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$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:
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$1.64 weeks, which is very close to our estimate of $2$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$500 years by rounding, they probably don’t care if the actual answer was $427.953$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!
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$0, which means there are no aliens, to $280000000$280000000, which means millions of different kinds of alien civilisations!
Know and apply standard form, significant figures, rounding, and decimal place value