How Many Digits Does a Number Have? The Mathematics of Magnitude

The largest known prime, 2¹³⁶²⁷⁹⁸⁴¹ − 1, has 41,024,320 digits. Nobody counted them. In fact, you can calculate that digit count on a phone calculator in a few seconds, without ever computing the prime itself.

The tool is the base-10 logarithm. It's the mathematical ruler for how big a number is, and it answers questions like "how many digits does 100! have?" or "how many bits do I need to store this?" almost instantly.

You Can Count Digits Without Knowing the Number

It seems obvious that to count digits you need to write the number out. But the number of digits depends only on the number's size, not on its individual digits. And size is exactly what logarithms measure.

The Formula

For any positive integer n:

Number of digits = ⌊log₁₀(n)⌋ + 1

Where ⌊x⌋ means "round down to the nearest integer."

Why? A number with d digits lies between 10ᵈ⁻¹ and 10ᵈ:

100 ≤ n < 1000  →  2 ≤ log₁₀(n) < 3  →  3 digits

The logarithm lands in the range [d − 1, d), so rounding down and adding 1 gives d. Check logs with the logarithm calculator or the base-10 logarithm table.

Example 1: How Many Digits Does 2¹⁰⁰ Have?

Use the power rule for logarithms, log(aᵇ) = b × log(a):

log₁₀(2¹⁰⁰) = 100 × log₁₀(2) ≈ 100 × 0.30103 = 30.103
Digits = ⌊30.103⌋ + 1 = 31

Indeed, 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376, which has 31 digits.

And the largest known prime? Subtracting 1 from a power of 2 never changes the digit count, so:

136,279,841 × 0.30103 ≈ 41,024,319.9
Digits = 41,024,319 + 1 = 41,024,320

Example 2: How Many Digits Does 100! Have?

For a product, logarithms turn multiplication into addition:

log₁₀(100!) = log₁₀(1) + log₁₀(2) + … + log₁₀(100) ≈ 157.97
Digits = 158

Adding 100 logarithms is tedious by hand, so mathematicians use Stirling's approximation:

log₁₀(n!) ≈ n·log₁₀(n/e) + ½·log₁₀(2πn)

For n = 100 it gives about 157.97, matching the exact sum. 100! has 158 digits. The factorial grows so fast that 70! already exceeds a googol.

Digits in Other Bases

The same idea works for any base b:

Digits in base b = ⌊log_b(n)⌋ + 1

The most useful case is base 2, where digits are bits:

Bits needed for n = ⌊log₂(n)⌋ + 1

So 255 needs 8 bits and 256 needs 9. A number with d decimal digits needs about 3.32 × d bits, because log₂(10) ≈ 3.32. That's why a 617-digit RSA modulus is a 2048-bit number. See this in action with the binary to decimal converter and the base-2 logarithm table.

An Insider Reference: The Googol

In 1920, American mathematician Edward Kasner asked his nine-year-old nephew, Milton Sirotta, to invent a name for 10¹⁰⁰. The boy suggested "googol." Kasner popularized the word in his 1940 book Mathematics and the Imagination, co-written with James Newman.

A googol has 101 digits. Milton also proposed the googolplex, 10^googol, which has a googol + 1 digits. There isn't enough matter in the observable universe to write it out in full, even with one digit per atom.

In the late 1990s, two Stanford PhD students named their search engine after a playful misspelling of the word: Google.

The Fractional Part Tells You the First Digit

The integer part of log₁₀(n) gives the number of digits. The fractional part tells you the leading digits.

For 2¹⁰⁰, log₁₀ ≈ 30.10300. The fractional part is 0.10300, and 10^0.10300 ≈ 1.2677. So 2¹⁰⁰ starts with 1.2677…, which matches the full number above.

This is the basis of Benford's law. In many real-world datasets, like populations, river lengths and financial figures, the leading digit is 1 about 30.1% of the time, because log₁₀(2) ≈ 0.301. The astronomer Simon Newcomb noticed it in 1881, when he saw that the early pages of logarithm tables were more worn than later ones. Physicist Frank Benford tested it on over 20,000 data points in 1938. Auditors now use it to spot fabricated numbers.

Two Concepts Worth Knowing

Order of Magnitude

The order of magnitude of a number is ⌊log₁₀(n)⌋, its power of ten. A number with 7 digits has order of magnitude 6. It's the fastest way to compare very different sizes.

Logarithm Rules

Logarithms turn multiplication into addition and powers into multiplication: log(ab) = log a + log b and log(aᵇ) = b log a. Those two rules make digit counting for giant numbers possible. See the algebra formulas.

Quick Answer: How Do You Find the Number of Digits in a Number?

The number of digits in a positive integer n is ⌊log₁₀(n)⌋ + 1. For powers, use log₁₀(aᵇ) = b × log₁₀(a). For example, 2¹⁰⁰ has ⌊100 × 0.30103⌋ + 1 = 31 digits. In base 2, the number of bits is ⌊log₂(n)⌋ + 1.

Try Them Yourself

• Logarithm Calculator: find log₁₀ of any number
• Base-10 Logarithm Table: logs at a glance
• Base-2 Logarithm Table: bits needed for a number
• Numbers to Words: hear how big your digit count really is
• Scientific Calculator: compute 100 × log(2) yourself
• How Much Is a Trillion?: making big numbers intuitive

How many digits does 3¹⁰⁰⁰ have? Use log₁₀(3) ≈ 0.4771, then check whether the first digit you predict from the fractional part is correct.