# Finding The Number of Digits

Some problems ask you to find the **number of digits** of an integer or variable. For example, the number 23 has two digits, and the number 1337 has four. But how many digits does an integer \(x\) have, if \(x^2\) has more than 2 but less than 4 digits?

There's not yet enough information to determine exactly what \(x\) is, but a range is ascertainable. If all that's known is that \(x\) is an integer and that \(x^2\) produces an integer that's \(3\) digits long, then \(x\) is on the range \([10,31]\) as \(9^2=81, 10^2 = 100, 31^2=961,\) and \(32^2=1,024\). \(_\square\)The wiki will discuss further and more rigorous ways to conduct this analysis.

Note that the numbers with precisely one digit are those integers in the range \( [1,9]\), the numbers with precisely two digits are those integers in the range \( [10, 99]\), and the numbers with precisely three digits are those integers in the range \( [100, 999]\), and so on. The two digit numbers are shown below:

However, determining the number of digits of an extremely large number can be somewhat tricky and is explored below.

#### Contents

## Large Numbers

A number will have precisely \(j\) digits if and only if it is in the range \(I_j = [10^{j-1}, 10^{j} - 1]\). For instance, the number \(5,000,000\) has \(7\) digits and is in the range \([10^{7-1},10^7-1] = [\text{1,000,000}, \text{ 9,999,999}].\)

Given an integer \(n\), one can determine \(j\), the number of digits in \(n\), by working with the inequality

\[ n\in I_j \implies 10^{j-1} \le n \le 10^j - 1.\]

Taking the base 10 logarithm gives

\[j-1 \le \log_{10} n < j,\]

so \(j-1 = \left\lfloor \log_{10} n \right\rfloor,\) where \(\lfloor x \rfloor\) is the floor function, denoting the greatest integer less than or equal to \(x\). Thus, \(j = \left\lfloor \log_{10} n \right\rfloor + 1\).

For any positive integer \(n\), the number of digits in \(n\) is \(\left\lfloor \log_{10} n \right\rfloor + 1\).

For example, take \(n = 5,000,000\), then \(\left\lfloor \log_{10} 5000000 \right\rfloor = 6,\) meaning that we expect this number to have \( \left\lfloor \log_{10} 5000000 \right\rfloor +1 = 7\) digits, which in fact it does.

## Examples

How many digits does \(100!\) have?

Using Stirling's formula \(n! \approx \sqrt{2\pi n}\left(\frac ne \right)^n,\) we can get the number of digits of \(100!\) as follows:

\[\begin{align} N & = \lfloor \log_{10} {\color{blue}100!} \rfloor + 1 \\ & = \left \lfloor \log_{10} {\color{blue}\sqrt{2\pi \cdot 100}\left(\frac {100}e\right)^{100}} \right \rfloor + 1 \\ & = \left \lfloor \frac 12 \log_{10}2 + \frac 12 \log_{10} \pi + 1 + 100(2) - 100 \log_{10} e \right \rfloor + 1 \\ & = \lfloor 157.970 \rfloor + 1 \\ & = 158.\ _\square \end{align} \]

**Cite as:**Finding The Number of Digits.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/finding-digits-of-a-number/