Playing with Arithmetic

We have easy access to calculators – smartphones, computers, carrying around a TI-83 if you like. Yet I find doing these small computations quickly and mentally is not only useful, but fun. It’s hard not to feel clever while quickly rattling off answers to seemingly difficult problems.

The catch, of course, is that carrying 1’s and collecting remainders is difficult without scratch paper. My suspicion is that this is due to a ruinous combination of the Zeigarnik effect and the limitations of an untrained short term memory, which most of us (myself included) have.

Working out problems in one’s head makes use of short term memory. If the total solution to a problem requires multiple intermediate solutions, for instance,

8 * 12 + 6 * 4 = 96 + 24 = 120,

then each intermediate solution poses two memory issues. First, the midway solutions occupy the 4 (+/-) 2 possible item spaces in short term memory and may begin to push out other, still relevant, information [1, p 65]. Second, the midway solutions are jobs “completed”. The brain is prone to retaining incomplete tasks in short term memory, so I speculate that deeming a task (even an intermediate one) as complete de-incentivizes the brain to remember it [1, p 67].

So we come to the point; I may be able to do the individual steps of a problem in my head, but keeping the numbers straight proves obnoxiously difficult. Tricks can reduce the number of steps and/or the total number of items stored in my short term memory, so I curated a list of some cool but basic tricks I use.

Trick # 1: Dividing by 5

Let’s say I am given a number that I need to divide by 5. Some numbers (eg. 25) are easier than others (eg. 77) to divide, but if I remember the simple fact that

1/5 = 2/10

I can make the problem much easier. After multiplying the original number by 2, all I have to do is move the decimal over one.

Trick # 2: Doubling and Halving

Another fun problem is multiplying two large numbers. For example, consider 55 * 24. Not hard with a pencil, but somewhat daunting to keep track of all the intermediate steps for an in-head calculation. Here remember that you can multiply one of the terms by 2 so long as you divide the other by 2. Using this can give us happier numbers to work with.

Note that while you can keep repeating this trick until you reach a number that isn’t evenly divisible by 2, hundreds and tens are a great stopping point, since they’re really easy to separate, multiply, and add back together.

Trick # 3: Multiplying Decimals

I’m not sure why, but decimals have never been within my mental-math comfort zone. For whole numbers or fractional forms, the placement of a decimal is very trivial. Always at the far right of the solution.

5 * 2 = 10.

5/6 * 2/5 = 10./30. = 1.0/3.

It’s so trivial, it looks strange to even write the decimal. But multiplying decimals is a tricky business, since multiplying two numbers with magnitudes between 0 and 1 has a solution with magnitude even smaller than the original numbers; the opposite effect of multiplying whole numbers.

Rather than carrying the decimal places through every step to figure out its placement the traditional way, count the number of places in each decimal and add them. Then forget about the decimals and simply multiply the non-leading-zero numbers as if they were whole numbers. After that multiplication, count, from right to left, the number of places until you reach the places sum you found in the first step. Place the decimal there.

Trick # 4: Finding Percents

Let’s say I am asked what 13% of 569 is. Not fun. Not in my head anyway. But I can easily find what 10% is. And I can find what 1% is. So, why not just add a series of 10%s and 1%s?

Trick # 5: Subtracting

Finding the difference of large, non-round numbers has the irritating quality of many intermediate steps despite the overall simplicity of the problem. In elementary school I learned the technique of “borrowing” from the digit in the next highest place, but doing this in my head (even now) gets confusing quickly.

To get around this, recall that you can add numbers to one quantity so long as you add the same amount from the other quantity without altering the overall result. So rather than carrying 1’s and keeping intermediate steps straight, manipulate the quantities given such that there is at least one round number. I like to make my round number the quantity I am subtracting by, rather than subtracting from.

Trick # 6: Squares Between (-100,100)

Of all the math problems one might run into daily, I will concede that this is the least common. Outside of a physics or math class, I can’t remember the last time someone asked me for the solution to 66^2. However, in my opinion this trick is by far the most fun and makes me feel super clever. For me, it also requires the most practice to do quickly. But if you are interested, as a treat, here is a trick for finding large squares.

First, for multiples of 10 separate out the digit in the tens place, square it, and then multiply the solution by 100.

To square a number with a 5 in the ones place, separate out the digit in the tens place, multiply it by itself plus one, multiply the result by 100, and finally add 25. It is, I grant you, more complicated, but it does work, and only has two intermediate steps.

The last trick can be used iteratively to find squares of numbers that have 1→4 or 6→9 in the ones place. It also is the trick which I think gives the clearest look under the hood of what’s actually happening. When I was first learning, I was taught that multiplication was a fast way to add. Then I was taught that exponents were an even faster way to add. I’m not sure about the precision of those statements, but they do make the following trick seem a lot more sensible.

To find the square of a number not ending in 0 or 5, take the square of the number before. Add to this the number before, and then that number plus/minus one to achieve (or get closer to) the desired square.

This can also work in reverse, of course. For example,

79^2 = 80^2 – 80 – 79 = 6400 -159 = 6441 – 200 = 6259

(Note that in these steps I often combine old tricks!)

If you’re looking for 82^2, then just repeat the process again after obtaining 81^2. This one does take some practice to do quickly. I have yet to be very good with it, but it is a fun challenge to try and keep all the pieces straight in my head!

Like anything else, practice with these tricks makes them come faster. I go in and out of practice and thus being able to do these smoothly. I hope, however, you can find a few tricks among these and have some fun playing small math games in your head. 

References:

[1] Ahrens Sönke. (2022). How to take smart notes: One simple technique to boost writing, learning and thinking. Sönke Ahrens.