Math Explained (Paused)

Slide Rules

Last time, we learned about logarithms and a couple of their fundamental properties. Today, I would like to mention a few others and talk about another tool our grandparents, and even some parents, used prior to the emergence of computers and scientific calculators. I introduce to you, the slide rule.

Believe it or not, we put men on the moon with these things!

What you see above is a relatively basic slide rule with a single moving part in the middle. In practice, you can encounter slide rules with multiple sliding pieces in the middle, called slides. The two stationary pieces are called the upper stator and lower stator, or summarily the body.

The ruler-like markings on the slide and body are called the scale, and many permutations of these exist. We can see four of these by looking at the left sides of both stators above: LL2, DF, D, and LL3. Each combination of scales allows you to perform a unique operation on two operands. For instance, the LL2 and LL3 scales are log-log scales used for calculating exponentiation.

The basic operation of a slide rule involves the movement of one of more slides to match up scale markings on the body. The combination of these can then be read to give you a result. You can find a great example of how this works from NPR here: http://www.npr.org/sections/ed/2014/10/22/356937347. If you are inclined to learn the many scales available and which operations they can be used for, you can find a great explanation at http://sliderulemuseum.com/SR_Course.htm. They even have virtual slide rules for you to play with!

Now, onwards with the math! Last time, I mentioned two important properties of logarithms. The first is that any exponent within a logarithm can be brought out as a multiplier of that logarithm.

log_n(a^b) = b • log_n(a) 

The second is that for any non-zero n, the logarithm base n of n is 1. You can see this by recalling that logarithms are the inverse or exponentials.

n^1 = n <=> log_n(n) = 1

What about zero, you might ask? Because there is no power that you can raise a number to to give 0, the logarithm of any base of zero is actually undefined in mathematics.

n^? = 0 <=> log_n(0) = ?

The next two properties of logarithms are actually the basis of how slide rules work: Logarithms turn multiplication into addition and division into subtraction.

log_n(a • b) = log_n(a) + log_n(b)

log_n(a ÷ b) = log_n(a) - log_n(b)

These are called the product rule and quotient rule, respectively. The product rule is actually the proof for how our first property above works:

log_n(a^b) = b • log_n(a) 

log_n(a^b) = log_n(a) + log_n(a) + ... (b times) 

log_n(a^b) = b • log_n(a) 

We can prove the product rule to be true as well by looking at exponentiation.

Let 

x = log_n(a) and y = log_n(b)

Then 

a = n^x and b = n^y

Let's multiply them together

a • b = n^x • n^y

Exponents have a property which allows us to combine this multiplication. I can prove this too if you would like me to, but let's avoid the proof rabbit hole for the time being.

a • b = n^x • n^y =n^(x + y)

Take the log base n of both sides

log_n(a • b) = log_n(n^[x + y])

Apply our two first two properties from last time

log_n(a • b) = [x + y] • log_n(n)

log_n(a • b) = x + y

Lastly, replace x and y

log_n(a • b) = log_n(a) + log_n(b)

The quotient rule can be proved very similarly by remembering that

1 / a = a^(-1)

As I mentioned above, the product and quotient rules are the basis of slide rules. Allow me to provide a quick example. Say I would like to calculate 2 • 3. With a slide rule, I would go through logarithm addition:

2 • 3 => LOG(2 • 3) 

Using the product rule, this becomes

LOG(2 • 3) = LOG(2) + LOG(3)

and I would shift the slide forward by LOG(2) and then another LOG(3). It will wind up in a position corresponding to some number, in this case LOG(6) or ~0.778. By looking at the other bottom edge of my slide rule, I would see that this corresponds to 6. Mathematically, what is happening is an exponentiation by the base of our logarithm, typically 10.

~0.778 => 10^(~0.778) = 6

If this example is unclear, feel free to ask questions or take a look at those links above!

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