, and many people probably will not have read this far, but some of those who have will be asking "but what actual use is all this?" More than you might think.

In the days before computers, log tables could be used to speed up complex calculations provided you only needed an approximate answer. (My calculation of "about two million" above was actually off by 38, to the nearest whole number,  because I had only cited the logarithmic value to three decimal places.)

We no longer need log tables for that now that almost everyone has easy access to computers (an iPhone has more computing power than NASA had when they sent men to the moon.) But logarithms they can be very useful to statisticians when you are looking to measure or predict proportionate relationships rather than linear ones.

Suppose you are an economist looking to chart the a demand curve which shows how sales of a product increase, other things being equal, when the price falls. If the relationship is a nice straight line on a graph - a fall in price of £5 per unit always causes a 100 units a week increase in sales, that sort of thing - you can estimate it with a simple linear formula.

But most real world relationships don't work like that and do not produce nice straight lines on a graph. As "The Economist" once wrote

"Beware of forecasters with rulers!"

Real world relationships are more often non-linear, and produce a curve when graphed.

Non-linear relationships, particularly those which take a proportionate form - e.g. whenever the price goes up by 10%, the volume of sales drops by 15% - are much easier to calculate and model by incorporating logged terms into the calculations.

I was lucky enough to have teachers who could make this sort of thing interesting. But it is more than interesting to be able to understand the different patters which numbers can take: it is genuinely useful in many numerate professions (including mine.)

Teachers like that are worth their weight in gold. A wise society recognises their value - and that of education. And does not encourage people to mock the value of numerate skills or any other type of learning.


Jim said...

I remember maths being difficult, then one day a stand in teacher taught us about "shortcuts", simple things like dividing by 3, if you take a number and you add all its constituent numbers together, then if the sum does not divide by 3 neither does the long number.

Basically 987,654 = 9+8+7+6+5+4 = 39 amd 3 + 9 = 12

and because i know 12 divides by 3 I can also tell you that 987,654 also does as well.

there are many different tricks to making maths easier, and a very good place, with a very good online teacher, I know of to show of these is
this you tube channel from tecmath

Its a great resource not only for people still in school, but also for anyone who wants to brush up on their maths

Jim said...

In fact that site will help out a lot of politicians, car manufacturers, and fuel sellers. See ideas that sound good, like lets ban petrol and diesel cars after 2040, need to be fully thought through.

Firstly the maths site will indeed help out people who we are going to need to vastly improve battery technology. We will also need to roll out lots and lots of electric charging points, or indeed re-tool every filling station with an alternate fuel like Hydrogen. also we would have to think about not just cars but emergency sevice vehicles and goods vehicles and if an alternate fuel source is viable (even trains which are driven by electric motors, but have diesel generators)

there weould need to be an entire new business model for all garages (servicing an electic motor car, is different than an Internal combustion one.

Think of a rule of 5s, I need something i can fill from empty to full in 5 mins, giving me a range of 500 miles, at over 50 Mph.

I am not saying its not doable, but such a huge change need to be fully thought through and they would naturally come about if market forces meant it was the best option, I think in cases like this the free market should be left to provide the options rather than ambitious laws.

Jim said...

LOl, yes with batteries, they are not simple, not simple at all, and yes to achive this is going to take eiter a massive change in battery technology, or a hyrdrogen fuel cell revoultion. and of course every fuel stop having liquid hyrogen to pump.

See poloticians always go for what sounds good, they often forget what is actually possible.

Chris Whiteside said...

Thanks for that, Jim.

Very interesting site. And a classic example of taking a positive and fun approach to maths rather than an "it's too hard and boring" one.

Chris Whiteside said...

So thank you, Jim, that post exemplifies exactly the sort of attitude I was suggesting we need more of.

Chris Whiteside said...

Some interesting and thought provoking comments too on what will be required to go over to electric cars on a huge scale over the next 23 years.

BTW, the original comment about batteries was from "undercover economist" Tim Harford rather than me, and neither of he now I suggested for a moment that the sort of improvement in deployment of battery technology required to abolish the internal combustion engine and replace both diesel and petrol cars with electric ones will be remotely simple.

He was actually making the point that it is the basics, inventions like the battery, rather than the glamourous inventions like cars or aircraft, which have the biggest impact and batteries were only described as relatively simple compared to say, a mainframe computer, a car or and aircraft (all of which contain batteries ...)

Jim said...

The divide by 3 trick proved very useful to me once in London, I bought 3 pints of Lager, they asked for £9.95. I knew they had either made a mistake or were trying to pull a fast one.
If they had asked me for £9.96 i would have thought "my thats expensive" but paid it anyway and following our drinks we would have moved on. As i questioned it, the landlord came over and said "something wrong, I simply said that 3 pints of lager cant be £9.95, he said you are not use to london prices, I said but £9.95 does not divide by three, so what is the price of one?
he went to the till, checked it and said "sorry, they are on the house"

Chris Whiteside said...

Now that really is proof that it is worth paying attention to your maths teacher :-)