Maths: why does 987654321 / 123456789 = 8.000000073

[john kane]

i have spent many time at work typing into my calculator stuff like this. anyone know why if you do 987654321 divided by 123456789 it comes to such an eerie number, so close to 8, but not quite?

[Styhn]

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

[felson]

because (http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-042JSpring-2005/CourseHome/index.htm)

[felson]

eh, i'll explain a little. if you compare successive lines in the picture, you can see what is happening. it's a consequence of the fact that eight is two less than ten, the base of the decimal system. you would get a similar pattern with 6 in octal, or 14 in hexadecimal.

[thylacine]

[ QUOTE ]
i have spent many time at work typing into my calculator stuff like this. anyone know why if you do 987654321 divided by 123456789 it comes to such an eerie number, so close to 8, but not quite?

[/ QUOTE ]

Consider 10/81 and 80/81 in decimal.

Also consider 1/9 in decimal, then square it.

[eviljeff]

I love this thread

[mbillie1]

because numbers are rigged, for more evidence see: fulltiltpoker.com

[vetiver]

[ QUOTE ]
because numbers are rigged, for more evidence see: fulltiltpoker.com

[/ QUOTE ]

Why are people trying to explain this with hexadecimals when this quote explains it entirely?

[David Sklansky]

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

[mrick]

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

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

Scary.

[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 /images/graemlins/smile.gif

[/ QUOTE ]

123456789 * 8 = 987654312, to be exact.

[GoodCallYouWin]

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