Arithmetic

2024-06-1322:39 Status:Incomplete LaTeX Guide

I could literally start from the very beginnings of arithmetic, but I’m assuming that, by reading this, you have a basic grasp of addition, subtraction etc. and a strong understanding of elementary (school) arithmetic. The following will be an index for some of the more advanced areas of arithmetic in high school (Alberta) and IB Program.

Notation (I’m not doing anything like this again)

Arithmetic Operations

$+$ (Plus sign) +

• Denotes addition and is read as plus; for example, 3 + 2.
• Denotes that a number is positive and is read as plus. Redundant, but sometimes used for emphasizing that a number is positive, specially when other numbers in the context are or may be negative; for example, +2.
• Sometimes used instead of ⊔ for a disjoint union of sets.

$-$ (Minus sign) -

• Denotes subtraction and is read as minus; for example, 3 – 2.
• Denotes the additive inverse and is read as negative or the opposite of; for example, –2.
• Also used in place of \ for denoting the set-theoretic complement; see [](https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols#) in § Set theory.

$\times$ (Multiplication sign) \times

• In elementary arithmetic, denotes multiplication, and is read as times; for example, 3 × 2.
• In geometry and linear algebra, denotes the cross product.
• In set theory and category theory, denotes the Cartesian product and the direct product. See also × in § Set theory.

$\cdot$ (Dot) \cdot

• Denotes multiplication and is read as times; for example, 3 ⋅ 2.
• In geometry and linear algebra, denotes the dot product.
• Placeholder used for replacing an indeterminate element. For example, saying “the absolute value is denoted by | · |” is perhaps clearer than saying that it is denoted as | |.

$\pm$ (Plus–minus sign) \pm

• Denotes either a plus sign or a minus sign.
• Denotes the range of values that a measured quantity may have; for example, 10 ± 2 denotes an unknown value that lies between 8 and 12.

$\mp$ (Minus-plus sign) \mp

• Used paired with ±, denotes the opposite sign; that is, + if ± is –, and – if ± is +.

$÷$ (Division sign) \div

• Widely used for denoting division in Anglophone countries, it is no longer in common use in mathematics and its use is “not recommended”.[1] In some countries, it can indicate subtraction.

$:$ (Colon) :

• Denotes the ratio of two quantities.
• In some countries, may denote division.
• In set-builder notation, it is used as a separator meaning “such that”; see {□ : □}.

$/$ (Slash) /

• Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, 3 / 2 or $\frac{3}{2}$.
• Denotes a quotient structure. For example, quotient set, quotient group, quotient category, etc.
• In number theory and field theory, $F/E$ denotes a field extension, where $F$ is an extension field of the field $E$.
• In probability theory, denotes a conditional probability. For example, $P(A/B)$ denotes the probability of $A,giventhat$B$occurs.Alsodenoted$P(A|B)$.

$\surd$ (Square-root symbol) \sqrt

• Denotes square root and is read as the square root of. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, √2.

$\sqrt[n]{x}$ (Radical symbol) \sqrt[n]{x}

• Denotes square root and is read as the square root of. For example, $\sqrt{3+2}$
• With an integer greater than 2 as a left superscript, denotes an nth root. For example, $\sqrt[n]{x}$

^ (Caret) \caret, \circumflex, \hat

• Exponentiation is normally denoted with a superscript. However, �� is often denoted x^y when superscripts are not easily available, such as in programming languages (including LaTeX) or plain text emails.
• Not to be confused with ∧

Equality, Equivalence, and Similarity

$=$ (Equals sign) =

• Denotes equality.
• Used for naming a mathematical object in a sentence like let $x=E$ , where E is an expression. See also ≝, ≜ or :=

$\stackrel{\mathrm{def}}{=},\triangleq ,:=$ (Copy paste)

• Any of these is sometimes used for naming a mathematical object. Thus, $x\triangleq E,x\stackrel{\mathrm{def}}{=}E$, and $x:=E$ are each an abbreviation of the phrase “let $x=E$ where $E$i s an expression and $x$ is a variable. This is similar to the concept of assignment in computer science, which is variously denoted (depending on the programming language used) =,:=,←,…

$\ne$ (Not-equal sign) \neq

• Denotes inequality and means “not equal”.

$\approx ,\cong$ (Approximately equal to) \approx

• The most common symbol for denoting approximate equality. For example, $\pi \approx 3.14159$.
• May denote an isomorphism between two mathematical structures, and is read as “is isomorphic to”.
• In geometry, may denote the congruence of two geometric shapes (that is the equality up to a displacement), and is read “is congruent to”.

~  (Tilde) ~

• Between two numbers, either it is used instead of ≈ to mean “approximately equal”, or it means “has the same order of magnitude as”.
• Denotes the asymptotic equivalence of two functions or sequences.
• Often used for denoting other types of similarity, for example, matrix similarity or similarity of geometric shapes.
• Standard notation for an equivalence relation.
• In probability and statistics, may specify the probability distribution of a random variable. For example, $XN(0,1$)
• Notation for proportionality. See also ∝ for a less ambiguous symbol.

$\equiv$ (triple bar) \equiv

• Denotes an identity, that is, an equality that is true whichever values are given to the variables occurring in it.
• In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer.
• May denote a logical equivalence.

Comparison

$<$ (Less-than sign)

• Strict inequality between two numbers; means and is read as “less than”
• Commonly used for denoting any strict order.
• Between two groups, may mean that the first one is a proper subgroup of the second one.

$>$ (Greater-than sign)

• Strict inequality between two numbers; means and is read as “greater than”
• Commonly used for denoting any strict order.
• Between two groups, may mean that the second one is a proper subgroup of the first one.

$\le$ (Less than or equal to sign)

• Means “less than or equal to”. That is, whatever A and B are, A ≤ B is equivalent to A < B or A = B.
• Between two groups, may mean that the first one is a subgroup of the second one.
• (≦ is a a rarely used symbol, generally a synonym of ≤.)

$\ge$ (Greater than or equal to sign)

• Means “greater than or equal to”. That is, whatever A and B are, A ≥ B is equivalent to A > B or A = B.
• Between two groups, may mean that the second one is a subgroup of the first one.

$\ll$ and $\gg$

• Means “much less than” and “much greater than”. Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several orders of magnitude.
• In measure theory, $\mu \ll \nu$ means that the measure $\mu$ is absolutely continuous with respect to the measure $\nu$

≺ and ≻

• Often used for denoting an order or, more generally, a preorder, when it would be confusing or not convenient to use < and >.
• Sequention in asynchronous logic.

Topics

Series Sequences Binomial Theorem Pascal’s Triangle System of Equations Logarithms Exponents