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December 9th, 2017, 06:01 AM  #1 
Newbie Joined: Mar 2017 From: Connecticut Posts: 8 Thanks: 0  Multiplication Algorithm
Hello all, I made a post a few months back asking about methods for squaring 2digit numbers after I recognized a pattern and you friendly folks told me I had actually stumbled upon an old method that had long fallen out of use. My question today focuses on multiplication in general. Last night, I was mindlessly doing 9 times tables when I realized that 9 times a number is the same as 10 times a number minus that number. For instance, I saw that 9x5 = 10x5  5. So then I tried it with a few other numbers like the 8 times table and the 7 times table (with an added step, as shown below) and found the same to be true. All in all, I wrote out the algorithm like this: (X)(Y) = 10Y  (10X)Y E.g.: (7)(15) = (10)15  (107)15 =150  (3)15 =150  45 = 105 I know that this method is probably not really useful and I still haven't tested it with a wide enough range of numbers to see if it's always true. However, I'm still curious as to whether anyone recognizes this as an actual method or not. Thanks! 
December 9th, 2017, 06:47 AM  #2  
Senior Member Joined: May 2016 From: USA Posts: 1,206 Thanks: 494  Quote:
It is a basic rule of arithmetic that if a, b, and c are any numbers, then $a \times (b + c) \equiv (a \times b) + (a \times c) \text { and } a \times (b  c) \equiv (a \times b)  ( \times c).$ This is called the distributive law. The funny looking equal sign means that the equality is always true. So let's look at your algorithm generally. $z = 10y  (10  x)y \iff z = 10y  (10y  xy) \iff z = (10y  10y) + xy \iff z = xy.$ It will always be true. Long ago, before hand calculators, people who had to do arithmetic a lot used your trick and others like them to do mental arithmetic. $28 \times 30 = 30 \times (30  2) = 900  60 = 840.$ $32 \times 28 = (30 + 2) \times (30  2) = 30^2  2^2 = 900  4 = 896.$ You said in your previous thread that you are not good at math. I very much doubt that to be true. I suspect that you find arithmetic boring and so your attention wanders and you make silly mistakes. Arithmetic IS boring, but the further you go in math, the more and more interesting it becomes. Stop persuading yourself that you are not good at math. Just recognize that computation punishes carelessness.  
December 9th, 2017, 02:07 PM  #3  
Newbie Joined: Mar 2017 From: Connecticut Posts: 8 Thanks: 0  Quote:
All I wanted to know was if this was an actual way to do multiplication in the past which you did answer for me. So thanks.  
December 9th, 2017, 02:16 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 2,200 Thanks: 1153  Quote:
So yeah, it's been known and used in the past.  
December 9th, 2017, 03:13 PM  #5 
Senior Member Joined: Aug 2012 Posts: 2,098 Thanks: 602  
December 9th, 2017, 10:18 PM  #6  
Senior Member Joined: May 2016 From: USA Posts: 1,206 Thanks: 494  Quote:
 
December 10th, 2017, 05:02 AM  #7  
Newbie Joined: Mar 2017 From: Connecticut Posts: 8 Thanks: 0  Quote:
 
December 10th, 2017, 05:04 AM  #8 
Newbie Joined: Mar 2017 From: Connecticut Posts: 8 Thanks: 0  

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