Why is $\frac{987654321}{123456789} = 8.0000000729?!$

Question

Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$.

I sent it in to Martin Gardner at Scientific American and he published it in his column!!!

My life has gone downhill since then:)

My questions are:

• Why is this so?

• What happens beyond the "$729$"?

• What happens in bases other than $10$?

Answer 1

In base $n$ the numerator is $$p = n^{n-1} - \frac{n^{n-1}-1}{(n-1)^2}$$ and the denominator is $$q = \frac{n(n^{n-1}-1)}{(n-1)^2}-1.$$

Note that $p = (n-2)q + n-1$ and for the quotient we get

\begin{align} \frac{p}{q} &= n-2 + \frac{(n-1)^3}{n^n} \frac{1}{1 - \frac{n^2-n+1}{n^n}} \\ &= n-2 + \frac{(n-1)^3}{n^n} \sum_{k=0}^{\infty} \left(\frac{n^2-n+1}{n^n}\right)^k. \end{align}

Indeed for $n=10$ this is

$$\frac{987654321}{123456789} = 8 + \frac{729}{10^{10}}\sum_{k=0}^{\infty}\left(\frac{91}{10^{10}}\right)^k $$

Answer 2

$$729=9^3$$ $$66339=9^3\cdot 91$$ $$6036849=9^3\cdot 91^2$$ $$...$$ $$987654321/123456789=8+9^3\cdot 10^{-10}\cdot\displaystyle\sum_{n=0}^{\infty}(91\cdot 10^{-10})^n$$

Answer 3

Let $$S_n(a)=1 +2a+\ldots +na^{n-1}=\frac{na^{n+1}-(n+1)a^n+1}{(a-1)^2},$$ $$T_n(a)=a^{n-1}+2a^{n-2}+\ldots +n=a^{n-1}S_n(a^{-1}).$$

Then $$ \frac{S_n(a)}{T_n(a)}=\frac{na^{n+1}-(n+1)a^n+1}{a^{n+1}-(n+1)a+n}.$$ For $a=10,n=9$ we have $$ \frac{S_n(a)}{T_n(a)}\approx\frac{8\cdot 10^{10}+1}{10^{10}}. $$

Answer 4

Just to add to the excellent answers above, some examples:

${987654321\,/\,123456789}\approx 8.00000007290000066339$

${{87654321}_9\,/\,{12345678}_9}\approx {7.000000628000056238}_9$

${{7654321}_8\,/\,{1234567}_8}\approx {6.0000052700046137}_8$

${{654321}_7\,/\,{123456}_7}\approx {5.00004260036036}_7$

${{\mathrm{fedcba987654321}}_{16}\,/\,{\mathrm{123456789abcdef}}_{16}}\approx {\mathrm{e.0000000000000d2f00000000000c693f}}_{16}$

Answer 5

$98765432 / 12345679 = 8$, exactly. You can see how the pattern works by multiplying out $12345679 * 8$ starting at either end.

This explains why your fraction is close to an integer. If you think the $729$ is interesting (I don't), it can be explained by some of the other answers here.

Edit:

What can we say about the fact that $12345679 * 8 = 98765432$? I have been aware of this 'factlet' for about 20 years, and remember it being used to 'demonstrate' calculators (which often had 8 digit displays back in the day).

I just recently realised that:

$$ \frac{1}{81} = \left(\frac{1}{9}\right)^2 = \left(\sum_{k=1}^{\infty}\frac{1}{10^k}\right)^2 = \sum_{k=1}^{\infty} \sum_{m=1}^{k-1} \frac{1}{10^m} \frac{1}{10^{k-m}} = \sum_{k=1}^{\infty} \frac{k-1}{10^k} $$

In other words, while $\frac{1}{9} = 0.1111111\ldots$ $$ \frac{1}{81} = 0.01 + 0.002 + 0.0003 + 0.0004 + 0.00005 \ldots $$

It is pretty easy to see that this infinite sum is going to converge to something starting $0.012345\ldots$. If you keep on adding, or work out $\frac{1}{81}$ by division, you get $$ 0.012345679012345679012345679\ldots $$ When you get to the point where you add $\frac{10}{10^{11}}$, the first carry happens, which leads to the 9 where you might expect an 8. After that every addition carries and the decimal expansion repeats every 9 digits (not every ten - because the amount we carry keeps on getting bigger and bigger).

