Simple arithmetic

• Miles Neale

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  December 11, 2014 at 9:44 am

  #1764142

  TomThomson (12/11/2014)

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  Miles Neale (12/10/2014)

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  You can in fact get three from this.

  1+1+1+1+1+1+1+1+1+1+1+1x0+1 = 3

  Here is how it works:

  1+1+1+1+1+1+1+1+1+1+1 = 11 and

  1X0 + 1 = 1 and

  11 and 1 is 12

  and 1 and 2 is 3

  🙂

  M.

  I like it. But the natural way to get 3 with the rules I think you are working to might be

  1+1+1+1+1+1+1+1+1+1 = 10 = 1 because 1+0 = 1

  1+1 = 2

  2 +1*0 = 2

  2+1 = 3

  Of course you can get all sorts of answers like that: if you were using octal notation instead of decimal you would get 5, base 7 gives you 6, base 6 gives you 3, again, and so on.

  Another interesting variant is to change the ring you are doing arithmetic in instead of collapsing numbers by adding their digits together: if for example you are looking at elements in the seven element field you will get 5, and in the 2 element field you will get 0.

  warning: all those calculations done in my head, not written down and checked properly, so the numbers may be wrong - the point is simply that there are lots of ways of changing the rules of arithmetic (as opposed to changing the syntax of the notation) to get lots of different answers.

  Miles' rule change (effectively a change so that 9+1 = 1 instead of 9+1 = 10) is one that has been the source of some amusing questions over the years, but it's far from the only possible rule change. Changing the ring used is of course a much less drastic change because it doesn't throw away any of the ability to cancel additions, ie you can still deduce that a+x = b+x implies a = b, whereas with a change like Miles' you can't (because 0+1 = 9+1 is true but 0 = 1 is false) and cancelling is not always possible.

  Thanks Tom, the natural way you presented is correct. Math in itself with the logic built-in is wonderful but also playful if you change the rules just a tad. It is like retuning the guitar and playing the altered chord pattern. It works but you have to rethink it just a little.

  M.

  Not all gray hairs are Dinosaurs!