If a student wrote down 1+1=10, most of us would mark it wrong. However, there is the possibility that the student could be right, if he is writing the answer in base-2. In fact, for any two digits, a+b=10 would be a mathematically correct answer, just in base-(a+b). So, 9+9=10 in base-18 and 2+2=10 in base-4.
What about speaking in more general terms?
23+47=72 in base-8
Changing the base makes this simple addition problem look like solving a linear polynomial.
[2(x)+3]+[4(x)+7]=[7(x)+2]
6x+10=7x+2
So x = 8 (makes sense, 8 is a our base.
Looking at bigger numbers, for what base is 123 + 256 = 412?
Now, you could cheat and just look at the unit digit. I notice that 3+6=9 on the left side of the equations, and from the right side I notice that 2 is congruent mod 7 to 9, therefore 7 must be our base. But, if I was not so crafty, I could consider the equation [x^2+2x+3]+[2x^2+5x+6]=[4x^2+x+2]
That works down to 3x^2+7x+9=4x^2+x+2
or 0=x^2-6x-7
So, 0=(x-7)(x+1)
Since we can't have a negative base we will ignore x=-1, Therefore x = 7.
So, in base-7, 123 + 256 = 412.
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