Now, $\frac{8}{81} = \frac{9}{81} - \frac{1}{81}$, or $$ \frac{8}{81} = 0.11111111\ldots - 0.012345679012345\ldots $$ Think of each '1' digit in $0.111\ldots$ as being a '10' in the next column. This means that we can work out $\frac{8}{81}$ as the "10's complement" of $\frac{1}{81}$, since we are subtracting a digit between $1$ and $9$ from $10$, to get another single digit which appears in the same place. So $\frac{8}{81}$ starts $0.098765\ldots$. The only break in the pattern is when you get to the digit '0' - subtracting 0 from 10 leaves you with 10, or a '1' in the next digit on the left, changing the 1 to a 2.

So $$ \frac{8}{81} = 0.098765432098765432098765\ldots $$

and therefore $$ 0.0123456790123456790\ldots * 8 = 0.0987654320987654320\ldots $$ and clearly this gets you that $$ 12345679 * 8 = 98765432 $$

Answer 6

I think @BorisNovikov's answer is the best here, but I had to do some work to understand it; this is my attempt at clarifying his answer, hopefully this can help others too:

In the following, we adopt the notation $123456789 = [ 1, 2, ..., B-1 ]_B$ in base $B=10$.

Boris introduced the following sum: $$ S_d(B) = [d, d-1, ..., 1]_B = 1 + 2B + ... + dB^{d-1} = \sum_{k=1}^d kB^{k-1} = \frac{dB^{d+1} - (d+1)B^d + 1}{(B-1)^2} $$ where $d$ is the number of digits, $B$ the base, and the final formula is obtained by derivation of the usual geometric sum.

From there, it is easy to see that: $$ [1, 2, ..., d]_B = d + (d-1)B + ... + 1B^{d-1} = B^{d-1} S_d(1/B) $$ and putting both results together, we have: $$ \frac{ [B-1, B-2, ..., 1]_B }{ [1, 2, ..., B-1]_B } = \frac{ S_{B-1}(B) }{ B^{B-2} S_{B-1}(1/B) } = \frac{1 + (B-2)B^B}{B^B - B(B-1) - 1} $$

Clearly both the numerator and denominator are dominated by the terms in $B^B$, so this fraction is equivalent to $B-2$ as $B$ tends to infinity, and we can see that the exact value is very close already for $B=10$ (see also @Mark Adler's post for other examples). Beautiful, isn't it? :)

Answer 7

I think one should also think about how this is related to 9 being before 10. Notice the two following striking facts:

$12345679*8=98765432$ and $12345679*9=111111111$

Answer 8

An identity:
$[B-1,B-2,...,1]_B - 1 = $ $=(B-2) ([1,2,...,B-1]_B+1)$
A way to show this general identity goes as follow (we shall use base B and exactly B ciphers):

1. $[0,B-1,B-2,...,2,1]+[0,1,2,...,B-2,B-1]=[1,1,...1,1,0]$
2. $[1,1,...,1,0]+[0,1,2,...,B-1]=[1,2,...,B-1,B-1]$
   So that:
   $[0,B-1,B-2,...,1]+2[0,1,2,...,B-1]=[1,2,...,B-1,0]+B-1=B[0,1,2,...,B-1]+B-1$
   Then in terms of B-1 ciphers numbers: $[B-1,B-2,...,1]=(B-2)[1,2,...,B-1]+B-1$
   and: $$[B-1,B-2,...,1]-1=(B-2)[1,2,...,B-1]+B-2=$$ $$=(B-2)([1,2,...,B-1]+1)$$ So that the division proposed to Martin Gardner, by you, was very close to $\frac{[B-1,B-2,...,1]-1}{[1,2,...,B-1]+1}=B-2$ for B = 10. In fact, posing $N=[0,B-1,...,1]$ and $D=[0,1,...,B-1]$, we get: $N/D-(B-2) = N/D - (N-1)/(D+1)=$ $=[N(D+1)-(N-1)D]/[D(D-1)]=(2D+N)/[D(D-1)]$ where the numerator has B ciphers and the denominator 2(B-1) ciphers, and the quotient will be a number with $B-3$ null ciphers after the decimal separator sign.