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Cubes and Cube Roots

Nov 15, 2022 by @turgon@dosgame.club

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

This post will be about cubes and cube roots using the K scale. Recall that slide rules love numbers in normalized scientific notation, that is, a coefficient between 1 and 10, times ten to some integer power. The exponent tells us exactly where the decimal goes.

Starting with the math and no rule, let's cube a normalized scientific notation number:

( x ⋅ 10 m ) 3 which we can easily rearrange into: x 3 ⋅ 10 ( 3 ⋅ m )

And, wow, that looks a lot like what we saw with squares. Right away we can see that the exponent has tripled, and we might have to deal with the cubed coefficient becoming large. It can't exceed 103 or 1,000, and we saw with squares the scale was split into two parts to account for either zero or one factor of 10. With cubes, the cube's coefficient could contain zero, one, or two factors of 10. That means we have three cases when we normalize:

{ x 3 ⋅ 10 ( 3 ⋅ m ) if 1 ≤ x3 < 10 ( x 3 10 ) ⋅ 10 ( 3 ⋅ m + 1 ) if 10 ≤ x3 < 100 ( x 3 100 ) ⋅ 10 ( 3 ⋅ m + 2 ) if 100 ≤ x3 < 1000

As with squares, if x3 is say, 15.3, we factor out a 10 from the coefficient and stuff it into the exponent by incrementing the exponent.

computing the cube root of 15.3 on my Pickett N1010-T using the middle subscale of K

If x3 had been 153 instead, we factor out two 10s from the coefficient and increment the exponent by two. In both examples the normalized coefficient is 1.53.

computing the cube root of 153 on my Pickett N1010-T using the right subscale of K

Unsurprisingly, a quick peek at the K scale reveals it is indeed split evenly into three parts, and just as with squares they look like miniature C scales. SO CUTE!

Cubing and squaring are very similar operations on the slide rule, it turns out. To cube, move the cursor to the operand on the D scale, and read the result's coefficient from the K scale. The exponent is tripled, then incremented if necessary. I like to think of the K scale as three subscales left to right: K0, K1, and K2, where the number corresponds to the remainder of exponents modulo 3. This remainder also tells you by how much to increment the answer's exponent. So there is a good visual cue: when the result is on the left of K, add zero; when on the middle, add one; and when on the right, add two.

computing the cube root of 1.53 on my Pickett N1010-T using the left subscale of K

In my photos above, I've calculated the cube roots of 1.53, 15.3, and 153, each of which has the same normalized coefficient. You can see the repetition of the coefficient on the K subscales, but the resulting coefficients on the D scale are quite different!

To take a cube root, first divide the exponent by 3 and use the remainder to determine which of K's subscales to use. Then move the cursor to the operand on the appropriate subscale of K, and read the result from D. The result's coefficient will always be in normalized form. The answer's exponent is: floor(m3)

One useful thing you can do prior to taking a cube root is to divide the operand's exponent by 3 using long division. The object is to find both the quotient (which will be the answer's exponent) and the remainder (which tells you which subscale to use). You can totally do this division on a slide rule, too! Although.. I haven't written a post on division yet. 😬 Coming soon to a theater near you!

See you next time..