Subtraction

Subtraction is carried out in a similar fashion, except the difference between the two digits is taken and the result multiplied by 9. For example, in subtracting $13$ from $31$ we have $$ 3-1 = 2 \mbox{ and } 2 \times 9 = 18, \mbox{ so } 31-13 = 18.$$ Similarly, for $28$ and $82$ we have $$ 2-8 = -6 \mbox{ and } (-6) \times 9 = -54, \mbox{ so } 28-82 = -54.$$ It works like magic. But why?

The underlying theory

Any positive whole number $a$ which is less than $100$ is written with two digits $x,y$: $$a=10x+y.$$ Its reverse is therefore $$b=10y+x.$$ For example, if $a=82$ and $b=28,$ we have $x=8$ and $y=2:$ $$82 = 10 \times 8 +2 \mbox{ and } 28 = 10 \times 2 + 8.$$ The sum of $a$ and $b,$ then, is $$a+b = (10x+y) + (10y+x) = 11x + 11y = 11(x+y).$$ Similarly, their difference is $$a-b = (10x+y) - (10y+x) = 9x + 9y = 9(x-y).$$ The result also holds if $a$ is a number that ends in $0,$ such as $a=10.$ In this case, $b=01 = 1.$

Triple digit numbers

Does something similar work for triple digit numbers? We can use the same method as above to derive the corresponding equations. A positive whole number $a$ with the digits $xyz$ is equal to $$a = 100x + 10y+z.$$ Its reverse will therefore be $$b = 100z+10y+x.$$ For the sum we get $$a+b = (100x + 10y+z) + (100z+10y+x) = 101(x+z) + 20 y.$$ For the difference we get $$a-b = (100x + 10y+z) -(100z+10y+x) = 99(x-z).$$

The general case

Can we extend these results to numbers with more digits? In the case of a four-digit reversible number $xyzw$ the equations become $$a+b = 1001(x+w)+110(y+z)$$ and $$a-b = 999(x-w)+90(y-z).$$ For a general $n$-digit number, written as $x_1x_2x_3...x_n$ the equations are $$a+b = 10^{n-1} \left(x_1 + x_n\right) + 10^{n-2}\left(x_2+x_{n-1}\right) + 10^{n-3}\left(x_3+x_{n-2}\right) + ...+\left(x_n+x_1\right)$$ and $$a-b = 10^{n-1} \left(x_1 - x_n\right) + 10^{n-2}\left(x_2-x_{n-1}\right) + 10^{n-3}\left(x_3-x_{n-2}\right) + ...+\left(x_n-x_1\right).$$

This looks a lot more complicated. There are limits, then, as to how far we can go in finding the sum and difference of two reversible numbers this way. But that does not detract from its usefulness within those limits!

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Further reading

Find out more about reversible numbers in Michael P. Greaney's Little book of reversible numbers.

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About the author

Michael P. Greaney is a writer with particular interests in astronomy and mathematics. Apart from the Little book of reversible numbers, he has written articles for a number of astronomical magazines and was a contributing author to the book Observing and Measuring Visual Double Stars (Springer, 2012).