BS: Who else plays little # games?

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 31 Aug 13 - 05:54 AM

OK, that's a proof but not quite what I had in mind. Perhaps I should have said something like demonstration.

Found it!

1/9 = 0.1111....
9 * 1/9 = 9 * 0.1111....
1 = 0.9999....

Subject: RE: BS: Who else plays little # games?
From: DMcG
Date: 31 Aug 13 - 05:41 AM

The proof follows from the definition I gave for equality, where we look at the difference between the two numbers. In this case
we want the difference between 1 and 0.9999recurring. Now, that recurring is a bit tricky, so let's take a series of finite approximations

1-.9 = 0.1
1-.99 = 0.01
1-.999 = 0.001
1-.9999 = 0.0001

You can see where this is going, I hope. For any epsilon you care to pick, let's say, 0.00000000000000000000000000000000001, or one with a hundred million leading zeros before the '1', I can produce a term in this series where the difference is smaller, because the 'recurring' means I can write as many nines as I like, even if it is a hundred million and one. And since that's my definition of equality, the two representations denote the same number.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 31 Aug 13 - 05:26 AM

DMcG

1 is exactly the same number, mathematically speaking, as 0.9999999999 recurring.

I know, but in the context of this thread which has a range of countributors from MtheGM who describes himself as "entirely unmathematical" to people such as Airymouse and your good self who are clearly highly mathematiccally educated (I'm somewhere inbetween) I think it has a certain "Well I never" factor.

I have seen a proof of 1 = 0.999 recurring but can't remember it. Do you have it to hand?

Subject: RE: BS: Who else plays little # games?
From: DMcG
Date: 31 Aug 13 - 05:03 AM

That's what comes of trying to translate maths into English. When I say 'for any number, epsilon', read 'for every number, epsilon'.

But since I've been obliged to repost, I'll take the chance to explain a little more. '1', '0.999 recurring', etc, are not, strictly speaking, numbers, but representations of numbers. Every (rational) number has lots of different representations: In fractional notation we could have 1/2, 2/4, 3/6 and so on, all representing the same number. We are used to that, so we are taught in schools to always use the simplest form, but all the others are equally valid. And it is the same in decimal notation: every rational number has two representations, and we always pick the simplest. So we normally write 0.5 rather than 0.499999recurring, but both represent exactly the same number in the way 1/2, 2/4 do.

Subject: RE: BS: Who else plays little # games?
From: DMcG
Date: 31 Aug 13 - 04:41 AM

Ah, but, O Molluscy one, 1 is exactly the same number, mathematically speaking, as 0.9999999999 recurring. I know it seems odd to 'outsiders', but that's how it is.    It's to do with the mathematical definition of equality. Now, I know it might seem that equality is easier to define than 'less than', but it happens that to mathematicians 'less than' is the more important relationship. And equality on numbers is defined like this.

Two numbers are equal if for any number, epsilon, that you care to pick, however small, the difference in the two numbers is less than epsilon.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 31 Aug 13 - 04:33 AM

Thanks for that Airymouse. Adds things I never got round to finding out epecially that the next number is 17. Your method of generating the second three digits from the first three has interesting consequences. 1/7 + 6/7 = 7/7 = 1 but try adding 0.142857* and 0.857142*.
I'm a bit lost by your bit about multiplying by a number larger than 7. Surely, for instance, 8 * 1/7 is 1 + 1/7 so the decimal part behaves just as before but the integer part can be anything.

Subject: RE: BS: Who else plays little # games?
From: MGM·Lion
Date: 31 Aug 13 - 02:40 AM

I have always liked, as wikipedia tells it, the

'famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan [1887-1920]. In Hardy's words:
"         I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."'

What are the two ways? Work them out. Even I did.

~M~

Subject: RE: BS: Who else plays little # games?
From: Rapparee
Date: 31 Aug 13 - 12:09 AM

License plates are something that drives me nuts. For example, I'll see one with a number like, oh, 4679. Right off I think, "That's a neat number becase 4 and 6 are even and 7 and 9 are odd and they come in a sort of sequence. Or when I got to choose my cell phone number I chose 1861 because that was the year the US Civil War started and I'd remember it. My wife got 4903 by the luck of the draw, and that's just two numbers off from our postal box number. Or I'll mentally add all of the numbers together (3496 might get 3+4+9+6=22=2+2=4 and square root of 4 is 2 but the cube root of 4 (first number is 3) is about 1.6 when I do it by multiplying numbers until I get the closest to 4 (it's really about 1.5874010519681994 -- I looked it up).

Or to go to sleep I'll mentally calculate the decimal answer to a problem like 35/17, which fractionally is 2 1/17th, but I have to arithmetically calculate the decimal equivalent of the fraction (1/17 x 100/1 = 100/17 = about slightly less than 6. Usually by then I'm asleep, dreaming of Fourier transformations or something. Try it yourself -- mental math can put you right to sleep, just as it did in grammar school.

