Show that these two numbers have the same number of digits I want to show that for n > 0

Rachel Ross

Rachel Ross

Answered question

2022-06-02

Show that these two numbers have the same number of digits
I want to show that for n > 0, 2 n and 2 n + 1 have the same number of digits.
What I did was I found that the formula for the number of digits of a number x is log 10 ( x ) + 1, so basically if I subtract that formula with x = 2 n with the formula with x = 2 n + 1, I should get zero.
log 10 ( 2 n ) + 1 ( log 10 ( 2 n + 1 ) + 1 ) = log 10 ( 2 n ) log 10 ( 2 n + 1 )
At this point, I don't know of a way to simplify this any further to make it equal 0. I thought about mentioning that log 10 ( x ) increases slower than x as x increases, which would mean the difference of the floor of the logs of two consecutive numbers may be close to zero, but that doesn't cut it to prove that 2 n , 2 n + 1 have exactly the same number of digits.
Are there any special floor or log properties I could use to make this easier? Any help is appreciated.

Answer & Explanation

hatasky815jd

hatasky815jd

Beginner2022-06-03Added 4 answers

Note the only way 2 n + 1 can have one more digit than 2 n is if 2 n ended in a 9 (actually ends is 999999 but that is not important). 2 n can never end in a 9
mllewestblam2

mllewestblam2

Beginner2022-06-04Added 2 answers

In order for them to have a different number of digits, 2 n + 1 must be exactly a power of 10. But that's impossible, since 2 n + 1 is odd.

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