Quartics

You might think quartics (fourth powers) would be difficult or at least messy to compute without succor of the log-log scales. Surprisingly, that's not the case at all. Only required are C, D and A, and just one slide movement to carry out such problems. The method is extremely fast and convenient.

Here's the scoop. We've already learned how to calculate squares in a previous installment, thanks to how the C and D scales are twice as long as A (and B, although it's not needed here). And, of course, we know at least two methods for multiplication. So, let's just rewrite the quartic as:

x⁴ = (x*x)²

It's as simple as that! For example, consider 2⁴ = (2*2)⁴. Go ahead and find 2*2 using canonical multiplication on C and D. Now, if you looked for the result under the index on the D scale, you'd have 4, of course. But if you instead checked what's under the hairline on the A scale, you'd see 16. (Since it lies in the second span of A, it must be a result with an even number of digits in the whole part, i.e., 16, not 1.6). I'll give you moment to try it out.

Notice: only one slide movement was required. And the scale transfer was trivial to keep track of; A must be the square of what lies on D. Shall we give it a whirl?

Exercise 1.1: Compute 2.73⁴.

For this first exercise, let's use canonical multiplication, as we saw in the explanation, above.

1. Set the cursor at 2.73 on the D scale.
2. Align the left index with the hairline.
3. Move the cursor to 2.73 on the C scale.
4. Read the result under the hairline on the A scale: 56.

Here's what it all looks like. As usual, you can click the photo to enlarge it.

Cursor to D:2.73, left index to hairline, cursor to C:2.73, result at A:56

Of course, we know the result must be 56 not 5.6, since it fell in the second region of the A scale. Moreover, 5600, the next possibility, doesn't make sense. So, 56 it is.

In this example, canonical multiplication worked well, since the argument (2.73) was less than sqrt(10), the dividing line of the two halves of a logarithmically measured scale. So, no index-swap was needed.

Exercise 1.2: Compute 4.5⁴.

Notice that 4.5 is to the right of the dividing line just mentioned. So, if you were to start this one out the same way as that in Exercise 1.1, you'd quickly see an index-swap would be needed. There's nothing particularly bad about that, other than it requires the expenditure of a few more calories.

Instead, let's use the old "multiply by a reciprocal" business with the CI scale to cut down on the movements.

1. Set the cursor at 4.5 on the D scale.
2. Align 4.5 of the CI scale with the hairline.
3. Move the cursor to the left index. (One index is guaranteed to be on-scale).
4. Read the result under the hairline on the A scale: 410.

My favorite Pickett 1006-ES places the CI scale on the reverse side of A, so I needed a duplex-flip. Not a big deal, though.

Cursor to D:4.5, CI:4.5 to hairline, cursor to left index

Duplex-flip, result at A:410

Since this lies in the first half of A, the result must have an odd number of digits. Only 410 is reasonable.

If you're using a slide rule with CI on the same side as A (like the popular UTO 610), then this is clearly the desired approach for minimal finger movement.

Who would have thought quartics could be so easy?

Next installment: Quintics