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An array whose entries appear in nondecreasing order under a specified total order. The displayed inequalities allow repeated values.

For example, $(2,5,5,9)$ is a sorted array of integers. Inserting $7$ at index $3$ gives $(2,5,5,7,9)$: with spare capacity, first shift $9$ one slot right. Appending $7$ would violate the sorted array property. Updating an entry also requires preserving the total order, or removing and reinserting it in a suitable position.

The useful combination is order plus direct indexed access: binary search eliminates a whole interval of candidates at each comparison, giving $O(\log n)$ search time complexity under constant-cost comparisons and the array cost model. A sorted array is therefore attractive for many lookups between changes, while keeping it sorted can still require $O(n)$ shifting work per insertion. A sorted linked list has the ordering information but does not supply the same constant-cost access to a middle index.

Storage of a multidimensional array with the first index varying fastest in consecutive positions. In a two-dimensional array, each column is contiguous. Traversing a row-major array by columns does not change its storage into column-major order.

Storage of a multidimensional array with the last index varying fastest in consecutive positions. In a two-dimensional array, each row is contiguous. Row-major order is a layout convention; a program can traverse that array by rows or by columns. See the explicit address calculation in spatial locality.

Simon Stevin's De Thiende, published in 1585, presents calculation with decimal fractions for practical users. Its notation explicitly labels units, tenths, hundredths, and smaller decimal places; it does not use our modern decimal point. This is a historical example of changing representation to make the same arithmetic operations easier to organize. ^{[ref] [ref]}

Integer arithmetic computes with integers. Addition, subtraction and multiplication preserve integer values; division can instead return a quotient and remainder through Euclidean division. For example, $102=20\cdot 5+2$ gives five complete groups and a remainder of two. The ceiling count for fixed-size batches requires six groups to hold all 102 items.

Mathematical integers have no fixed size bound. A computer representation may have one, so an exact arithmetic formula still requires sufficient space for its inputs and intermediate values. The floor function and ceiling function define mathematical bounds independently of a chosen representation.

A figurate number is an integer illustrated by a regular geometric arrangement of points. Triangular numbers and square numbers are basic examples: growing a prescribed shape turns a counting problem into a sequence of integers.

A sum of squares adds the squares of numbers $a_i^2$ for specified real numbers $a_i$. It is nonnegative and vanishes exactly when every term is zero. For the coordinates of a vector, it is the squared Euclidean norm; for regression residuals, it gives the residual sum of squares minimized by ordinary least squares.

An inequality compares mathematical quantities using an ordering, such as $a<b$ or $a\le b$ for real numbers. A strict inequality excludes equality; a non-strict inequality allows it. For example, $4<5$ is strict, while $4\le 4$ is true because equality is allowed. Adding the same real number preserves either comparison; multiplying by a negative real number reverses its direction.

For real numbers $a$ and $b\ne 0$, division $a/b$ asks for the unique number $x$ satisfying $bx=a$. Equivalently, $a/b=a{b}^{-1}$ uses the multiplicative inverse of $b$. This description separates the role of the denominator from the result: the quotient undoes a specified multiplication.

For example, $3/2$ is the number whose double is three, so it is exactly $1.5$. A second interpretation asks how many portions of size $2$ fit into an amount $3$; the answer is one and a half portions. Equal sharing and measuring with a specified portion size lead to the same quotient, although the units assigned to the result can differ.

In the natural numbers, exact division need not be possible. Euclidean division instead keeps a remainder: $17=5\cdot 3+2$, with $0\le 2<5$. Thus seventeen objects form three complete groups of five with two left over. As a rational number, the same quotient is $17/5=3+2/5=3.4$. The remainder and the fractional part describe the same unfinished group in different ways.

Division by zero cannot retain the unique-solution rule. For $a\ne 0$, the equation $0x=a$ has no solution; for $a=0$, every real number $x$ solves $0x=0$, so there is no unique quotient. This explains both failures through multiplication, rather than assigning $a/0$ a value that would contradict ordinary arithmetic. ^[ref]

Numbers are mathematical objects used to express quantities and carry out operations. Familiar families include the natural numbers, integers, rational numbers, real numbers, and complex numbers; specifying the family determines which operations and solutions are available.

By William J. Devenport.

Online text: devenport.aoe.vt.edu/aoe3054/manual/expt3/text.html

Cited passages: Sections 2B–C and 4, goal 1: apparatus, pressure connections, datum levels and independent checks.

Antibody affinity concerns one paratope binding one epitope. A Fab fragment supplies a single-site comparison. ^[ref] In the dilute chemical equilibrium model below, smaller dissociation constant $K_d$ means larger antibody equilibrium occupancy at the same free antigen concentration. State the antigen, temperature and buffer solution: a dissociation constant is not a context-free score. It does not alone specify how fast attachment or detachment occurs.

Avidity is the combined strength of multivalent attachment. An intact IgG can make two contacts when accessible epitopes have suitable spacing. ^[ref] If one contact breaks while the other remains, the detached arm stays nearby and can bind again before the whole antibody escapes. That geometric advantage does not require changing either paratope's intrinsic antibody affinity. Two arms binding two separate freely moving antigens do not automatically provide the same tethering advantage. Epitope spacing, accessibility and surface density matter; avidity is not obtained by adding two dissociation constants.