Mathematics Glossary

Link to article: Mathematics Glossary.

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text=A glossary of real-life math terms relevant to the wiki.]]
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Here is a selection of real-life mathematics terms and associated definitions that may be relevant to the wiki. To return to the main hub, click [[[math-department-hub|here]]]; for further information on technical terms used throughout the site, see [[[technical-words|Technical Words]]].

Have a math term that's used on the site that you want defined here? Feel free to add it! Furthermore, if you are using one of these terms in your works, feel free to copy any parts of the definitions below and include them as well.

++ Structures

[https://en.wikipedia.org/wiki/Axiom Axiom]: A simple statement that is taken to be true in order to help form the foundations of a mathematical field. For example, the law of non-contradiction holds that a logical statement cannot be true and false at the same time, while one axiom of geometry states that given two points on a plane, it is always possible to construct a straight line passing through these two points. Also known as a **postulate**.

[https://en.wikipedia.org/wiki/Equation Equation]: The establishment of two or more **expressions** being equal to each other, e.g. [[$2x - 5 = 3$]], [[$a = 2b = 3c$]], or [[$z_{n+1}^2 = z_n^2 + c$]] ([[[SCP-9001]]]). In some cases, a **variable** present in an equation (or group of equations) can be solved for in terms of the other components present; for example, solving the first equation for [[$x$]] gives [[$x = 4$]], and solving the second for [[$c$]] gives [[$c = a/3 = 2b/3$]]. 

[https://en.wikipedia.org/wiki/Expression_(mathematics) Expression]: An arrangement or combination of mathematical symbols, such as [[$x^2 + 2$]], [[$2(5 - 3)(5 + 3)$]], or [[$4 \sin(\pi \theta)$]]. Notably, expressions by themselves can't be "solved" for anything; **equations**, inequalities, and **formulas** //can// potentially be solved in cases where they relate multiple **variables** to each other.

[https://en.wikipedia.org/wiki/Formula#In_mathematics Formula]: An **equation** or [https://en.wikipedia.org/wiki/Inequality_(mathematics) inequality] that relates one **expression** or **variable** to another. For example, [[$V = \frac{4}{3} \pi r^3$]] is the formula for the volume [[$V$]] of a sphere in terms of its radius [[$r$]].

[https://en.wikipedia.org/wiki/Hypothesis Hypothesis]: Also known as a **conjecture** in math. These are unproven statements that are supported by observations or evidence. As these are not yet proven true or untrue, they are not yet **theorems**, even if there is evidence pointing towards a result. Examples include the [https://en.wikipedia.org/wiki/Continuum_hypothesis continuum] [[[SCP-9153|hypothesis]]], the [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann hypothesis], and the [https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture].

[https://en.wikipedia.org/wiki/Theorem Theorem]: A statement that has been shown to be true by a mathematical **proof**. Theorems are the logical consequences of a set of **axioms** and are often used as stepping stones to prove other theorems as well. A classic example is the [https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem], which relates the three sides of a right triangle and can be proven using geometric principles.

[https://en.wikipedia.org/wiki/Variable Variable]: A symbol (e.g. a Latin or Greek letter) representing a quantity or some other mathematical object. For example, the radius of a circle or sphere is usually denoted [[$r$]], while the time elapsed since the start of an experiment is often labeled as [[$t$]]. Particularly in a research context, independent variables are inputs/causes, while dependent ones are outputs/effects; for example, when providing a voltage to a circuit connected to a light bulb, the voltage applied would be an independent variable, with the brightness of the bulb being a dependent one.

++ Constructs

[https://en.wikipedia.org/wiki/Euclidean_geometry Euclidean Geometry]: The local, normal spatial geometry you're used to. It is often defined by the parallel **postulate**: given a line and a point not on that line, there exists exactly one line in the same plane that is parallel to the first and passes through the given point. The opposite is [https://en.wikipedia.org/wiki/Non-Euclidean_geometry Non-Euclidean Geometry], which is used to describe many spatial anomalies (e.g. [[[SCP-1898]]]); as a real-life example, the 2D surface of the Earth is non-Euclidean due to its curvature and the ability to loop back to one's starting point on it.

[https://en.wikipedia.org/wiki/Fractal Fractal]: A geometric shape that is detailed no matter how far you zoom in on it. Many fractals are self-similar, i.e. zooming on a smaller component of it yields a similar geometry to what you have when zoomed out. A classic real-life example is the recursive pattern of a [https://haggisthesheep.wordpress.com/2010/01/04/fractal-snowflakes/ snowflake]; on the wiki, certain mathematical anomalies like [[[SCP-3125]]] and [[[SCP-9001]]] possess fractal geometries that reflect their anomalous effects on reality.

[https://en.wikipedia.org/wiki/Infinitesimal Infinitesimal]: Colloquially, an infinitesimal quantity is very small; formally, an infinitesimal is in fact "infinitely small", i.e. closer to 0 than any non-zero real number. These don't exist in the standard number system, but there are extensions used to define them due to their usefulness in mathematics (e.g. calculus). They are the opposite of **infinite** quantities.

[https://en.wikipedia.org/wiki/Infinity Infinity]: Something which possesses no bound, i.e. the quantity in question is larger than any number. Also denoted as [[$\infty$]]. For standard numerical purposes, consider infinity as a direction instead of an actual destination, as by definition, you can never reach it. For the purposes of counting the number of elements in infinite sets, there is no one infinity: there are in fact an [[[Final Lecture|endless sequence]]] of infinities, each one bigger than the last.

