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987654321 / 123456789

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Re: 987654321 / 123456789

#101

Post by _Microft »

a13n wrote: post #96
tetris11 wrote: post #32

I like calculator quirks like this. I remember as a kid playing with the number pad and noticing a geometric center of mass in number sequences ┌───┬───┬───┐ │ 7 │ 8 │ 9 │ ├───┼───┼───┤ │ 4 │ 5 │ 6 │ ├───┼───┼───┤ │ 1 │ 2 │ 3 │ ├───┼───┼───┤ │ 0 │ . │ │ └───┴───┴───┘ I remember seeing that (14787 + 36989) / 2 would produce 25888, in that the mean of geometric shape traced by the two sequences would average out in the…

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Re: 987654321 / 123456789

#102

Post by sockaddr »

TheTon wrote: post #80
jedberg wrote: post #44

This was by far the most interesting part to me. I've never considered that code and proofs can be so complementary. It would be great if someone did this for all math proofs! "Why include a script rather than a proof? One reason is that the proof is straight-forward but tedious and the script is compact. A more general reason that I give computational demonstrations of theorems is that programs are complementary to…

As a kid, I was marginally decent at competitive math. Not good like you think of kids who dominate those type of competitions at a high level, but like I could qualify for the state competition type good. What I was actually good, or at least fast at, was TI-Basic, which was allowed in a lot of cases (though not all). Usually the problems were set up so you couldn’t find the solution using just the calculator, but i…

I used to do the same thing. I'd scan for problems on the test amenable to computational approaches and either pull up one of my custom made programs or write one on the spot and let it churn in the background for a bit while I worked on other stuff without the calculator.

Re: 987654321 / 123456789

#103

Post by adrian_b »

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.

Re: 987654321 / 123456789

#104

Post by debtta »

Joker_vD wrote: post #15

Somewhat interesting, 123456789 * 8 is 987654312 (the last two digits are swapped). This holds for other bases as well: 0x123456789ABCDEF * 14 is 0xFEDCBA987654312. Also, adding 123456789 to itself eight times on an abacus is a nice exercise, and it's easy to visually control the end result.

On an 8 digit calculator the common variant of this was

    12345679 * 8 = 98765432

Re: 987654321 / 123456789

#105

Post by BoorishBears »

_Microft wrote: post #101
a13n wrote: post #96

Earlier quoted context omitted.

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Re: 987654321 / 123456789

#106

Post by debtta »

gus_massa wrote: post #62

The other replies are good, but let's add another one anyway. 0.987654321/0.123456789 = (1.11111111-x)/x = 1.11111111/x - 1 where x = 0.123456789 You can aproxímate 1.11111111 by 10/9 and aproxímate x = 0.123456789 using y = 0.123456789ABCD... = 0.123456789(10)(11)(12)(13)... that is a number in base 10 that is not written correctly and has digits that are greater than 9. I.E. y = sum_i>0 i/10^i Now you can consider…

An easier way to evaluate sum i/10^i is by squaring sum 1/10^i

If you multiply term by term every term has coefficient 1 of course. There are n terms with exponent n+1, made from the n sums of the first exponent and the second exponent.

Eg 1+5, 2+4, 3+3, 4+2, 5+1.

So (1/9)^2 = (sum 1/10^i)^2 = 1/10 sum i/10^i

The derivative trick is more useful generally, but this method gets you the solution to 0.12345678.. in an quick way that's also easier to justify that it works.

Re: 987654321 / 123456789

#107

Post by qnleigh »

> 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.

Re: 987654321 / 123456789

#108

Post by ghoul2 »

kapitar wrote: post #94
deepsun wrote: post #88

Earlier quoted context omitted.

That would work in any base, I even think we would find way more interesting coincidences in base 12 (as Sumerians preferred), because it's divisible by 2,3,4,6. It's unfortunate that we have 5 fingers.

If you count the sections of your four fingers with your thumb, you can count up to 12 on one hand!

I have always counted to 20 on one hand. even as a kid. base, lower joint, upper joint, top. times 5 - including the thumb: my motor memory is trained so that i switch seamlessly from keeping the curse on top of the finger using my thumb, and then, once i cross 16, switch to using the index finger to "cursor" the thumb.

Re: 987654321 / 123456789

#109

Post by susam »

ghoul2 wrote: post #108
kapitar wrote: post #94

Earlier quoted context omitted.

If you count the sections of your four fingers with your thumb, you can count up to 12 on one hand!

I have always counted to 20 on one hand. even as a kid. base, lower joint, upper joint, top. times 5 - including the thumb: my motor memory is trained so that i switch seamlessly from keeping the curse on top of the finger using my thumb, and then, once i cross 16, switch to using the index finger to "cursor" the thumb.

Same here. I have always counted 20 on one hand, so 40 with both. That's how my parents taught me to count when I was little. I used this method so often as a kid that, even though I don't count like this anymore, every number up to 40 still has its own place on my fingers.

It was only as an adult that I realised nobody around me counted this way. You are the first person I have found who talked about this method, so I am glad to find this comment of yours.

Re: 987654321 / 123456789

#110

Post by firefax »

tetris11 wrote: post #32

I like calculator quirks like this. I remember as a kid playing with the number pad and noticing a geometric center of mass in number sequences ┌───┬───┬───┐ │ 7 │ 8 │ 9 │ ├───┼───┼───┤ │ 4 │ 5 │ 6 │ ├───┼───┼───┤ │ 1 │ 2 │ 3 │ ├───┼───┼───┤ │ 0 │ . │ │ └───┴───┴───┘ I remember seeing that (14787 + 36989) / 2 would produce 25888, in that the mean of geometric shape traced by the two sequences would average out in the…

now there's some solid ascii, great work sir
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