Fun With Arithmetic

Fun With Arithmetic

I had a long drive today for an event that my wife's band was doing, and I found myself trying to keep my brain amused. I was trying to figure out how far an object falls under gravity in a particular period. The equation you know: x=1/2at^2. But doing the arithmetic in your head (at least, in my head) is a bit of a struggle. An object that takes 5 seconds to fall means 1/2 x 32 x 25.

Well, 16 x 25 is pretty easy: that's the same as 16 x 100 divided by 4, or 400. For any number that you are multiplying by 25, just add two zeroes, and divide by 4. For any number multiplying by 20, add two zeroes, and divide by five.

What about 16 x 36? This isn't as hard as it sounds. Any time that you multiply a number by a power of two, you can just divide the power of two by two, and multiply the other number by two--which almost everyone can do in his or her head. So 16 x 36 is 8 x 72, or 4 x 144, or 2 x 288, or 576.

Since in the equation above, 1/2a is always 16, this sure speeds up doing this type of calculation in your head. For a nine second fall, you now have 16 x 81, which is the same as 8 x 162, or 4 x 324, or 2 x 648, or 1296.

While not quite as simple, you can use this same divide by two approach with other otherwise hard to manipulate numbers: 22 x 23 is the same as 2 x 11 x 23, or 11 x 46. Any two digit number multiplied by 11 is pretty easy to do in your head, since you are only having to remember two different numbers to add up in different columns: 46, and 460, or 506. Try this with something like 44 x 48, which is the same as 2 x 2 x 11 x 48, or 2 x 2 x 528, which is 2 x 1056, or 2112.

As long as you can factor one of the numbers down to 2, even if the other number is three, four, or even five digits, most people can remember enough digits to do this computation without writing anything down.

While not quite as easy, you can use the same approach if you can factor one of the numbers down to something with a 3 in it. For example, 21 x 23 is the same as 3 x 7 x 23. Most people can do 7 x 23 in their head, by remembering that 7 x 3 is 21, and 7 x 20 is 140, so 7 x 23 is 161. Now just multiply by 3: 3 x 1 is 3, and 3 x 160 is 480, so 483.

Of course, never miss the special cases, such as multiplying by 25 or 20 (mentioned earlier), or where both numbers are powers of 2. For example, 16 x 512. Remember that 16 is 2 to the 4th power, and 512 is 2 to the 9th power. You can then reduce 16 x 512 to 2^4 x 2^9, which is 2^13. If you have powers of two memorized, even part way up, this speeds up the process. You know that 2^10 is 1024, so doubling goes to 2^11, or 2048, doubling again is 2^12, or 4096, and doubling again to 2^13 is 8192.

You may think this is a sign of a rather peculiar person to spend time figuring out tricks like this in the age of computers, but I think it is rather cool to not be dependent on these gadgets. I confess that I have long found these sort of tricks intriguing because of reading about Carl Friederich Gauss, the mathematician and scientist after whom the unit of magnetic force measurement is named. When he was quite young, one of his teachers needed to keep the class busy for a while, and told them all to add up all the numbers from 1 to 100. Gauss immediately raised his hand and gave the answer: 5050. How? By noticing that 1 + 100 = 101, as did 2 + 99, and 3 + 98. There were a total of 50 pairs that summed to 101, so 50 * 101 = 5050. Anytime that you can foil a teacher's busywork plans, that's a virtue!