Tuesday, July 2, 2019

Today's Useless Skill

Some years ago, my wife and I were on a long car trip.  She wanted the stereo off so she could sleep.  Bored out of my mind, I tried to find an easy way to do two digit by two digit multiplication in my head.  Some people count sheep; I calculate the squares of two digit numbers in my head to fall asleep.

1. Refactor the multiplicand to something simpler.  For some it is  so obvious that everyone immediately knows how,  40 x 40 is (4 x 10) x (4 x 10) = 4x4*100: 1600.  For all even numbers factor by two because multiplying by 2 twice is easy to do in your head.  22 squared is 11 squared (121) x 2 (242) x 2 = 484.

2. Non-prime odd numbers can be easy,  27 squared is 3x9x3x9.  9x9 (81) x 3 (243) x 3 (729).

3. Some multiplies have tricks.  If you are squaring a power of two: 32 is 2^5.  So 32 squared is 2^(5*2), which every software engineer of my generation has memorized:  1024.  Ditto for 64 squared: 2^(6*2)=4096.

4. For odd primes: (n+1)^2 = n^2 + 2n + 1.  21^2 is 20^2 (400, See #1).  400 + 2*20 + 1 = 441.

5. Some single digit multipliers have their own tricks.  n * 25 = (n * 100)/4 (again, easy to do in your head).  n * 5 = (n * 10)/2.

6. In time, syou do ome calculations so often that they burn into your memory.  The radius of Mars is .45 of Earth.  For calculating surface gravity relative to Earth, when I was a kid, I calculated 45 squared so many times that 2025 is in permanent memory.  Similarly, from calculating relativistic effects in high school physics, ,99 squared is .9801, so 99 squared is 9801.

7. If you need to do column by column in your head, it is simplest to do each multiply separately and add: 23 x 23 = (3x3) + (3x20) [69] + (20x3)[60 + (20x20) = 529.

8. Some operations are artificially simple to do: 1 digit by 2 digits.  Any two digit number ending with a one in ones column.  51 squared=51+2550=2601.  Similarly trivial if the ones column is 2.  62 squared=124(2 x 62)+120 (2 x 60) + 3600 (60 x 60).  The trick is to keep repeating the partial sums as you add them in your mind.

And eventually you can work this into a sequence of pounding your hoof!

1 comment:

jdege said...

Any number ending in 5 is easy:

(x5)^2 = (x*(x+1)) 5

25^2 = (2*3) 25 = 625
35^2 = (3*4) 25 = 1225
45^2 = (4*5) 25 = 2025

And then there's the off-by ones:

(x+1)^2 = x^2 + 2*x + 1
(x-1)^2 = x^2 - 2*x + 1

16^2 = 15^2 + 2*15 + 1 = 225 + 30 + 1 = 256
14^2 = 15^2 - 2*15 + 1 = 225 - 30 + 1 = 196