Dealing with Big and Small Numbers

How are we going to learn to guess? It's actually something we can learn. For example, say you ask me, "How many people are on earth?"
So I look on Wikipedia and find an estimate that on May 31, 2009 the earth's population was 6792,467,727. Wow down to the person: 6 billion blah blah blah and TWENTY SEVEN! Good job Census Bureau. But really now, that's a pretty useless number. Most people would have guessed 6 billion and really 7 billion would have been a better guess BUT and this is the point I'm trying to get to: the twenty seven people are a useless detail when estimating.

In fact! The 6 or 7 is pretty useless. When estimating the only important word is BILLION.
Don't get me wrong there will be times when we'll keep the 6 or 7 but remember the point of these Fermi questions is to get good estimates without looking things up and without using a calculator. So say we choose 7 billion and then we have to divide it by 2, ok? 7 / 2 is ... I don't know (actually I do but humor me) that's tougher math than 6 billion / 2. 6 / 2 = 3 SO THREE IS CLOSE ENOUGH. We will always chose the easy math or just drop numbers whenever we want.

Ok? I know that this will take some getting used to for anyone in the audience who wrongly thinks that the details matter. Another example: we were all in some high school math class and the teach asked someone to put some number through their calculator. Say you were calculating the area of a circle of radius 2m. The formula for area is
So some kid puts it through their calculator and says 12.56meters squared. The teacher says that's correct BUT then some annoying kid (that kid was usually me, by the way) shouts "No, it's actually 12.56637061...blah blah blah . Ok so maybe the annoying kid was more accurate but come on! The amount of time it takes to write those useless numbers out and he had to have used his calculator. There's no way he did it in his head.Hugo StrangeUs, here at Building Batman however, will always say things like
Who cares about the details. We are just estimating to get a guess. So then we'd say the area of the circle is 3*2*2 = 12. Close enough for the girls we go out with, ok?

And back to the world population of 6 or 7 billion.
How many zeros are in a billion? There's nine. So then the world's population is 7000,000,000. That's a lot of zero's. From now on we're going to use something called scientific notation. Don't be scared of it. It's actually really easy. How many times would you need to multiply 10 to the 7 to get 7000,000,000? Just count the zeros. Nine times. So we write this as 7 * (10 nine times) and the "nine times" is just taking the power. So 7 * (10 to the power 9) =
Because I'm writing this on a blog I'm going to write taking

as 10^9, ok?CalculatorWhy this "trouble"? Well, for two reasons: 1) if we wrote out all the zeros sometime we'd miss count or forget a zero and then we'd be screwed because 2) it tells us the size. Really the 7 in 7 * 10^9 isn't so important the important number is the 9 because the 9 tells us the size or the order of magnitude. If we change the 7 to a 6 so what? Our guess would be off by a little bit but it's only a guess so little bit here, a little bit there doesn't matter. We aren't looking for the "real" answer. We're looking for a good estimate. On the other hand the 9 makes a big deal. Turn the 9 to and 8 and only 10% of the population survives!!!

We can do the exact same thing with really small numbers. Think about 0.00097. Again we just count the zeroes (4 not 3). 0.00097 = 9.7 * 10^-4.

Good so far? If I did this too confusedly and you need something explained better just leave a comment.

Ok so three last things:

1) The rules for multiplying and dividing with scientific notation. When you multiply the coefficients and add the exponents.
For example (4*10^8) * (9*10^3) = (4*9) * 10^(8+3) = 36 * 10^11 = 3.6 * 10^12 is about 4*10^12. Cool?
And for dividing we divide the coefficients and subtract the exponents. Like (9*10^13) / (3*10^7) = (9/3) * 10^(13-7) = 3*10^6.

2) 10^0 = 1. Anything to the power zero is 1.
Bruce Wayne3) We're going to be taking alot of averages. We'll be guessing things and be saying things like, "well... 10,000 is definitely too big but 1 is way too small." So we will want to take the average between them. BUT notice the average of 10,000 and 1 is (10,000 + 1)/2 = 5,000.5. Well that's stupid. If you round 5,000.5 it's just 10,000. That's just the big number!!! 5000.5 is like 5000 times bigger than our small lower bound and only a factor of two smaller than our upper bound. Averaging isn't going to work.

When dealing with really big numbers and really small numbers taking the normal/everyday average isn't the right thing to do. What we really have to do is take the geometric mean. This is a good average for when you are dealing with big and small numbers. The actual definition of the geometric mean is that you multiply the two numbers and take the square root. But square roots are hard and we are doing this all in our heads so the greatest trick in the whole world which I learned from Guesstimation is how to take the approximate geometric mean. To take the approximate geometric mean just average the coefficients and average the exponents.
The geometric mean of 1 (10^0) and 10,000 (10^4) is ((1+1)/2) * 10^((0+4)/2) = 1*10^2 = 100. 100 is a good in between number if 10,000 is your high guess and 1 is your low guess.
Another example, the geometric mean of 4*10^17 and 8*10^5 is about 6*10^11 (because 6 is the average of 4 and 8 and 11 is the average of 17 and 5).
Oracle BarbaraIf the sum of the exponents is odd. There is a rule. You must decrease the exponent by 1 making it even and multiply the coefficients by 3. For example, the geometric mean of 1 and 10^3 is 3*10^1 = 30. The reason for this is because when you average the exponents you get a something and a half. An exponential of 0.5 or 1/2 is exactly the same as square root. And the square root of 10 is 3.16 blah blah blah or just 3. If that's confusing, don't worry about why. Just follow the simple rules for geometric mean and we will get some really magical answers from very little information at all.

That was a long post but we need to know all that. Tomorrow I'm going to do a non-superhero one with you and when you see this in action it will make a lot of sense and you will probably be wowed (maybe not but I was).

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