Normal Number — Definition, Formula & Examples

A normal number is a real number whose digit expansion in a given base contains every possible string of digits with the same limiting frequency you would expect from a random sequence. For instance, in base 10, each digit 0–9 appears exactly 1/10 of the time, each pair of digits appears 1/100 of the time, and so on.

A real number $x$ is normal in base $b$ if, for every positive integer $k$ and every string $s$ of $k$ digits in base $b$ , the limiting frequency of occurrences of $s$ in the base- $b$ expansion of $x$ equals $b^{-k}$ . A number that is normal in every integer base $b \geq 2$ is called absolutely normal.

How It Works

To check whether a number is normal in base 10, you examine its decimal expansion and count how often each digit appears in the first

N

digits, then take the limit as

N \to \infty

. Each single digit should appear with frequency

1/10

, each two-digit block with frequency

1/100

, and each

k

-digit block with frequency

10^{-k}

. Proving normality for specific well-known constants is extremely difficult. Although almost all real numbers are normal (in the measure-theoretic sense), no one has proved that

\pi

e

, or

\sqrt{2}

is normal in any base. The Champernowne number

0.123456789101112\ldots

was one of the first explicit examples proven normal in base 10.

Example

Problem: The Champernowne number

C_{10} = 0.12345678910111213\ldots

is formed by concatenating all positive integers in base 10. Verify the expected frequency of the digit 7 in the portion covering all one-digit numbers (1 through 9).

Step 1: List the digits contributed by the one-digit numbers 1 through 9. Each contributes exactly one digit, giving 9 digits total.

\text{Digits: } 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, 8,\, 9

Step 2: Count occurrences of the digit 7. It appears exactly once among these 9 digits.

\text{Frequency of 7} = \frac{1}{9} \approx 0.111

Step 3: For a normal number in base 10, the expected limiting frequency of any single digit is

1/10 = 0.1

. In this small sample the frequency is slightly above

0.1

, but as more integers are concatenated, the frequency of every digit converges to exactly

1/10

. This convergence is what Champernowne proved in 1933.

\lim_{N \to \infty} \frac{\#\{\text{occurrences of 7 in first } N \text{ digits}\}}{N} = \frac{1}{10}

Answer: The digit 7 appears with limiting frequency

1/10

C_{10}

, consistent with normality in base 10.

Why It Matters

Normal numbers sit at the intersection of number theory, ergodic theory, and information theory. Understanding digit distribution is relevant to pseudorandom number generation and to open problems about famous constants like

\pi

. Determining whether specific irrational numbers are normal remains one of the notable unsolved problems in mathematics.

Common Mistakes

Mistake: Assuming that an irrational number must be normal because its digits never repeat.

Correction: Irrationality only means the expansion is non-terminating and non-repeating. A number like

0.101001000100001\ldots

is irrational but clearly not normal, since most digits are 0.