Slide Rule Multiplication!

Nov 08, 2022 by @turgon@dosgame.club

Alright, it's finally multiplication time! Recall from my last post that slide rules love numbers in normalized scientific notation, that is, a coefficient between 1 and 10, times ten to some power. Let's start without the slide rule and see what happens when we multiply two numbers in this form.

Let's suppose we have a multiplicand of $x\cdot {10}^m$, and a multiplier of $y\cdot {10}^n$. If we multiply them together, we get:

$x\cdot {10}^m\cdot y\cdot {10}^n$, which we can rearrange (thank you commutativity) as:

$x\cdot y\cdot {10}^m\cdot {10}^n$, and then by exponent rules:

$x\cdot y\cdot {10}^{(m+n)}$

Since both $x$ and $y$ are greater than or equal to $1.0$, their product must be, too. But also, since $x$ and $y$ are less than $10$, their product can't exceed $10\cdot 10=100$. That presents two possibilities: either the product is less than $10$, in which case it gives us the coefficient of the result and we can simply add $m+\mathrm{n}$ to get the exponent of the result.

Or, the product is between $10$ and $100$. In that case, We can factor out a $10$ from the product, and increment the exponent to get an exponent of $m+n+1$ and keep everything in normal form. The result looks like: $\frac{x\cdot y}{10}\cdot {10}^{(m+n+1)}$, which is messy, but we don't need to worry about it on the slide rule.

Let's do a quick example, and then we'll look at our slide rules. Suppose we want to multiply $3.14$ and $450$. Oh, sorry, I meant $3.14\cdot {10}^0$ and $4.5\cdot {10}^2$.

$3.14\cdot 4.5=14.13$, which is bigger than $10$. And summing the exponents gives us $0+2=2$. We know this is the second case and that means the result is $1.413\cdot {10}^3$.

Break out your slide rules!

To multiply using the C and D scales, align the C's left margin to the multiplicand (3.14) on D. Then move the cursor to the multiplier (4.5) on C, and use the hairline to read the result from the D scale. When the product exceeds 10 as it does in this example, the multiplier will be on the part of the C scale that's sticking out of the rule and we can't reach it with the cursor!

my post versalog 2, showing 3.14 ⋅ 4.5 using C's left margin gives a result that is off the scale

close

Oh but remember, the C and D scales repeat! That is, the right margin and left margin are equal, and we usually see them both labeled as 1 as a reminder. Whenever the result of multiplication is off scale, switch to using the right margin: Align C's right margin to the multiplicand (3.14) on D. Then move the cursor to the multiplier (4.5) on C, and use the hairline to read the result from D. It's a smidge over 1.41, just as we calculated before.

Resetting to use the right margin has the same effect as the division by 10 in our earlier arithmetic, so we need to increment the sum of exponents by 1. In turn that tells us where the decimal place goes. Thus the answer is around $1.41\cdot {10}^3$.

my post versalog 2, showing 3.14 ⋅ 4.5 using C's right margin and the result is 1.41

close

By the way, you can expect to get three digits of precision from your slide rule calculations with an error margin of around plus or minus 2%.

If you don't have your own slide rule, you can play around with these ideas using a virtual slide rule. In my photos for this post I used a Frederick Post 1460 Versalog 2, and there's a virtual version of that one here: https://www.sliderules.org/react/hemmi_versalog_ii.html

Slide away!

Next post: square and square roots