Digit
ArithmeticA digit is any one of the ten symbols (0-9) used to write numbers in the base-10 number system.
Definition
A digit is one of the ten symbols we use to write numbers: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$. Every number is made from these ten digits.
Example
The number $472$ has three digits: $4$, $7$, and $2$. The number $5$ has one digit. The number $1{,}000$ has four digits.
Key Insight
Just like the $26$ letters of the alphabet can make every word in English, the $10$ digits can make every number in our number system.
Definition
In the base-$10$ (decimal) positional numeral system, a digit is any of the symbols $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Each digit's value depends on its position (place value) within the numeral. The number of digits in a positive integer $n$ is $\lfloor \log_{10}(n) \rfloor + 1$.
Example
In $3{,}058$: the digit $3$ is in the thousands place (value $3{,}000$), $0$ is in the hundreds place (value $0$), $5$ is in the tens place (value $50$), and $8$ is in the ones place (value $8$).
Key Insight
The word "digit" comes from the Latin "digitus" (finger), because humans historically counted on fingers. Different bases use different digit sets: binary uses only $\{0,1\}$, hexadecimal uses $\{0\text{-}9, A\text{-}F\}$.
Definition
In a base-$b$ positional system, a digit is any element of $\{0, 1, \ldots, b-1\}$. The value of a numeral $d_n d_{n-1} \ldots d_1 d_0$ in base $b$ is the sum of $d_k \cdot b^k$ for $k$ from $0$ to $n$. The digit sum of $n$ and divisibility rules for $3$, $9$, and $11$ derive directly from properties of powers of $10$ modulo these values.
Example
Divisibility by $9$: a number is divisible by $9$ if and only if its digit sum is divisible by $9$. For $729$: $7+2+9=18$, and $18/9=2$, confirming $729 / 9 = 81$.
Key Insight
The choice of base $10$ is anatomical, not mathematical. Base $12$ (dozenal) would make divisibility by $3$ and $4$ as easy as divisibility by $2$ and $5$ is in base $10$.