Multiplying with Paper
Division is fairly straightforward: take some number of things, maybe X of them, and separate them into groups of another number, Y. The number of groups of Y is the answer to X/Y.
Essentially, division of X by Y is the answer to the question “how many Ys do I need to add together to get X?”
People, for the most part, seems to understand this. However, if my recollection of 9th grade math class is accurate, everyone flips out when you change “add” to “multiply.”
This is effectively what logarithms are: the answer to the question “How many Ys do I need to multiply together to get X?”
Let’s try some examples to get the feel for it:
We can reformulate this into “How many 10s do I need to multiply together to get 100?” the answer, of course it 2, as 10*10 = 100.
This one is a little harder if you don’t know your cuberoots. Again, we reformulate it into “How many 7s do I need to multiply together to get 343?” Well, 7*7*7 = 343, so the answer is 3.
I’m going to leave a couple more examples. If you’re still a little uncomfortable with it, go through each one and make sure they make sense.
Something interesting is going on with the last three examples. 8*32 is 256, and 3 + 5 is 8:
If you multiply the numbers inside the parentheses, and add the numbers on the right, (and keep the base the same) the equation holds. Hmm. We may have spotted a pattern. Let’s try it again:
Does the pattern hold?
I won’t formally prove it here, but this pattern is always true, no matter the base.
At some level, this should make sense. 100 can be expressed as (10*10) and 1,000 as (10*10*10). When we take the log of a number in base 10, we count the number of 10s, so what happens when we take the log of 100,000? Well, count the 10s:
One two three four five. There are five 10s. We could count like that, or we could group the tens by the products. The only difference here is the parentheses:
Now we count using the parentheses: One two. One two three. Two plus three is five, so there are five 10s.
Notice that by counting using the parenthesis first, we are essentially computing the log of the products, then adding them:
This can be written more generally as
This is one of those things you memorized in high school math, then forgot minutes after the test. You figure it might be useful in the field of math, but you don’t really see the point.
Now I’m going to use it to make a calculator. Out of paper.
First, let’s familiarize ourselves with a logarithmic scale:
The distance from the left side to a number on the scale is the log of that number.1
Notice that the difference between 1 and 2 is significantly larger than the difference between 2 and 3, which is larger than 3 to 4. If you find this is confusing, look at this log graph, and think about what happens when you take the log of successively higher numbers.
Now the clever bit.
By placing two log scales next to each other, we can visually add the two log values:
Note that the distance from the far left of the bottom slide to the 3 on the top slide is log(3) + log(2). We just added the distances by placing them side by side.
From our identity (below), the sum of the log of two numbers is the log of the product of those two numbers.
So, log(3) + log(2) = log(6), and our picture verifies that:
From this, we can see, visually, that 2*3=6. But what if we want to know two times a different number, maybe 4? Well find 4 on the top slide, and look at the corresponding number on the bottom. The distance from 1 to 4 on the top slide is log(4), and the distance from 1 to 2 on the bottom slide is log(2), so it’s sum is log(8), and again, the picture verifies this.
If you look closely, you can see that the entire top slide is doubled. This is the amazing power of logs.
But we can go one step further. What If we wanted to triple the entire top slide? Easy, just move the top slide so the 1 lines up with the 3 on the bottom slide. Can you see why this would result in the top slide being tripled?
This isn’t just some cool trick to impress your friends. This is a true calculator!
All that’s left is to build one.
The first step is to create a log scale, and plot the points one through 10 on it.3
One way to do this would be to plug the different values into a calculator and measure the distance out with a ruler. However, I’m using a piece of paper, and it already has convenient lines that will serve as my units.
The values of log10(1 to 10) are all between 0 and 1. There are 33 lines on a piece of paper. I want to use the entire stretch of the paper, so I’ll multiply the value of each log by 33, then count off the lines.4
Here is a handy excel sheet if you are playing along:
And, we have our log scale:
Now I’m going to fold another piece of paper around it so that the log scale can slide around. This paper will also serve as my second log scale:
Now we’re ready for some calculations!
This device, called a slide rule, was invented in the 17th century by Reverend William Oughtred, and is, in my opinion one of the coolest human inventions ever.5
1If you’re wondering what happened to the base,2 it doesn’t matter. If I created this scale using base 10, I could stretch or squeeze it to be in another base, meaning everything will be proportional.
2Also if you’re not wondering what happened to the base
3It actually doesn’t have to be one through 10, but that seems like a good stopping point.
4This should make sense, as log10(10) is one, which will put it at line 33, and log10(1) is zero, which will put it at line 0.
5Right behind the icebox and the freezer.
Sorry for not posting last week, I had some exams and school got the best of me.