Maths: why does 987654321 / 123456789 = 8.000000073

[Dan.]

[ QUOTE ]
I have worse news, sir. Try adding the numbers in ROULETTE.

1+2+3+4+...+35+36

Scary.

[/ QUOTE ]

*GASP!*

[jessica1994]

987654321/123456789 = 8.0000000729000006633900060368490...

which is very nearly 8, except for the isolated 729, 66339, 6036849 ... Now, 729 = 9^3. This isn't a coincidence, as we will see.


Explanation:
Using our prime factor finder program or built-in function, we find that:


729 = 9^3 = 9^3 * 91^0
66339 = 9^3 * 91 = 9^3 * 91^1
6036849 = 9^3 * 91^2 = 9^3 * 91^2

so we can conjecture that:


987654321/123456789 = 8 + 9^3*1E-10*SUM[N=0,N=INFINITE,(91*1E-10)^N]

where the sum goes from N=0 to N=infinite. This can be proved very easily using the well-known summation formula for geometric progressions, as this happens to be one.

[john kane]

thanks for all the replies and apologies for not replying sooner. the link above has certainly helped me see there is an actual pattern, rather than it being an eerie one-off.

thanks and gl.

[Duke]

[ QUOTE ]
I'd rather someone tell my why cubes add up to squares. Using geomety, not algebra.

[/ QUOTE ]

I'm not convinced that they do.

[pzhon]

[ QUOTE ]
I'd rather someone tell my why cubes add up to squares. Using geomety, not algebra.

[/ QUOTE ]
(1+2+...+n)^2 = 1^3 + 2^3 + .. + n^3.

Proof without Words: Sum of Cubes
Alan L. Fry
Mathematics Magazine, Vol. 58, No. 1 (Jan., 1985), p. 11

I don't have that in front of me, but typically these proofs are inductive. You find some way to illustrate why n^3 = (1+2+...+n)^2-(1+2+...+(n-1))^2. If you fit the smaller square inside the larger, the remainder is an L shape, the intersection of two strips that are n x (1+2+..+n), or a strip that is n x (1+2+...+n) plus a strip that is n x (1+2+...+(n-1)). Break the strips into subrectangles indicated by the sum and pair n x k and n x (n-k), forming n pairs (pair n x n with n x 0) fitting into nxn squares.

[MelchyBeau]

[ QUOTE ]
[ QUOTE ]
I'd rather someone tell my why cubes add up to squares. Using geomety, not algebra.

[/ QUOTE ]

I'm not convinced that they do.

[/ QUOTE ]

he is talking about sum(i^3,i=1,k)=(sum(i,i=1,k))^2

[CORed]

[ QUOTE ]
Uhm.. Because 123456789 * 8 is almost equal to 987654321? Not trying to sound smug but that's all there is to it I think [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

123456789 * 8 = 987654312, to be exact.

[GoodCallYouWin]

Someone is clearly trying to send us messages through the fabric of spacetime itself.