Subject: RE: BS: Who else plays little # games?
From: Airymouse
Date: 30 Aug 13 - 11:09 PM

Snail's 1/7 2/7 etc. is deeper than it looks: After you know the first 3 digits 142 you get the next three digits by subtracting from 9 (e,g. 8=9-1, 5=9-4, etc. and multiplication merely rotates the digits unless you multiply by a multiple of 7. (142857)(3) = 428571. If you multiply by a number larger than 7 the number still rotates, but you have to help by joining a few digits by adding e.g., (142857)(23)=3285711 and you have to add the first and last digits to see the rotation. What makes this work? The key is that when you divide 7 into 1 by long division it repeats after 7-1 digits, because you get all the possible nonzero remainders (1,2,3,4,5 and 6). Are there other such numbers (like 7, which yields .142857...)? Yes and they are all prime numbers The next one is 17
1/17   repeats after 16 digits. Most pocket calculators show only the first 8 digits 0588235 but 9-0 =9 etc so the full 16 digits are .0588235294117647. Are there infinitely many such numbers? This is an unsolved problem. The statement is true if Artin's conjecture is true.

Subject: RE: BS: Who else plays little # games?
From: Rapparee
Date: 30 Aug 13 - 09:41 PM

My current license plate is 1B S5762.

In 1957 I was starting seventh grade. In 1962 I was starting my last year of high school. The 1BS is self-explanatory and, according to some people, very appropriate.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 30 Aug 13 - 07:55 PM

I raher like this (but it's easier to check it with a calculator) -

1/7 = 0.142857*
2/7 = 0.285714*
3/7 = 0.428571*
4/7 = 0.571428*
5/7 = 0.714285*
6/7 = 0.857142*

where * means recurring. The six digits repeat forever.

Subject: RE: BS: Who else plays little # games?
From: DMcG
Date: 30 Aug 13 - 12:28 PM

And one thing I pointed out to my daughter recently was that, short of a major improvement in life maintenance, she is just about to pass through the last of the only three ages with a specific property:

1-to-the-power-1 => 1
2-to-the-power-2 => 4
3-to-the-power-3 => 27
4-to-the-power-4 => ... unlikely!

Subject: RE: BS: Who else plays little # games?
From: DMcG
Date: 30 Aug 13 - 12:24 PM

There's a whole bunch of 'magic' tricks that rely on that property of nine (or, for the really obscure ones, the digit N in numbers of base (N+1)) In fact, if I remember correctly, Martin Gardner had a chapter on that in his book 'Mathematical Puzzles and Diversions' (Or perhaps his other book 'More Mathematical ...') One concerned a sort of telephone dial with weird symbols, but they were just flummery to hide the numerical basics.

Subject: RE: BS: Who else plays little # games?
From: Nigel Parsons
Date: 30 Aug 13 - 08:28 AM

Any multiple of 9 will have digits which also add up to a multiple of 9. And will the same thing happen by adding the digits of 6561²? No -- their sum appeared to be 27 and the digits of 27, 2+7=9
This is an extension of the fact that any multiple of 3 has digits which add up to a multiple of 3.

Subject: RE: BS: Who else plays little # games?
From: MGM·Lion
Date: 30 Aug 13 - 06:31 AM

Thank you BWL. This is now getting a bit out of my league: but I note that the number you produce is the square of 'the number I first thought of', as the catch-problems used to say.. Is that why the phenomenon of the added digits repeats. And will the same thing happen by adding the digits of 6561²? No -- their sum appeared to be 27: the cube of the square root of 18/2 Now, where does that fit into the relationship?

See what I mean about not being any sort of mathematician. One of those would presumably have extrapolated some general principle from all this.

But not me tho... LoL

Subject: RE: BS: Who else plays little # games?
From: Bee-dubya-ell
Date: 30 Aug 13 - 06:14 AM

81 squared is 6561. 6+5+6+1=18. Rotate your computer screen 180° and... 81!

Subject: BS: Who else plays little # games?
From: MGM·Lion
Date: 30 Aug 13 - 02:44 AM

Although entirely unmathematical, (a great regret to me but there it is), I like to play little games with numbers. I enjoy reading of the number games and discoveries by Charles Dodgson [aka Lewis Carroll] recorded in such places as Martin Gardner's "The Annotated Alice" - but, then, Dodgson was a distinguished Oxford mathematician.

E.g. it occurred to me yesterday that my present age, 81, is a nice number, because it is a square. Moreover, its digits add up to the number that it is the square of. Do any other squares do that?

And does anyone else find such things of as much interest and entertainment as I do? Or have any examples to contribute?

~Michael~