Approximation and Errors

Hong Kong

Stage 1 - Stage 3

Lesson

At 5.10am, July 16th 1945, Enrico Fermi showed the world what a true Mathemagician is capable of. Fermi, a nuclear physicist regarded as the “Father of the Atomic Bomb”, stood in the middle of the desert of New Mexico, ready to watch the world’s first nuclear detonation, together with the other scientists who had helped create this terrifying new weapon. When the bomb went off, the other scientists gazed in awe at the burning light, as the explosion consumed the desert. Fermi, on the other hand, did something a little odd: rather than taking in the sight of the mushroom cloud, he was paying attention to some scraps of paper that he was dropping from his hand, as they fluttered in the breeze created from the explosion. He carefully noted where they fell on the ground, and silently calculated some numbers in his head (calculators didn’t exist back then). When he was finished he had an estimate of the power of the bomb he had made – about 10000 tonnes of TNT. Remarkably, when scientists analysed the bomb site later, they discovered that Fermi’s calculation was quite close to the correct answer!

Although Fermi was no doubt a genius, he did not perform this magical calculation by any superhuman mathematical ability: rather, he had invented a method for getting close-enough answers to problems by rounding numbers up or down so that they were simple enough to work with in his head. Today, problems solved by this method are called “Fermi Problems”.

A classic Fermi Problem might look something like this: How long would it take to count to a million? Some of you may have tried this as kids, and probably gave up. Now you’ll know how long it would take you if you kept going, and you won’t even need a calculator to figure it out!

Let’s assume you count one number per second:

- In one minute there are 60 seconds.
- In one hour there are 60 minutes, so you count to 60 \times 60. Using our times tables, 6 \times 6 is 36, and the two 0’s means the answer also has two 0’s: 3600 seconds in an hour.
- In one day there are 24 hours. But 24 \times 3600? Yuk. Let’s change those to some nicer numbers. How about we make 24 into a 20, and 3600 into 4000! But wait, are we allowed to do that? Doesn’t that make our answer wrong? Technically, yes, but it all depends on how accurate the answer needs to be. In this case we only want an estimate, rather than an exact answer, so rounding numbers is perfectly fine. So, to finish the calculation, the number of seconds in a day is 20 \times 4000. 2 \times 4 is 8, and there are four 0’s, so the answer is roughly 80000.
- Now, how many seconds are there in a week? 80000 \times 7 ,which equals 560000.
- So, how many weeks to count to 1000000? That would be 1000000 divided by 560000. We would definitely need a calculator for this one. However, if we round the 560000 to 500000, we get a nice simple answer of 2! Therefore, according to this calculation, it takes around 2 weeks to count to a million.

The remarkable thing about this kind of calculation is that it usually ends up “close enough” for the purposes of simple curiousity. For example, using your calculator, you would have found out that the actual answer is 1.64 weeks, which is very close to our estimate of 2 weeks. The reason it works is that the rounding up and the rounding down balance each other, resulting in an answer which is close to the correct one.

Fermi Problems may sound like nothing but a game, but they have real scientific applications. There are many situations where people only want a general idea of something, rather than a very specific answer. For example, a computer company might want to figure out roughly how long its computers will last before breaking. If they get an answer of roughly 500 years by rounding, they probably don’t care if the actual answer was 427.953 years.

Have a go at some of these Fermi Problems, or even better, make up some of your own! Each time you need to multiply numbers together which look difficult, just round them to the nearest nice numbers which you can use your times tables for. Many of these problems require you to make guesses of things which you do not know exactly, like how long per day you brush your teeth. Once again, the point is only to make a close-enough estimate, so don’t worry too much about how accurate your guess is!

- How long would it take you to walk around the world? (Use the internet to find the circumference of the Earth)
- How much water do you drink in one year?
- If you made a pile of all the hair from all the haircuts in your life up until now, how tall would it be?
- How many hours of your life have you spent brushing your teeth?
- If bank robbers tried to steal \$1000000 in coins, how many of them would it take to carry it all?
- How many jelly beans fill a one liter jar? (Use this next time they have a guess-the-number contest!)
- How many people of the same age as you are there in your city? (Use the internet to find the population of your city)
- How many of those people are sitting in Maths lessons right now?
- If you took all of your Maths books and stacked them on top of each other, how tall would it be?

- How many of those people are sitting in Maths lessons right now?

Did you know?

Possibly the most difficult Fermi Problem is The Drake Equation, which estimates how many alien civilisations there could be in our galaxy. This can be calculated by using scientific estimates of the number of planets in the galaxy and likelihood that any particular planet would have aliens on it.

Astronomical experts disagree significantly about their estimates in this equation, getting a result of anywhere from close to 0, which means there are no aliens, to 280000000, which means millions of different kinds of alien civilisations!