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Subtraction Without Borrowing

Subtraction Method
"Borrow" vs. "Shift"

Most can remember subtraction using the classic "borrowing" method, such as,
 
63 - 28 = (50 + 13) - (20 + 8) = (50 - 20) + (13 - 8)* = 35.

Here we "borrowed" from the 60, adding 10 to the 3 for 13, and leaving 50 behind to subtract the 20 from. This all looks more familiar if one writes it out in the traditonal way, vertically, crossing out the 6, writing the 1 by the 3, etc. You know, "borrowing."

      13
  5  63
-    28
     35

This is a way to do this, but not the only one. We also can change the problem a bit, and into one that gives the same answer. We do this by "shifting" the numbers an equal amount, in this case by adding 2 to both. Here it is.

63 - 28 = (63 + 2) - (28 + 2) = 65 - 30 = 35.

Or vertically,

  63 ---- "+2" ------>  65
- 28 ---- "+2" -----> - 30
 ???                           35

That's it. One might ask how we can decide to choose 2 as the "shift" size here. It's because that's the amount that gets 28 to 30, a nicer number to subtract.
 
Here it is illustrated on a number line.
---------20---------40---------60---------80---------
                  28                        63
                    30                        65

*Note:  There's an unsubtle difference (pun intended, of course) in the subtraction involved here, as "borrowing" only results in the same problem repeating itself. Try it out. But "shifting" by +2 makes quick work of it.

13 - 8 = 15 - 10 = 5.

---10---20---30---
   8 13
   10 15
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