Borrowing: An in depth explanation to a simple process

Chris HansenChris Hansen Regular
edited March 2013 in Tech & Games
I believe it is quite safe to say somewhere in your life you came across the concept of borrowing numbers in mathematics. It seems like such a simple idea. If I have to subtract 616 from 1005 then we come to this point:
1005
-  616
  389

This looks simple enough. As we can see 1005 is larger than 616 so we will not come to a negative number but what is the action taken to insure that we can subtract the 6 from the 5? Lets take a closer look:

this
1005
-  616
  389

can also be represented as:
1000 + 0 + 0 +5
  -600- 10- 6
  300+80+9
I am going to take a moment to explain why it can be represented as such. If we separate each digit in the number 1005 or 616 (or 389) to its proper place we have 1000 in the thousands place, we leave a zero at the hundreds and tens places respectively, as it they are already occupied by zero, and finally we put a 5 in the ones place. Doing this with 616 we have 600 in the hundreds place, 10 in the tens place and 6 in the ones place. If we add up each of the digits we expand so to speak our result is the starting number.

If we examine this property we notice we can do this:
1005         = 1000+0+0+5  = 0+900+90+15  
- 616   =      -600-10-6  =     -600-10-6 
  389   =      300+80+9 =      300+80+9

The last set is a representation of distributing the 1000 across the other digits in the number 1005. We place 900 in the hundreds place, 90 in the tens place and 15 in the ones place. Add them up and what do we get? 900+90+15=1005. Now if you see what we did you notice that instead of 5-6 we get 15-6 and then we simply add all the separated digits back up to complete the problem and get our answer.

Now the fact remains that you will probably never have to separate the digits of a number to figure out how much of something you need if its a simple problem like what I have had above, but the mathematical implications are where this is really interesting as you can basically take any number separate its digits out and make them different digits as long as they all add up to the original number.

Now sally forth with your acquired knowledge and post something interesting you figured out with it below.

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