Formula

Closed form of a repeated-digit number

Arithmetic • Number System • Digit Problems

Formula

n digits dd \dots d = d \times \frac{1 0 ^{n} - 1}{9}

The number written as

n

copies of the digit

d

equals

d \times \frac{1 0 ^{n} - 1}{9}

; in particular the repunit of

n

ones is

\frac{1 0 ^{n} - 1}{9}

How to use it

1Convert every repeated-digit number to this form before combining them, so the powers of $10$ cancel algebraically instead of being multiplied out.
2The two that occur most often are $n$ threes $= \frac{1 0 ^{n} - 1}{3}$ and $n$ twos $= \frac{2 ( 1 0 ^{n} - 1 )}{9}$ .
3A repunit of $2 n$ digits factors as $\frac{( 1 0 ^{n} - 1 ) ( 1 0 ^{n} + 1 )}{9}$ , which is what links a double-length repunit to a single-length one.

Express the number made of five

4

s as a fraction and evaluate it.

Step-by-step

1
$4 \times \frac{1 0 ^{5} - 1}{9}$
Why: The digit is $4$ and it is repeated $5$ times.
2
$4 \times \frac{99999}{9} = 4 \times 11111 = 44444$

Answer

44444

Watch out

The exponent counts digits, not value: a

20

-digit repunit carries

1 0^{20}

while a

10

-digit one carries

1 0^{10}

, and it is that mismatch, not the digit, that has to be tracked when the two are combined.

Builds on

Also known as

repunit, repdigit, 111...1 formula, number made of repeated digits, (10^n-1)/9