Halving and Rehalving, as Well as Doubling and Redoubling, as a Calculator-Free Calculating Strategy


I don’t like being too dependent on calculators. The future might bring an EMP (electromagnetic pulse) that would fry all such gadgets (and cell phones, cars, computers, etc.), and I want to be ready for a post-calculator world, if that happens.

My overarching strategy for doing math in my head is this: don’t have just one single strategy. Instead, devise one, on the fly, based on the problem you are trying to solve.

I do, however, use a few “go to” strategies for certain things, such as finding 25% of something, or multiplying by eight, or similar problems. This involves looking for, and take advantage of, powers of two, as well as their reciprocals. 25% is 1/4, which is simply halving twice, and multiplying by eight is three doublings, since 8 = 2³. With practice, doubling or halving numbers repeatedly and silently, in one’s head, becomes much faster and easier. If done out loud, it becomes easier still, and on paper, it’s extremely easy.

I intend to do more blog-posts in the future with calculator-free calculation strategies, but not all at once — instead, these techniques will be posted one at a time. However, these postings will stop immediately, in the event of an EMP.

On Calculator Use and Abuse


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Calculators are important tools.

Well, so are guns.

Everyone acknowledges that, if guns are going to be used at all, certain safety precautions are essential. While it generally is not a matter of life and death, the same thing is true of calculators.

I have seen — multiple times — students multiply or divide some number by one, using a calculator, and then be genuinely surprised when they got the same number with which they started. I have also seen a bright student calculate the mass of an atom, and get an answer larger than the mass of the earth. When I informed him of this, his reaction was predictable: “But that’s what the calculator says!”

These are examples of calculator abuse. Aside from avoiding errors, such as the “planetary atom,” it’s also important not to abuse calculators because such an act is insulting to one’s own brain.

Some people treat calculators as if they are omniscient and infallible. They aren’t. They’re small, simple, narrowly-focused computers. The human brain is also a computer, but a far more advanced one. The calculators are our tools, not the other way around.

Here are some tips to prevent calculator abuse:

1. Know how to do arithmetic without a calculator. If you don’t know, learn.

2. Use calculator-free mathematics when appropriate. If you need to know what six times eight is, for example, it insults your own brain to consult a calculator. Don’t do it!

3. Calculator-free mathematics is not to be called “mental math,” for the simple reason that ALL math is mental. Also, if you ever meet anyone claiming to have invented that insipid phrase, kick that person immediately.

4. If you have contact with children, promote the use of calculator-free math to them. This is most important, of course, for parents and teachers. It is no service to a child to raise them to be dependent on a calculator.

5. Finally, when you do use a calculator, don’t be too trusting of the little thing. Check your answers, constantly, by using estimation. Say, for example, you’re multiplying 109 by 36. That’s “a bit over a hundred” times “a bit under forty.” Since forty hundreds is four thousand, the answer has to be in the ballpark of 4,000 — and if a calculator disagrees, the calculator is wrong, probably because a human pressed at least one incorrect button. You will press incorrect buttons on occasion — we all do — and it’s important to have a method in place to detect such errors. This estimation-method is both simple and effective.

All these principles boil down to this: be smarter than your calculator. They aren’t actually very intelligent, so this is neither difficult, nor unreasonable.