Irrational Numbers

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Cezanne123

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Sep 26, 2006, 7:50:37 PM9/26/06
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Stuck on a little problem...

Prove that log(2)-base10 is irrational, and conclude that there exist infinitely many positive integers k for which the first 11 digits of the base 10 expression of 2^k (counting digits from the left) are 77777777777.

How do you show this is true?

Daniel Mayost

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Sep 26, 2006, 8:23:01 PM9/26/06
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For the first part:

Suppose that log_10 2 = a/b, where a and b are integers. Then we have:

10^a = 2^b

But this can't be, because the left side is a multiple of 5 while the right
side is not.

Now for the second part:

Since log_10 2 is irrational, given any two numbers x, y in the interval
(0,1) with x<y, there exist infinitely many k for which the decimal part
of k log_10 2 lies between x and y. So take x to be the deimal part of
log_10 77777777777, and take y to be the decimal part of
log_10 77777777778. Any k for which k log_10 2 lies between x and y
will have 77777777777 as its first eleven digits.

--
Daniel Mayost


In article <4008838.11593146676...@nitrogen.mathforum.org>,

Dave L. Renfro

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Sep 26, 2006, 9:34:08 PM9/26/06
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Cezanne123 wrote:

> Prove that log(2)-base10 is irrational, and conclude that
> there exist infinitely many positive integers k for which
> the first 11 digits of the base 10 expression of 2^k
> (counting digits from the left) are 77777777777.

Daniel Mayost has already posted a proof, so I thought
I'd point out some interesting extensions that might
not be well known (at least, aside from experts in
Diophantine approximation). Macon/Moser (full reference
below) give fairly elementary proofs for the following
results:

1. Every finite sequence of decimal digits appears as
the first digits of some power of 2.

2. Every finite sequence of decimal digits appears as
the first digits of infinitely many powers of 2.

3. Let k and n be positive integers. Let N(k,n) be the
number of powers of 2 that are less than 2^n and whose
first digits are identical to the digits of k. Then
LIM(n --> oo) N(k,n)/n is equal to (log_10)[(k+1)/k].

Since (log_10)[(k+1)/k] = ln(1 + 1/k) / ln(10), it follows
that this limit is approximately equal to (0.4343)/k.

4. Let m be a positive integer that is not a power of 10.
Then each of #1, 2, 3 holds for powers of m in place
of powers of 2.

Note that the limit for the corresponding version of #3
is independent of m.

They also formulate some simultaneous first digit results
for powers of a fixed finite set of positive integers,
such as if k and k' are given positive integers, then there
are infinitely many values of n (in fact, a positive relative
frequency of such n's, analogous to #3 above) such that the
first digits of 2^n is k AND the first digits of 3^n is k'.

Nathaniel Macon and Leo Moser, "On the distribution of first
digits of powers", Scripta Mathematica 16 (1950), 290-292.

Dave L. Renfro

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