Positional notation ($N={\sum }_{j=0}^n{d}_j{b}^j$)

In positional notation, a numeral digit contributes its value multiplied by a weight determined by its position. Fix an integer radix $b\ge 2$. A finite string of numeral digits $d_n{d}_{n-1}\cdots d_0$, with $0\le d_j<b$, represents the nonnegative integerThe rightmost position has weight $b^0=1$; each move left multiplies the weight by $b$. This weight is the place value. For a nonzero integer, the canonical string has $d_n\ne 0$. Zero is written with the single numeral digit $0$. Leading zeros are permitted in a fixed-width display but omitted in the canonical form: $00205_{10}=205_{10}$. A negative integer is written with a separate minus sign and the positional notation for its absolute value; the minus sign is not a numeral digit.^[ref]

For example,The zero is essential inside this string: $25_{10}=25$, not $205$. It records that the tens position contributes nothing while keeping the hundreds and units in their proper positions. The same integer has binary representation

Repeated Euclidean division constructs the representation exactly. Set $N_0=N$. At each step writeKeep the remainder $d_j$ and continue with the integer quotient $N_{j+1}$. A positive quotient strictly decreases, so eventually it is zero. Substituting the successive equations back into the first gives $N=d_0+b{d}_1+b^2{d}_2+\cdots$: the numeral digits are obtained from right to left. The remainder modulo $b$ uniquely fixes $d_0$, and then the quotient uniquely fixes the remaining string by the same reasoning. This proves existence and uniqueness once leading zeros are excluded. The needed unique quotient and remainder are precisely the conclusion of the Euclidean division theorem.^[ref]

This finite expansion concerns nonnegative integers. Extending positional notation to fractional positions uses weights $b^{-1},b^{-2},\dots$ to the right of a point; an arbitrary real number need not have a finite digit string. In particular, the decimal representation $0.1$ represents $1/10$, while the base-two string $0.1_2$ represents $1/2$. The radix is part of the interpretation.

The benefit is compact arithmetic: $k$ positions can encode every nonnegative integer below $b^k$. To see the upper endpoint, fill all positions with $b-1$; the finite geometric series gives $(b-1)(1+b+\cdots +b^{k-1})=b^k-1$. Carrying in addition exchanges $b$ units of one place value for one unit of the next. Thus a short string stores a large count and also supplies an algorithm for computing with it.

The digit sum of a nonnegative integer in a given base is the sum of its numerical digits. In base $b$, replacing each power $b^j$ by $1$ shows that the integer and its digit sum are congruent modulo $b-1$.

A positive integer whose sequence of numerical digits is neither weakly increasing nor weakly decreasing from left to right. Equivalently, one adjacent pair strictly rises and another strictly falls. Numbers in both monotone families have all digits equal, so their overlap consists of the repdigits.

A mixed-radix positional notation using factorials as place values: $n={\sum }_{i\ge 1}{a}_{i}i!$ with $0\le a_i\le i$. For lexicographic order on permutations, division by $(k-1)!$ selects the next symbol when $k$ symbols remain; the remainder gives the rank within that block.

A nonnegative integer whose positional notation reads the same forwards and backwards, with no leading zeros. This property depends on the base: $585$ is a palindromic number in both decimal representation and binary representation, since $585=(1001001001{)}_2$.

A palindromic number in binary representation, with no leading zeros. Every positive binary palindrome is an odd number, because its first and last numerical digits must both be one.

A palindromic number in decimal representation, with no leading zeros.

A positive integer whose repeated reverse-and-add process never reaches a palindromic number in the chosen base. Each step adds the current number to the number obtained by reversing its numerical digits. Failure to find a palindromic number after a fixed iteration limit establishes only a candidate, not a proof that the process never succeeds.

A numerical digit is one symbol in a positional notation string, denoting an integer from zero through one less than the radix.

One of the ten numerical digits $0,1,\dots ,9$ used in decimal representation.

A positive integer whose first $\lceil k/2\rceil$ decimal numerical digits have the same sum as its last $\lceil k/2\rceil$ digits, where $k$ is the number of digits. For odd $k$, the shared middle digit cancels from the equality. Counts and value sums can therefore be assembled from two half-length digit dynamic programming tables indexed by base-p digit sum.

A positive integer whose adjacent decimal numerical digits differ by exactly 1. Its digits form a walk on the path graph $0-1-\cdots -9$, with a nonzero starting digit. Counting pandigital numbers of this type can use digit dynamic programming with the final digit and a bit mask of visited digits as the state.

Repeatedly sum the numerical digits of a nonnegative integer in base $b$ until a single numerical digit remains. For $n>0$, the result is $1+((n-1)\mathrm{mod}(b-1))$, while the result for zero is zero. This follows because $b\equiv 1(\mathrm{mod}b-1)$, so summing numerical digits preserves the integer congruence class modulo $b-1$.

A number whose representation contains every numerical digit from a specified alphabet. The convention must specify the base, whether zero is required, and whether repetitions are allowed. In many counting problems, “0-to-9 pandigital” means that all ten decimal numerical digits occur exactly once; other problems require only that each occur at least once.

A prime number that is also a pandigital number for the specified base and digit set.

