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  • Why does this pattern occur: - Mathematics Stack Exchange
    Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
  • Why is $\\frac{987654321}{123456789} = 8. 0000000729?!$
    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 si
  • number theory - $12345679$ and friends - Mathematics Stack Exchange
    As an additional note, it is in fact true that the multiples of $123456789$ (note! not $12345679$ which I explained above) that are relatively prime to $9$ also result in a permutation of those digits (including 0 when you reach 10 digits):
  • Why is $\\frac{987654321}{123456789}$ almost exactly $8$?
    I just started typing some numbers in my calculator and accidentally realized that $\\frac{123456789}{987654321}=1 8$ and vice versa $\\frac{987654321}{123456789}=8 000000072900001$, so very close to
  • real numbers - Approximation for the infinite counting decimal . . .
    This number is called the Champernowne constant Some of it's approximation is $$\frac{10}{81}=0 \overline{123456790}$$ An even better approximation is $$\frac{60499999499}{490050000000}$$ Share Cite
  • Why does $987,654,321$ divided by $123,456,789 = 8$?
    Consider the product $$9\cdot123456789=1111111101 $$ This pattern is due to the fact that $9$ is one less than the basis of the numeration so that the products with individual digits (from the right $81,72,63,54\cdots$) have an increasing unit digit, while the tens digit increases
  • Sum of digits, sequence (no theory) - Mathematics Stack Exchange
    Now we use the decimal expansion of $1 81 = 0 \overline{012345679}$ to see that $10^{2012} 81 = 123456790\cdots7901234 \overline{567901234}$ (we can see in various ways that this number has $2011$ digits before the period not counting leading zeroes - resulting in $223$ repetition of $123456790$ and then three digits more which must be $123
  • Mathematical Intuition Behind Schizophrenic Numbers?
    Nowhere in the wikipedia page, nor Darling's or Pickover's writing, is justification given for this behavior I noticed the relation between $$\sqrt{123456790}\approx11111 1111$$ $$11111 1111^2=123456789 87654321\approx123456790$$ but that doesn't explain the bizarre pattern shown I was wondering if anyone has or could point me to an explanation
  • number theory - repeating unit of 1s - Mathematics Stack Exchange
    Start with just 1 Now add in the next nine repunits, from 11 to 1,111,111,111 The ones place adds up to 9, which must be added to the previous 1 to give 10 and thus carry a 1 over to the tens place The preceding places give 123,456,789, which together with the carried 1 gives 123,456,790 So the sum of the first ten repunits is 1,234,567,900
  • How do we find a fraction with whose decimal expansion has a given . . .
    We know $\frac{1}{81}$ gives us $0 \overline{0123456790}$ How do we create a recurrent decimal with the property of repeating: $0 \overline{0123456789}$ a) Is there a method to construct such a





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