4+1+2024=13=4

Happy "Casting Out Nines" Day!

On weekends my father, Paul, would pick me up from the Washington Heights apartment of my mother, Bunny, and he and I would head to the 168th Street subway station.

As soon as we’d gotten ourselves a seat together on the train, Dad would take out these little pieces of paper he carried with him — stamped with his name, address, and phone number on top. The train would start rumbling south, and he’d pencil in the first math problem for me to solve. Then he’d hand me the paper and the pencil, and I’d set to work.

We didn’t just love numbers — we were numbers. Hunched over my calculations, we each looked like a “6.”

Photo by Kalawin.

Dad was a math teacher — well, sometimes, when he wasn’t subtracting himself from a job by mouthing off to his principal. He never lost his position as my math teacher. And every time I solved a new problem, we celebrated it as if I’d just just won the Fields Medal.

In our math as in our communism, it was important to be absolutely correct. So I’d always check my calculations by “casting out nines.”

Basically, you start by adding all the numerals across each line of addends (such a cool word!). So in the top line just above, 3+4+5=12. But you want to get to a single digit — so here you’d go on to add the 1 and the 2, getting to 3. Then you do the same with the other two lines of addends. After which you add all those totals down — in the above case, getting to a grand total of 7.

Next you do the same with the sum. 1+9+7+8=25. But you want to keep going until you get to a single digit. So 2+5=7.

It matches — 7=7!! Each successful computation, as verified by casting out nines, served as further proof that Dad was right: about math, about revolution, about the evils of class society (with its painful divisions). It all checked out!

Only there was a catch. It turns out that casting out nines doesn’t actually prove you were correct! It only shows that you might have been correct.

Let me try to explain.

Say that — perhaps distracted by my intense desire to run up to the very front of the first car of the subway train, so I could peer out the window at the onrushing tunnel and pretend (be still, my beating heart!) to be the conductor — I messed up and came up with a sum not of 1,978 but of 1,852. Well, in this case casting out nines would still get me to 7 — even though my answer was wrong!

Yes, the sad truth is that casting out nines can tell you only one thing for sure: if the numbers don’t match, then you did something wrong. But if the numbers do match … well, maybe you’re okay.

It should have been an early lesson for me in the fallibility of systems. Because if casting out nines couldn’t be totally trusted, maybe the same should go for, say, the leadership of the the Soviet Union. But alas, I didn’t make that connection.

Okay, I honestly can’t remember what this all means.

Now you may be wondering: Why is this method called “casting out nines”? Which nines are being “cast out”?

Well, it turns out that when you’re casting out nines, anytime you run into a nine, you can just turn it into a zero.

Take the third line of addends above: 946. When I did my adding-across thing, I could have zeroed out the 9 and just added 4+6=10, then 1+0=1. So I’d get to the same result as before, while saving a precious nanosecond of calculating time.

Or in the sum, 1,978, I could “cast out” the 9 and add together the rest: 1+7+8=16, and then 1+6=7.

Now, you could nitpick and point out that by casting out nines here, I actually haven’t reduced the number of steps. But I’m hoping you won’t do that, because it wouldn’t be nice!

Dad’s pockets would be stuffed with my completed calculations by the time we arrived at 272 East 7th Street. If you add together all the numbers in that address — 2+7+2+7 — you get a sum of 18. And 1+8=9. But then if you cast out that nine, you end up with … zero. Nothing. Nada.

I know that might sound bleak — and yet I can assure you, it was anything but. Dad would frequently tell me about how the invention of zero was a transformative one for humanity. Zero is beautiful. Zero is powerful. Zero holds the place for all of us — it wraps its infinite arms around our sometimes puny-feeling dreams. Zero says, I know you were cast out, but now you are here — and believe me, you belong.

Photo by Kalawin.