The radix is the base of positional notation. In the ordinary integer-base case it is an integer $b\ge 2$, and successive place values differ by a factor of $b$.

The place value is the weight of a position in positional notation. A numeral digit $d_j$ at that position contributes $d_j{b}^j$ to the represented number.

A nonnegative integer's decimal expansion uses digits $0\le a_i\le 9$. Testing divisibility reduces this polynomial in ten modulo the divisor: for example $10\equiv -1(\mathrm{mod}11)$ makes divisibility by eleven equivalent to vanishing of the alternating digit sum.

A hereditary base representation expands not only the number in base $b$, but also every exponent recursively in that same base. Changing the base at every depth defines the Goodstein base-change operation; replacing it by omega defines an ordinal rank of a Goodstein term.

Start at base two. At each nonzero step, change every base occurrence in the hereditary base representation to the next base and subtract one. Although the natural-number terms can grow, their ordinal ranks of a Goodstein term decrease strictly. Keep zero fixed once it is reached.

Every Goodstein sequence reaches zero. Before zero, the ordinal rank of a Goodstein term of each step is strictly smaller than the preceding one. An infinite sequence would contradict well-foundedness of the ordinals below epsilon zero.

Replacing every base occurrence by omega in a hereditary base representation gives a Cantor normal form below epsilon zero. Hereditary base change preserves this ordinal expression, and subtraction of one lowers it strictly. The ordinal is a termination measure even when the natural-number value increases.

The Goodstein function records the least step at which the Goodstein sequence from a given initial number reaches zero. With the initial step numbered zero, its values on $0,1,2,3$ are $0,1,3,5$. Goodstein's theorem makes this effective termination-time function total.

For an integer base $p\ge 2$, a nonnegative integer has a finite positional representation with digits $k_i\in \{0,\dots ,p-1\}$. Repeated Euclidean division gives the digits uniquely; appending zero high-place digits has no effect. The base need not be prime for uniqueness, but a prime base is useful in Lucas theorem and modular binomial coefficients.

The binary numeral system is positional notation with radix two. Each binary digit is zero or one, and its place value is a power of two. A one includes that weight in the sum; a zero skips it. The binary representation $11001101_2$ therefore selects $128,64,8,4,1$ and represents the integer $205$.

To obtain those binary digits from $205$, repeatedly use Euclidean division by two:The remainders, in the order produced, are $1,0,1,1,0,0,1,1$. Read them in reverse order to get $11001101_2$. The final zero is the quotient that stops the conversion, not an additional leading binary digit. Starting with zero itself needs the special single-digit representation $0_2$.

A useful independent check reads the binary digits from left to right. Start with $v=0$ and, for each binary digit $d$, replace $v$ by $2v+d$. For $11001101_2$, the intermediate values are $1,3,6,12,25,51,102,205$. This reconstructs the integer without reversing a list and explains why appending a zero on the right doubles the value.

Binary is well suited to computation because each binary digit has just two possibilities. A stored bit can encode either possibility; groups of bits encode larger integers. Practical integer arithmetic can use a radix that is itself a power of two, grouping several binary digits into each stored numeral digit.^[ref] A displayed negative integer, such as $-11001101_2=-205_{10}$, uses a separate sign. A computer's particular signed storage convention is an additional encoding choice, so the displayed sign should not be confused with an extra binary place.

A binary digit is a zero or one in the base-two positional representation of an integer. The sum $s_2(n)={\sum }_j{b}_j$ determines the Thue–Morse sequence and is the Hamming weight of the binary representation.

The decimal numeral system is positional notation in base ten.

A decimal representation is positional notation with radix ten and numeral digits zero through nine. For an integer, the nonnegative part is a finite decimal expansion; a negative integer has a separate minus sign. For example, $205=2\cdot 100+0\cdot 10+5$, whereas $250=2\cdot 100+5\cdot 10+0$. The numeral digits alone do not determine the value: their positions do.

Fractional positions extend the same place value rule to the right of the decimal point, with successive weights $1/10,1/100,1/1000,\dots$. ^[ref] For example,The matching binary representation is $11001101.011_2$, since its fractional part is $0\cdot 2^{-1}+1\cdot 2^{-2}+1\cdot 2^{-3}=3/8$. These representations terminate in both bases; finiteness in one radix does not imply finiteness in another.

Leading zeros in the integer part and trailing zeros in the fractional part leave the value unchanged: $00205.3750=205.375$. A zero appended to the right of an integer string has a different effect: $2050=10\cdot 205$. Keeping the decimal point fixed prevents confusing these two operations.

Every integer has a finite decimal expansion, but a noninteger rational number may need infinitely many fractional numeral digits, as in $1/3=0.333\dots$. A terminating decimal has a zero tail by convention. Infinite decimal strings have a further ambiguity that finite canonical integer strings do not: $0.999\dots =1$, because the geometric series ${\sum }_{j=1}^{\infty }9\cdot 10^{-j}$ sums to one. An infinite decimal tail denotes a limit, not a sufficiently long rounded approximation.

A decimal representation with only finitely many nonzero fractional digits. A reduced fraction has this property exactly when its denominator has no prime factors other than two and five. The zero tail is distinguished from the genuine repeating block measured by decimal period from multiplicative order.