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A binary representation writes a number in positional notation with radix two and binary digits zero and one. To the left of the binary point the place values are $1,2,4,8,\dots$; to its right they are $1/2,1/4,1/8,\dots$. For example,Binary notation changes the numeral, not the underlying number. The same value is $5.25$ in decimal. A finite binary fraction represents a dyadic rational, while an infinite binary expansion can represent any real number in the unit interval. Storing a finite binary string additionally requires an interpretation such as unsigned interpretation or two's complement.

For a positive integer $n$, its bit length is the number of binary digits in its usual representation:Equivalently, $\ell (n)$ is the unique positive integer satisfying $2^{\ell (n)-1}\le n<2^{\ell (n)}$. For example, $13=1101_2$ has bit length four, and $8=1000_2$ has bit length four too. The inequalities provide an exact characterization without a floating-point logarithm.

Many integer programming interfaces define the bit length of zero as zero. That convention counts no significant one-bit; it differs from the written binary numeral $0$, which has one character. A fixed-width machine word can also contain leading zero bits that do not contribute to the value's bit length.

An integer is a whole number, positive, negative, or zero. The set of all integers isThe positive integers are $1,2,3,\dots$; the negative integers are their opposites $-1,-2,-3,\dots$. Zero is neither positive nor negative. The nonnegative integers include zero and all the positive integers. On a number line, consecutive integers are one unit apart, negative integers lie to the left of zero, and positive integers lie to its right. ^[ref]

The integers let counting describe changes in both directions. If a counter starts at $-3$ and increases by five, it reaches $2$: $-3+5=2$. Decreasing that result by four gives $2+(-4)=-2$. In addition on the number line, the sign of the added integer chooses the direction and its absolute value gives the distance. The additive inverse $-a$ reverses the change made by $a$.

The integers have closure under a binary operation for addition, subtraction and multiplication: whenever $a,b\in \mathbb{Z}$, the results $a+b$, $a-b$ and $ab$ are still integers. Together these operations give the integers their ring structure.^[ref] For instance, $-3+5=2$, $-3-5=-8$, and $(-3)\cdot 5=-15$. General division has no such closure: $1/2$ is a rational number but not an integer. Some divisions are exact, such as $(-6)/3=-2$; division by zero is undefined. Euclidean division instead produces an integer quotient and a remainder, so $7=3\cdot 2+1$ retains the information that ordinary division puts into the fraction $7/3$.

An integer is the value, whereas its written numeral is an encoding. The decimal representation $205_{10}$ and the binary representation $11001101_2$ name the same integer. Changing positional notation changes how that value is displayed and how calculations are organized; it does not change the value itself.

A number board game associates positions and moves on a board with numbers. A linear track can connect spoken counting, written numerals, spatial position and the number of moves needed to reach a destination. These are related cues to numerical magnitude, rather than merely decorative labels.

Geetha B. Ramani and Robert S. Siegler tested such a game with low-income preschool children. Their numbered track had ten consecutive positions; a corresponding comparison board used colors without the numbers. Children played across four sessions, with numerical assessments immediately afterwards and nine weeks later. The numbered-game group improved on the four measured numerical tasks, with retained gains at follow-up.^[ref]

The comparison is useful because it controls aspects of game playing and adult interaction while changing the numerical activity. It tests a particular intervention and set of outcomes; it does not imply that every board layout teaches the same skills. The relation between positions on the track and a number line also gives a concrete hypothesis about why a linear arrangement might support number-line estimation.

A numeral is a written expression representing a number. The number five can be written as the decimal numeral $5$, the binary numeral $101_2$, or the Roman numeral V. The expressions differ while the represented integer is the same.

This distinction matters when an algorithm examines numerical digits. A palindromic number, for example, is defined through its representation in a specified radix, rather than through the appearance of an abstract number. The integer five is a palindrome as the one-digit decimal numeral $5$ and as the binary numeral $101_2$, whereas ten is written $10$ in decimal and $1010_2$ in binary, neither of which is a palindrome.

Numeral identification is recognition or naming of written numerals. When shown 7 and asked its name, a child maps a visual symbol to a conventional numerical label. This differs from counting seven objects or choosing whether 7 is larger than 4: related components of numeracy place different demands on the learner.

Separate assessments make those differences visible. Geetha B. Ramani and Robert S. Siegler asked preschool children to name numerals as one of four numerical tasks before and after a number board game intervention. A naming score therefore measures performance under that task's symbols and response rules; it does not by itself establish understanding of every property of the represented numbers.^[ref]

A two-variable formula enumerates a set $A$ in a theory $S$ when membership supplies a standard numeral stage whose instance is provable, while nonmembership makes every standard-stage instance refutable. The notion contains these proof obligations, not merely a syntactic complexity restriction on the formula. With bounded completeness, a bounded formula whose existential closure defines a computably enumerable set provides such an enumerator.^[ref]

Numerical magnitude comparison is a task in which a person decides which of two represented numbers is larger or smaller. The inputs might be written numerals, spoken number names or sets of objects. Choosing 8 rather than 3 as the larger number tests a numerical relationship; simply naming the symbol 8 instead tests numeral identification.

A task records both the decision and, when appropriate, reaction time. These outcomes depend on the representation and instructions, so successful comparison of two familiar digits need not establish equal skill with unfamiliar notation. Geetha B. Ramani and Robert S. Siegler assessed comparison of pairs of numerals separately from naming and counting in their number board game experiment.^[ref]

Robinson arithmetic is a finitely axiomatized first-order theory in the language $0,S,+,\cdot$. Its seven axioms are the universal closures ofThey describe successor, addition and multiplication without an induction schema. Each calculation on particular standard numerals can still be verified: repeated use of the recursion equations proves, for example, $\overline{2}+\overline{3}=\overline{5}$. This finite computational strength is enough for representing computable finite calculations and for the diagonal lemma used in crude incompleteness theorem. Peano arithmetic adds induction for arbitrary arithmetic formulas.^[ref]