## A Short, Short Discourse on Digit Sum

Posted by mark schwartz on January 16, 2017

__Introduction__

My daughter and I play with numerology; it’s just play, nothing serious. We play with calendar dates, prime numbers, birthdays, etc., looking for patterns and such. Most recently she texted me that she added the digits in my wife’s birthday and continued to add them until there was a single digit, the result was 4. She wasn’t aware of digit sum nor casting-out-9s. So, I played back.

__The Story__

I first took a look through some of my old math texts, dating from about 1850 to 1900. I looked there because in those days, having a way to check your work was important and digit sum was popular, but not noted in all texts. What I don’t know is whether instructors may have taught it although it wasn’t in the text. I did find a few (I have about 75 old texts) that actually demonstrated how addition can be checked using digit sums. Oddly though, none of those that presented how to check addition indicated that digit sum can be used to check subtraction, multiplication and division as well. Yes, it can.

But, let’s take a look at why digit sum works. It’s based on what is called casting-out-9s. In essence, given a number – 23 – if you cast out 9s (which can be done by subtracting 9 until you have a single digit), you get 23 – 9 = 14, then 14 – 9 = 5. Notice that if you simply added the 2 and 3 in the number 23, you also get 5, so what simplifies getting a digit sum is simply add the digits repeatedly until you get a single digit. For the number 268, at first you get 16, then 7.

Why does this work? Let’s get basic. Using 23 again, this is really 2(10) + 3(1). Rewriting this in what I call ‘slow motion’ math, it becomes 1(10) + 1(10) + 3(1), then 1(9) + 1(1) + 1(9) + 1(1) + 3(1). If the ‘1(9)s’ are now ‘cast out’, the result is 1(1) + 1(1) + 3(1), giving 5(1).

This of course is not a rigid proof but rather a demonstration of casting-out-9s.

For example, 235 + 568 = 803. The digit sum for 235 is 1; the digit sum for 568 is 1, and the sum of these is 2. This equals the digit sum of 803, so it checks. I realize that in today’s technical world, this procedure isn’t likely to be taught nor used but in olden days without calculators, it seemed reasonable to check your work.

Now back to what I sent back to my daughter. I generated the digit sum for the birthdays for all the members of our family and then generated the digit sum for the sum of them and lo and behold the result was 1! Of course, the family is unity!

Told you it was a short discourse, but couldn’t resist sharing it. Can’t wait to see what she comes up with next.

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