# Thread: Number of possible digits in a binary representation of a number

1. ## Number of possible digits in a binary representation of a number

Hi all,

Suppose that when n is written in decimal notation it has 100 digits. Which
possibilities are there for the number of digits if n is written in binary notation?

Any ideas?

Thanks

2. Hello, sirellwood!

Suppose that when $\,n$ is written in decimal notation it has 100 digits.
What possibilities are there for the number of digits if $\,n$ is written in binary notation?

$\text{If }n\text{ has 100 digits: }\:10^{99}\:\le\: n\:<\:10^{100}$

$\text{Suppose }2^k\text{ is the largest power-of-2 contained in }n.$

. . $\text{Then: }\;10^{99}\:\le\: 2^k \:<\:10^{100}$

$\text{Take logs: }\;\log(10^{99}) \:\le\;\log(2^k) \:<\:\log(10^{100})$

. . . . . . . . . . . . . $99 \:\le \:k\log 2 \:<\:100$

. . . . . . . . . . . . . $\dfrac{99}{\log 2} \:\le \: k \:<\:\frac{100}{\log2}$

. . . . . . . . . . . . $328.87 \:\le\:k \:<\:332.19$

. . . . . . . . . . . . . . $329 \:\le\:k\:\le\:332$

$\text{Therefore, }n\text{, written in binary, has from 329 to 332 digits.}$

3. Originally Posted by Soroban
Hello, sirellwood!

$\text{If }n\text{ has 100 digits: }\:10^{99}\:\le\: n\:<\:10^{100}$

$\text{Suppose }2^k\text{ is the largest power-of-2 contained in }n.$

. . $\text{Then: }\;10^{99}\:\le\: 2^k \:<\:10^{100}$

$\text{Take logs: }\;\log(10^{99}) \:\le\;\log(2^k) \:<\:\log(10^{100})$

. . . . . . . . . . . . . $99 \:\le \:k\log 2 \:<\:100$

. . . . . . . . . . . . . $\dfrac{99}{\log 2} \:\le \: k \:<\:\frac{100}{\log2}$

. . . . . . . . . . . . $328.87 \:\le\:k \:<\:332.19$

. . . . . . . . . . . . . . $329 \:\le\:k\:\le\:332$

$\text{Therefore, }n\text{, written in binary, has from 329 to 332 digits.}$
The text contains a mathematical question about the number of digits in the binary representation of a number with 100 decimal digits, and a detailed solution is provided.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: Number of possible digits in a binary representation of a number

1. ## Number of possible digits in a binary representation of a number

Hi all,

Suppose that when n is written in decimal notation it has 100 digits. Which
possibilities are there for the number of digits if n is written in binary notation?

Any ideas?

Thanks

2. Hello, sirellwood!

Suppose that when $\,n$ is written in decimal notation it has 100 digits.
What possibilities are there for the number of digits if $\,n$ is written in binary notation?

$\text{If }n\text{ has 100 digits: }\:10^{99}\:\le\: n\:<\:10^{100}$

$\text{Suppose }2^k\text{ is the largest power-of-2 contained in }n.$

. . $\text{Then: }\;10^{99}\:\le\: 2^k \:<\:10^{100}$

$\text{Take logs: }\;\log(10^{99}) \:\le\;\log(2^k) \:<\:\log(10^{100})$

. . . . . . . . . . . . . $99 \:\le \:k\log 2 \:<\:100$

. . . . . . . . . . . . . $\dfrac{99}{\log 2} \:\le \: k \:<\:\frac{100}{\log2}$

. . . . . . . . . . . . $328.87 \:\le\:k \:<\:332.19$

. . . . . . . . . . . . . . $329 \:\le\:k\:\le\:332$

$\text{Therefore, }n\text{, written in binary, has from 329 to 332 digits.}$

3. Originally Posted by Soroban
Hello, sirellwood!

$\text{If }n\text{ has 100 digits: }\:10^{99}\:\le\: n\:<\:10^{100}$

$\text{Suppose }2^k\text{ is the largest power-of-2 contained in }n.$

. . $\text{Then: }\;10^{99}\:\le\: 2^k \:<\:10^{100}$

$\text{Take logs: }\;\log(10^{99}) \:\le\;\log(2^k) \:<\:\log(10^{100})$

. . . . . . . . . . . . . $99 \:\le \:k\log 2 \:<\:100$

. . . . . . . . . . . . . $\dfrac{99}{\log 2} \:\le \: k \:<\:\frac{100}{\log2}$

. . . . . . . . . . . . $328.87 \:\le\:k \:<\:332.19$

. . . . . . . . . . . . . . $329 \:\le\:k\:\le\:332$

$\text{Therefore, }n\text{, written in binary, has from 329 to 332 digits.}$
user
The text contains a mathematical question about the number of digits in the binary representation of a number with 100 decimal digits, and a detailed solution is provided.

TYPE: SINGLE_QUESTION_WITH_ANSWER