Piece of cake. Like many things, they’re tricks and/or shortcuts. This is no different. Oh, and by the way, you can extend the same principles to multiply triple digits.By the way, I know someone from an Ivy League University that can multiply double digits by double digits mentally.
Hint - think before calculators, think old style adding machines - circa 1970 and prior to.
Great idea - that works too.Originally Posted by seriousplayer
I was thinking more of the 1920’s version of adding machines in a business office.Think about the sounds they made while they would grind out the answer.
Analogy - Shipthecookies always comments on AP play off CSM. He can’t (don’t think) hear how they work, but he knows how they work.
The old style adding machines - you can literally hear how they work as they grind out the various calculations.
My mother helped my Dad in the store, before calculators and computers and Atari video games for that matter, for decades. When a customer bought goods, she would dutifully punch the numbers into the machine to provide a receipt. After the customer left and because she didn’t trust the machine, she would work it out again with pencil and paper to make sure the machine got it right. She was pretty good with numbers herself. I listened to that machine for years, learned actually quite easily what it was doing. I’m pretty sure I was the only kid in high school who could square numbers in his head virtually instantaneously. The same concepts can be used for any 2 numbers under 100, and with some discipline, any 2 numbers under 1000.
I’m sure someone will ask how. It’s easier than you think.
Either - principle is the same. Initial example would be squaring since the example(s) would be easier. Then, anyone should be able to extend the skill to multiplying any 2 numbers.
Note the human brain should be able to handle any 2 - 2 digit numbers, some people 3, and only a very few beyond that.
There can be tricks and/or shortcuts but the key is auditory ability and memory. Being able to listen to the numbers and do the calculation. Not sure if old adding machines train you to do that. It is not going to do you any good if you have weak auditory ability.
In general terms, the old style adding machines were very loud, and you could literally count the number of operations it was performing in order to determine the answer. So, after asking for the answer, 8+8 or 8*8, it required 1 operation. The example mentioned in posts 9&10 was 18*18, which cranked 4 times which meant 4 operations to solve - but what was it calculating?
To my way of thinking, the easiest numbers to multiply by are 10, and units of 10. So, for the human mind to quickly calculate, without benefit of pencil and paper, or any other type of aid, would it be easier to multiply
18*18, or
(10+8)*(10+8)
So, if 10’s are the primary (P), and 8’s (the single digit) are the secondary (S) as in
10+8 *
10+8
and if the old machines needed 4 operations to correctly calculate, the inescapable answer is
(P*P) + (P*S) + (P*S) + (S*S) or
(10*10) + (10*8) + (10*8) + (8*8) or
100+80+80+64.
Exactly the same principle is applied to correctly calculate 28*18 or any other 2 digit numbers, and which can be extended to 2-3 digit numbers, or any combination of 3 and 2 digit numbers. My parents store had 3 of those machines, and I could literally visualize how the machine was calculating.
Ok, I see what you did. You FOIL (First, Outside, Inside, Last) it. I did mines mentally different. I know the formula for squaring any two-digit integer are
mn x mn
18 x 18
10 x m(mn+n)+n^2
m=1
n=8
Since it is 18^2 (18 x 18). I only need to mentally solve the middle term and add a zero at the end. 1(18+8)+64= 260+64=324.
Last edited by seriousplayer; 01-01-2021 at 01:42 PM.
I never took trig or calculus in high school, or advanced math. Maybe I should have - think I would have been a natural at it. So, I’m confined to the parameters of pure logic coupled with determination and intersection of spatial relationships, to which the above noted problem is a combination of.
I like to solve puzzles. Extending that, I’ve always been interested in finding the absolute easiest way to perform a function, willing to invest time up front to reduce my future time expense so I could goof off elsewhere. My favourite example was my month end billing procedure when my business was busiest and at its height.
1. Walk downstairs to my basement office - 20 seconds
2. Open computer, select printer, Tray 2 - 10 seconds
3. Insure adequate supply properly sized paper for Tray 2 - 10-15 seconds
4. Open master billing file, enter date once, enter month and year number once (2020-12) - 10 seconds
5. Hit Print
Everybody had been converted to flat rate billing, same invoice number per month prefixed by acct code, same invoice date - absolute simplicity built into a macro, easily modified as required.
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