987654321 / 123456789

Posted by ColinWright 10/26/2025

987654321 / 123456789(www.johndcook.com)

637 points | 107 commentspage 2

trotro 10/30/2025|

Feels like a Temu version of Ramanujan's constant [0].

[0] https://mathworld.wolfram.com/RamanujanConstant.html

qnleigh 10/31/2025||

> Why include a script rather than a proof? One reason is that the proof is straight-forward but tedious and the script is compact.

Yes the script lets you check that the result is correct, but a proof lets you see why it's correct. A good proof might even give you a sense of how you could have discovered the result yourself, or how you might generalize it.

ok123456 10/30/2025||

For the even bases, the "error" appears to be https://oeis.org/A051848.

pp = lambda x : denom(x)/ (num(x) - denom(x)*(x - 2))

[pp(2),pp(4),pp(6),pp(8)]

[1.0, 9.0, 373.0, 48913.0]

madcaptenor 10/30/2025|

And if you see the description there it traces back to https://oeis.org/A023811, which is more obviously relevant

danielbarla 10/30/2025||

I also spent hours messing around with calculators as a kid. I recall noticing that:

11 * 11 = 121

111 * 111 = 12321

1111 * 1111 = 1234321

and so on, where the largest digit in the answer is the number of digits in the multiplicands.

ozb 10/30/2025||

More general analytic proof: https://math.stackexchange.com/questions/2268833/why-is-frac...

ukuina 10/30/2025|

That question was asked 8 years ago. Coincidence? I think not!

adrian_b 10/31/2025||

As a young child, a half of century ago, when I have received an electronic pocket calculator (with 8-digit numbers and without transcendental functions) I was taught that I can do a quick check whether it functions correctly by multiplying 12345679 with 8 (using thus all non-null digits), when the result must be 98765432. Obviously, an additional check is the corresponding division that reverses this operation.

gowld 10/31/2025|

Did you ever encounter a dysfunctional calculator?

That test wouldn't detect a dead left side on the 2nd from-right digit

adrian_b 10/31/2025||

Obviously, that was not intended to be a full-functionality test, but it would detect any frequently-encountered display defect (or keyboard defect).

Calculator displays are multiplexed, so the usual defects are either one digit that never displays anything, or one segment that stays blank on all digits.

The defect mentioned by you is frequent only on displays with independent digits (like some digital clocks), not on calculators.

I do not know whether on calculator LCD displays there are frequent cases when a single segment can become defect.

At the time about which I am talking, calculators had either green vacuum fluorescent displays (like mine) or red LED displays. With such displays, the normal defects were either in the driving circuits or in the connections to the multiplexed display, so they affected either all segments of a digit or the same segment in all digits. I have never seen a case when the actual light-emitting segment of a digit of a VFD or LED display was defect.

veganjay 10/30/2025||

Reminds me of an old calculator trick:

Pick an integer between 1 and 9. Multiple it by 9. Take that number and multiply it by 12345679. (Skip the 8)

>>> 3 * 9

>>> 12345679 * 27

333333333

This all works because:

>>> 111111111 / 9

12345679.0

usr1106 10/31/2025||

Sounds interesting, but site is behind the great Cloudflare firewall. Cannot see it on this phone.

yohbho 10/30/2025|

For smaller bases, does this converge to base - 1 ?

Base 3: 21/12 = 7/5(dec.)

Base 2: 1/1 = 1

Base 1: |/| = 1 (thinking |||| = 4 etc.)

incredible1 10/30/2025|

[dead]

More comments...