Turning it up to 10.1

Nov 04, 2022 by @turgon@dosgame.club

Welcome to the next post in my series about decimal placement on slide rules!

In this post I'll discuss the role played by numeric formats.

Let's first introduce some terminology. Recall that when $x$ is the base- $b$ logarithm of a number $k$ , it solves the equation $k = b^{x}$ . The slide rulers of old had names for the parts of that logarithm, $x$ , to help them reason about decimal placement. The whole number part before the decimal place is called the characteristic, and the part after the decimal place is called the mantissa. Unfortunately, those were overloaded terms across math, science, and engineering back then, but I'll leave that side quest to you and carry on.

Right away we see that the logarithm $x$ is the sum of the characteristic and the mantissa. Let's call those $c$ and $m$ respectively, noting $x = c + m$ . If we substitute this into our original equation $k = b^{x}$ , we get $k = b^{(c + m)}$ , and by exponent rules then $k = b^{c} \cdot b^{m}$ .

Now, since the characteristic $c$ is a whole number, the $b^{c}$ term is a nice round power of the base. In base 10 for example, $10^{c}$ will always be things like thousands, millions, hundredths, billionths, and so on. So on slide rules in base 10, the characteristic $c$ tells you exactly where to put the decimal place. This is key, and my future posts on decimal placement per operation will largely demonstrate how to find the result's characteristic under various operations.

On the other hand, the $b^{m}$ term also has some properties worth noting. Since $0 \leq m < 1$ , it follows that $1 \leq b^{m} < b$ . Again, in base 10, that means $10^{m}$ is at least $1.0$ and strictly less than $10$ . If you look at your C and D scales, you can see they increase from 1 to 2 to 3 and up to 8 to 9 to .. one?? What on earth!

Using the ideas we discussed above, let's reason through what's happening on the slide rule's scales right there at C's right margin where $m$ is almost $1.0$ and $10^{m}$ is almost $10$ . Let's pretend $10^{x} = 9.95$ , which has a characteristic of $0$ and a mantissa around $0.998$ . On my Versalog's C scale, $9.95$ is the very last tick mark on the far right.

If we want to increase $10^{x}$ from $9.95$ to $10.1$ , we'll have to add a little bit to $x$ . Notice this takes us past the end of the C scale. What should we add to $x$ to increase $10^{x}$ from $9.95$ to $10.1$ ? Let's use our exponent rules: $10.1 = 10^{(x + t)}$ for some new $t$ , so $10.1 = 10^{x} \cdot 10^{t}$ . But we know $10^{x} = 9.95$ , so we substitute that and divide both sides by it: $\frac{10.1}{9.95} = 10^{t}$ . That leaves approximately $1.0151 = 10^{t}$ , and taking ${log}_{10}$ of both sides gives $t = 0.0065$ .

Wonderful, but what the heck does it mean? Well, let's add $x$ and $t$ together and see what we end up with: $x + t = 0.998 + 0.0065 = 1.0045$ , a logarithm with characteristic $1$ and mantissa $0.0045$ !

Ok, so now that we're familiar with characteristic and mantissa values with base 10, and the exponent rule we used earlier, let's put $10.1$ into normalized scientific notation. Scientific notation factors out a leading coefficient and the order of magnitude from a number to approximate numbers of any size. In this case, $10.1$ is not very big but it does have a power of ten, so in scientific notation it is written as: $1.01 \cdot 10^{1}$ . In normalized form, the leading coefficient doesn't exceed $10.0$ .

That's amazing, because it corresponds neatly with the concept of a logarithm's characteristic and mantissa! Observe that $log (10.1) = 1.0045$ , with $c = 1$ and $m = 0.0045$ . Now notice that $10.1$ in normalized scientific notation is $1.01 \cdot 10^{1} = 10^{m} \cdot 10^{c}$ for those $c$ and $m$ values. Marvelous!

To bring this home, we wanted to get at the question of how to deal with numbers that are smaller than $1.0$ or bigger than $10$ . We've learned that in normal form, the leading coefficient is always between $1.0$ and $10$ , because the mantissa of the logarithm of the number is between $0.0$ and $1.0$ . And we've learned that when the mantissa exceeds $1.0$ , the characteristic increments and the mantissa restarts at $0.0$ . The same is true on the slide rule: you never really go off scale, you just shift the decimals around appropriately. That is, using the normalized scientific notation not only solves the problem of keeping your numbers on scale, it also gives you the tools you need to track the decimal of your results. This is WHY slide rule tutorials out there tell you that scales repeat infinitely.

At the end of the last post, I encouraged you to make your own rulers and try adding numbers with those rulers. Do you still have those rulers out and ready? This time try thinking of the numbers on the rulers as mantissas that you're adding together. Just be sure to think of 1, 2, 3.. on the ruler as 0.1, 0.2, 0.3, etc.. Try adding some mantissas and use your calculator or WolframAlpha to find $10^{x}$ !

Until next time!

Next post: slide rule multiplication!