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The intergalactic medium is the gas between galaxies. Its diffuse, ionized component contains much of the ordinary matter outside stars. Lyman-alpha absorption measures its small neutral-hydrogen fraction, while hotter phases require other tracers.

Stellar parallax is the apparent change in a star's viewing direction caused by the observer's changing position. Nearby stars shift more against a distant reference background. In an ideal circular Earth orbit, the conventional annual stellar parallax $p$ uses a baseline of one astronomical unit, approximately the Earth-Sun distance. Two opposite observing positions, about six months apart, are separated by approximately two astronomical units. Their maximum angular separation is $2p$, so $p$ is half that excursion, not the full January-to-July shift. The observed excursion depends on the projected baseline and observing dates. ^[ref].

For the perpendicular baseline in the diagram, a right triangle giveswhere $p$ in the final expression is in radians. Combining the small-angle approximation with the parsec convention givesThus $p=0.05$ arcseconds implies $d\simeq 20$ parsecs, while the ideal six-month excursion is $0.10$ arcseconds. Geometry supplies this distance without assuming a luminosity or a particular stellar atmosphere.

In practical astrometry, many observations separate the annual stellar parallax signal from proper motion, the star's own changing sky position. The Gaia DR3 catalogue reports stellar parallax and its uncertainty in milliarcseconds: $50\mathrm{mas}=0.05$ arcseconds. A catalogue angle is a fitted measurement with measurement error. ^[ref].

For a positive stellar parallax with small standard deviation ${\sigma }_p$, first-order propagation gives ${\sigma }_d/d\simeq {\sigma }_p/p$. Thus $p=50\mathrm{mas}$ with ${\sigma }_p=1\mathrm{mas}$ gives $d\simeq 20$ parsecs with ${\sigma }_d\simeq 0.4$ parsecs, assuming systematic offsets are negligible. This linear estimate requires ${\sigma }_p\ll p$; a reciprocal is nonlinear and does not generally preserve symmetric uncertainty intervals.

A point-like astronomical source has an angular extent negligible for the calculation or measurement being considered. The ideal mathematical limit concentrates its luminosity at one source-plane position. In gravitational lensing, this approximation permits divergent formal magnifications on a caustic; a finite source averages the magnification over its emitting area. An unresolved source need not have negligible size compared with the lens's caustic scale.

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.