Giveaway Calculators Then and Now

In my previous post, I described how to calculate a cube root on an ordinary calculator. Ordinary calculators are so inexpensive that they are given away as promotional knick-knacks. Here is one I received a few years ago at a university’s Campus Day – it’s not only a calculator, it’s also a ruler with inches and centimetres:

Now you might think that giveaway calculators first appeared around 1990 when advances in integrated circuits brought down the cost of a calculator microprocessor to less than a dollar. And you would be wrong. Here is a giveaway calculator from the 1950s, over half a century ago:

Like its modern counterpart, it’s not only a calculator, it’s also a ruler with inches and centimetres. Like the digital calculator above, it is shown just after someone worked out the cube root of 10 – and quickly too. Slide the cursor (the window) so that the red line is over the 10 on the K scale, then see the answer on the D scale directly above – 2.15 – accurate to three digits. What the digital calculator loses in speed (because you have to poke all those buttons), it makes up in accuracy (with an extra four or five digits of precision).

Back of giveaway slide rule calculator.

One difference between then and now: lots of people use digital calculators today, but analog calculators were mainly aimed at tech-heads like engineers — this one even came with a soft plastic case perfect for engineers’ shirt pockets.

Calculate 2π. Line up 1 on the C scale to π on the D scale. Now, 2 on C lines up with the answer on D: 6.28. Bonus: follow the red line to the A scale for 2π squared, 39.5.

Calculate Cube Roots on an Ordinary Calculator

I discovered a relatively compact way of calculating a cube root on an ordinary calculator. The procedure is built on the formula for geometric series and on the calculator’s hidden “K” – constant – register.

By “ordinary calculator” I mean one of those inexpensive calculators that you can find in endless sizes, shapes, and styles, but all of which use identical calculating logic. They all include, among other things, an 8-digit display, square root key, and two clearing keys, AC and CE. They also all include an internal K register. I suspect that over 99% of people who use ordinary calculators have no idea what the K register is, or that it even exists. This is because I suspect those people who are geeky enough to figure out the K register don’t use ordinary calculators. Instead they buy and use fancy scientific/ programmable/ graphing calculators.

Now, to calculate a cube root. First enter the number, then press [√] [√] [×], then repeatedly press [√][√][=] eleven or twelve times at which point the number in the display stabilizes. That number is the cube root correct to seven digits. For example, take the cube root of 10:

Enter:               See displayed:
[1][0]                10
[√][√][×]         1.7782793
[√][√][=]         2.0535247
[√][√][=]         2.1287512
… 7 more iterations …
[√][√][=]         2.1544339
[√][√][=]         2.1544342
[√][√][=]         2.1544342

The answer differs only in the last digit from the accurate approximation 2.1544347.

This algorithm depends on the geometric series for 1/3:

1/3 = 1/4 + 1/16 + 1/64 + … + 1/4ⁿ + …

Packaged into exponential form:

x^1/3 = (x^1/4)(x^1/16)(x^1/64) …

And calculated with the assistance of the calculator’s K register, which caches the factor x^1/4, while the product builds up in the displayed result.