Geekspeak: Why Life + Mathematics = Happiness

The idea behind statistical sampling is the same as used in surveys of, for example, voters. The nation’s intended voting pattern could be found by asking all 30 million voters about their plans for the voting booth. More practically, a representative sample of voters is questioned. The sample might consist of just 1,000 people carefully selected to represent all the localities and social groups in the country.

The same approach can be used to estimate your vocabulary. Sample the ‘population’ of words by opening the dictionary at random 100 times. Each time, look at the first entry at the top of the page. Do you know the meaning of this word? If the answer is yes, add one to your word score. At the end of the exercise, divide your score by the sample size of one hundred to get an estimate of the fraction of words in the dictionary that you know. Multiply that fraction by the total number of words in the dictionary to make an estimate of your vocabulary size.

This method works, but you need to be careful: how many times should you dip into the dictionary at random to get a good estimate? Say you do the test twice and find that you know the first word, but not the second. That means that you know 50% of the words in the tiny bit of the dictionary you examined.

But common sense tells you that this estimate is unreliable. It is true that you might know half the dictionary, but it is also possible that you know 10% or 90% of all the words. The two words you chanced upon might have been unusually uncommon, or unusually common. Two out of however many thousand words the dictionary defines is not a representative sample.

Do the trial 10 times, and confidence in the result is greater; 100 times, even better. If you did the trial 1,000 times and found that you knew 500 words, you could argue quite strongly that you really do know about half of all the words in the dictionary.

To complete the estimate of your vocabulary you’ll need to know the total number of words in the dictionary – preferably without having to count them. This is quite easy: look up the number of the last page in the dictionary, and take that as the number of pages. Next, open the dictionary at random and count the number of different words listed on that page. Multiply the number of pages by the number of words per page, and you have an estimate of the number of words in the dictionary.

I thought I’d better test myself using this statistical sampling technique. The dictionary I used has about 60 entries on each page, and over 800 pages. That’s around 48,000 words altogether.

I opened the dictionary 125 times, and made a tick on a piece of paper if I knew the meaning of the word at the top of the page, and a cross if I didn’t. Like me, you’ll probably find it hard to stop yourself jumping ahead to other entries if the first is unfamiliar. Don’t – that’s cheating, and invalidates the statistical sampling!

The result: there were 25 words whose meaning I didn’t know. On that basis, my passive vocabulary is 48,000 multiplied by 100/125. That’s around 40,000 words. It sounds high, but it includes all the possible extensions of the stem of each word. For example, take the word ‘abstract’. The dictionary will include ‘abstractedly’, ‘abstractedness’, and so on. The number of stem words I know is a lot less than 40,000.

Still, I’m feeling pretty good about myself, so I’m going to exercise my gigantic male vocabulary by introducing the next chapter:

‘The, er, next chapter is, er, fucking interesting…’

SPEAK GEEK

‘IT IS A TRUTH UNIVERSALLY ACKNOWLEDGED THAT A SINGLE MAN IN POSSESSION OF A GOOD FORTUNE MUST BE IN WANT OF A WIFE.’

Some authors are instantly recognisable from their vocabulary. For example, everyone recognises the style of Jane Austen, and many would say that her writing’s distinguishing feature is its abundance of long words. But is this true? A bit of statistical analysis can reveal the answer.

The four longest words used by Jane Austen in Pride andPrejudice have 16 or 17 characters. They are ‘superciliousness’, ‘communicativeness’, ‘disinterestedness’ and ‘misrepresentation’. But just looking at the longest words is not enough: we need to examine the distribution of word lengths over her entire vocabulary, as shown in the graph below:

For comparison, here is the ‘fingerprint’ of the writer Ian McEwan, showing that his vocabulary includes many shorter words:

And, what about this book? In this work I intend to speakwith candour, and without misrepresentation or superciliousness, of the accomplishments of the irreproachable retrospections…

2 (#litres_trial_promo)

PUMPING IRON

Are you powerful as a washing machine?

For the price of a few cupfuls of oil, men can be transformed into mechanical supermen. Next time you pass major roadworks or a large building site, watch the hydraulic rams on a mechanical digger pushing the bucket and scooping up one-ton heaps of sand, all at the twist of the wrist of the driver. Those rams become extensions of his limbs: he is a superman.

Mechanical power is often quantified as horsepower, a word coined by the eighteenth-century engineer James Watt, the man whose work changed steam engines from profligate steam guzzlers into much more efficient and powerful machines.

In Watt’s day, ponies or horses were used to turn a windlass that hoisted buckets of coal up a mineshaft. He would have wanted to know how many horses would be needed to lift a bucket in a given time. Watt knew that a horse could pull with a force of about 180 pounds, and that it could walk a total distance of around 180 feet each minute while pulling the load. That became his definition of horsepower: it’s the power needed to move a force of 180 pounds through a distance of 180 feet every minute.

To get a feeling for one horsepower, think of it this way: an average man weighs around 180 pounds, so with a suitable pulley and rope, a horse could hoist him 180 feet into the air in about 1 minute. The Eiffel Tower in Paris is 986 feet high. If we could position our hoisting pulley at the top of the Tower, our man would be dangling almost one-fifth of the way up after 1 minute.