[https://en.wikipedia.org/wiki/Logarithm Logarithm]: The inverse of an [[[technical-words#toc2|exponential]]] function. For example, the base 10 logarithm function takes on a value of [[$2$]] at an input of [[$100 = 10^2$]], [[$3$]] at [[$1000 = 10^3$]], and so on. In other words, it takes exponentially more effort to yield linearly higher outputs. For real-life examples, see the [https://en.wikipedia.org/wiki/Decibel decibel] scale and the frequencies of the [https://en.wikipedia.org/wiki/Electromagnetic_spectrum electromagnetic spectrum].

[https://en.wikipedia.org/wiki/Radix Radix]: Also known as the "base", this is the number of unique digits (including zero) of a positional numerical system. The decimal system (or base 10), is the default and most common system used. Binary (or base 2) is most often used for computing systems as computer bits only operate on two values: on (1) or off (0). Other often used bases are octal (base 8) and hexadecimal (base 16, with letters A through F being used for the last six digits), which computers use for a number of purposes (e.g. you may recognize hexidecimal from CSS color codes). As an example, the number [[$110$]] in octal is [[$1 \cdot 8^2 + 1 \cdot 8^1 + 0 \cdot 8^0 = 72$]] in decimal (denoted [[$110_8 = 72_{10}$]]), while the number [[$110$]] in binary is [[$1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 6$]] in decimal (i.e. [[$110_2 = 6_{10}$]]).

++ Patterns

[https://en.wikipedia.org/wiki/Accuracy_and_precision Accuracy]: How close a group of similar measurements are to the true value of the measurement. For example, a scattered group of shots towards a shooting target is accurate if the average position of the group is on the bullseye, even if they are spread out and thus not **precise**.

[https://en.wikipedia.org/wiki/Asymptote Asymptote]: The value, line, or behavior something tends to, but never reaches, as its input tends towards **infinity**; for example, an **exponentially** decaying function might get closer and closer to zero as time elapses, but it never properly reaches zero; as such, the function is said to asymptotically approach zero.

[https://en.wikipedia.org/wiki/Correlation Correlation]: A notable connection or relationship between two quantities. A positive correlation means that they both tend to go up/down together, e.g. when shark attack incidents go up, ice cream sales also go up; a negative correlation means that one tends to go up when the other goes down and vice versa, e.g. the less hours one sleeps before an exam, the more mistakes they tend to make. Some correlations arise from one **variable** influencing the other (or vice versa, or even both), such as the second example; other correlations have a underlying cause for both, such as the first example (warmer outdoor temperatures); others beyond that might simply be coincidences.

[https://en.wikipedia.org/wiki/Dyscalculia Dyscalculia]: A learning disability that causes difficulty dealing with numbers, arithmetic, and/or other concepts and procedures in mathematics. In effect, it is the math equivalent of dyslexia.

[https://en.wikipedia.org/wiki/Limit_(mathematics) Limit]: The value of a function or sequence's output as its input gets arbitrarily close to/approaches either a finite value or positive/negative **infinity**. For example, the limit of the function [[$f(x) = 1/x$]] as [[$x$]] approaches [[$\pm \infty$]] is [[$0$]] (in this case, [[$y = 0$]] is an **asymptote** of [[$f$]]); the limit of [[$g(x) = x^2$]] as [[$x$]] approaches [[$2$]] is [[$4$]] (which also equals [[$g(2)$]] in this case); the limit of [[$h(x) = 1/x^2$]] as [[$x$]] approaches [[$0$]] is [[$+ \infty$]].

[https://en.wikipedia.org/wiki/Accuracy_and_precision Precision]: How close a group of similar measurements are to each other. As an example, a tight group of shots on a shooting target is precise, even if the group is far away from the bullseye and is thus not **accurate**.

[https://en.wikipedia.org/wiki/Mathematical_proof Proof]: A deductive argument that shows the logical soundness and veracity of a mathematical statement (e.g. a theorem). As a simple example, say we want to prove that the sum [[$n + m$]] of two odd integers [[$n, m$]] must be even. By definition, you can write these odd numbers as [[$n = 2a + 1$]] and [[$m = 2b + 1$]] for some integers [[$a, b$]]. Thus, their sum can be written as the expression [[$(2a + 1) + (2b + 1)$]], which can be simplified and factored as [[$2(a + b + 1)$]]. Since [[$a + b + 1$]] is also an integer, the sum we acquired is even by definition. In higher level mathematics, proofs are long, involved, and involve many complicated steps and procedures; fortunately, the Foundation is no stranger to complexity.

[https://en.wikipedia.org/wiki/Proportionality_(mathematics) Proportional]: The relation between two values if the ratio between them is constant, i.e. [[$a = kb$]]. For example, "for every hour that passes in the real world, one year passes within the anomaly". Is considered a type of positive **correlation**. Two values are instead **inversely proportional** (and thus negatively correlated) if one is proportional to the reciprocal of the other, i.e. [[$a = k/b$]]; for example, travel time is inversely proportional to your speed. Linear relationships [[$a = kb + c$]] are like proportions, but may also be offset by some constant [[$c$]].

[https://en.wikipedia.org/wiki/Measurement_uncertainty Uncertainty]: The statistical "wiggle room" of a measurement. No measurement is exact: in addition to variations caused by operator skill and the particular measurement procedure, any quantity you measure has some inherent uncertainty due to the limited precision of the measuring tool used. For example, a standard ruler might have tick marks every millimeter (tenth of a centimeter), meaning that any measurement made with it will have a minimum uncertainty of one millimeter. You can write an uncertainty as, say, [[$(2.6 \pm 0.1) \text{ cm}$]] or [[$(30.531 \pm 0.015) \text{ kg}$]], but it's more concise to write it using parentheses notation, i.e. [[$2.6(1) \text{ cm}$]] or [[$30.531(15) \text{ kg}$]]. These work well for scientific articles where precise measurements of an anomaly are relevant.

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