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## Pocket Calculator Power

Published on Thursday, April 04, 2013 in , , , , ,

My regular readers might be looking at the title and picture and wondering whether they're on the right blog. I normally post about doing math and memory in your head, not on a calculator.

However, using your brain in conjunction with a simple 4-function calculator, you can get much more out of them than you may have ever thought possible.

Since you only see functions for addition, subtraction, division, and multiplication (and sometimes a square root function) on a 4-function calculator, most people limit their use to just those few functions. However, even a simple pocket calculator has a few hidden features that, when combined with an understanding of varous aspects of math, allows you get much more out of it.

Before you try these out, make sure you're using an actual 4-function calculator, as more complicated calculators act differently. Many calculator apps on mobile devices appears to be 4-function calculators in one orientation, and scientific calculators in another orientation. Unless you discover for yourself otherwise, these calculator apps are generally always working as a scientific calculator, even when it appears otherwise.

Over at Ted's Math World, there's a very complete course in using a 4-function calculator, which includes these sections:

1: Introduction to Programming a Four-Function Calculator

2: Integer Powers

3: Integer Roots

4: Trigonometry

5: Compound Interest

6: Logarithms

7: Extra Decimals for Square Roots

8: Some Arithmetic Shortcuts

Even if you don't go through every section, at least go through the introduction section, as you may learn about some hidden features of your pocket calculator. Ted's Math World also features a very simple continued fraction approach to square roots, and in the Integer Roots section, you can learn how to enter this into your calculator.

Eddie's Math and Calculator Blog also has a course on calculator usage called Calculator Tricks. Surprisingly, there is very little crossover with the above course, and this one gets as far as dealing with 2 by 2 matrices! Eddie's course is available at these links: Part 1, Part 2, Part 3, Part 4, Part 5.

Back in 1974, when 4-function calculators were just starting to become affordable and popular, Popular Science wrote up an excellent guide, including many common real-word uses, such as photography, cooking, and shopping. True, you might have apps on your mobile device that handle similar functions today, but it's still good to know how to handle them yourself. The article, titled New Tricks For Pocket Calculators, can be found in the December 1974 issue of Popular Science, on page 96, page 97, page 98, page 118, and page 119.

Go through these resources, and you'll start to get a good idea of just how much more powerful your 4-function calculator can truly be!

Don't forget to keep an eye out for the occasional individual tips, as well. For example, here's a quick way to find any root on a 4-function calculator, as long as you have a square root button available:

One kind of math that doesn't get much coverage on calculators is modular arithmetic. If you're not familiar with modular arithmetic, BetterExplained.com and Martin Gardner (page 1, page 2) have excellent introductions.

Surprisingly, even many models of scientific calculators don't have basic modulo functions. In the few places I have seen methods for working out the modulus on a calculator, the methods were similar to the ones taught in this xkcd.com forum thread.

That method is certainly useful, but I never cared for the back-and-forth nature of it. I developed another method (other people must have come across this, but I've never found a reference to it) which takes you straight to the answer. Let's say you're trying to figure out what 83 mod 13 equals. Simply enter 83 - 13 = on the calculator, and you'll see 70. Hit equals again, and you see it drop down again to 57. Keep hitting the equals button until you come to a positive number that is less than 13, and that's your answer! In this case, the answer is 5.

For any number x mod y, just start with by entering x - y =, and then keep hitting the equals button until you wind up with a non-negative number that's less (LESS - not LESS THAN OR EQUAL TO) than y, and that's the final answer.

This answer works well when the numbers are relatively close, or at least have the same number of digits. What happens, though, if you have to work out something like 96,528 mod 17?!? In this case, we use powers of 10 to help. What number starts with 17, ends in 1 or more 0s, and is less than 96,528? It's easy to see that 17,000 fits the bill, so we start with 96,528 - 17,000 =, and keep hitting the equals button until we get a non-negative number that is less than 17,000. After this, we wind up with 11,528. Now, drop a zero from 17,000 to get 1,700, and repeat the process starting with 11,528 - 1,700 =, resulting in 1,328. Repeating this with 170, we work our way down to 138. Finally, we go through this process one last time with 17, and we come to our final answer, which is 2.

So, when working through any modular problem, you can not only take the number itself out, but add an appropriate number of zeroes to the end, and them out by the hundreds, thousands, millions, or whatever scale is needed! This approach may take longer, but it goes to the answer directly, and helps you understand the process of modular arithmetic.

You can also use a similar process with addition in order to find congruent numbers. What numbers are congruent to 2 modulo 6? Start with 2 + 6 =, and you'll get 8. Hit equals again, and you should get 14, then 20, and so on. Each of the displayed results are numbers that are congruent to 2 modulo 6: 2, 8, 14, 20, 26, 32, etc.!

Give a little thought, a little fun, and a little effort to your simple 4-function calculator, and you just may be surprised by what you can do with it!