Connect the same pulley rope to the 60 hp engine in your car, and you could hoist the same man to the top of the Eiffel Tower in less than 6 seconds, although he might not have much stomach for the view once he arrived at the top.

What about men instead of horses? A simple way of measuring your power in horsepower would be to tie the pulley rope around your own waist, take the strain, and see how long it takes you to hoist the 180-pound man through 180 feet.

A 180-pound man is too much for most of us to lift, so let’s replace him with a small child weighing, say, a quarter of that, 45 pounds. If you managed to hoist the child through 180 feet in 1 minute, you would have a power of one-quarter of a horsepower.

In reality, even if someone was willing to lend you their child in the service of science, most people would have difficulty in performing the task in less than 2½ minutes. So, a man’s power is nearer one-quarter divided by two and a half, which is one-tenth of a horsepower.

Nowadays power is usually measured in watts rather than horsepower. We’ve just changed from using Watt’s own term, horsepower, to using his second name as our standard unit of power. There are 746 watts in one horsepower.

Power comes in many forms, but it’s always a measure of the rate at which energy is delivered somewhere. For a car it’s the rate at which mechanical energy is delivered at the engine’s flywheel. For a gas cooker it’s the rate at which heat energy is delivered by the burner to the bottom of the pan, and for a light bulb it’s the rate at which electrical energy is supplied to the bulb.

It would be quite legitimate, and possibly more meaningful, to rate, say, a 75-watt light bulb as one-tenth of a horsepower. A label of 1/10 hp on the bulb would indicate that one horse turning a windlass connected to an electrical generator could light ten such bulbs. More sobering, it shows that one athlete turning a windlass or a treadmill could manage to keep just one 75-watt bulb burning.

I’m guilty of sometimes having up to 300 watts-worth of light bulbs switched on in my house during the evening. In a pre-fossil-fuel era, I would have needed four slaves continuously walking a treadmill to keep them alight. But don’t be too smug – to boil your 3 kW electric kettle you would need forty slaves.

The use of watts as the unit of power has probably contributed to the divorce between our understanding of machine power and of human power. Labelling the power of commonplace machines and devices in manpower instead of watts would keep us much more aware of the gearing provided to our lives by fossil fuel.

In a world without fossil fuel, the unit of power might be the ‘slave’. Perhaps the obscenity of slavery disappeared only because we invented other means of getting cheap energy.

If a slave is equal to 1/10 hp, your car has a 600-slave engine, your water heater is rated at 40 slaves, and your fridge is about 1 slave. Yes, there is one slave pedalling a generator 24 hours a day, seven days a week to keep your food cool.

A few more examples are in the table, starting with the least powerful and working upwards. The cost of the energy delivered by any of these machines is phenomenally low. Imagine that someone offered to pay you to climb onto a treadmill to generate power to boil a kettle for a pot of tea. How much money would you expect to receive?

Most kettles are 3 kW, which is equivalent to 40 slaves. It will take about one and a half minutes to boil with that power input. With only you on the treadmill it’s going to take 40 times as long. You will have to tread the mill for 1 hour to boil the water to make the tea.

If you get paid the current minimum legal wage, you’ll get £5.35 when the kettle boils. Compare that with what you’d pay for electricity generated by fossil fuel or nuclear power – about 10 pence for each thousand watts for each hour. The kettle will use 3 × 1.5/60 = 0.075 kilowatt-hours, costing you about 0.75 pence. So human power, even at its cheapest, is about 700 times as expensive as using fossil fuel.

Even worse, after an hour on the treadmill you will need a change of clothes: you’re going to need the washing machine. That means more work on the wheel. There is no escape.

Domestic washing machines have a motor with a power of up to 250 W, about one-third of a horsepower, which is more than twice the power that you could generate, even if you rigged the washing machine to some form of treadmill or pedalling system.

A typical wash cycle can take over an hour, and the motor’s energy consumption will be about 0.3 kilowatt-hours, costing you just 3 pence. More significant is the electricity used to heat the water in the machine. Getting the water temperature up to 50 or 60 degrees Celsius will use over 1 kWh of electricity – three times the amount used for turning the drum.

Climb back on your treadmill again: you’ll be there for twelve hours to heat the water to wash your clothes. And then you’re going to need another change of clothes…

SPEAK GEEK

MORE THAN 60% OF THE ENERGY FROM BURNING PETROL IN YOUR CAR IS WASTED.

There is an unassailable limit to the proportion of the heat energy that can be converted into mechanical power by any kind of engine. The unconverted energy is then dissipated: in the case of a car, out through the radiator and in the hot gases from the exhaust pipe.

To figure out that wasted energy, you take the temperature, T

, of the hot gases made by burning the fuel and subtract the temperature, T

, of the exhaust gases leaving the machine. Then divide this difference by 273+T

(don’t ask – too geeky). Multiply that by 100 to get the percentage of the energy that could be converted to mechanical power. The wasted energy is the difference between 100% and the number you just calculated. As a formula, it looks like this:

For a car engine, T