Mathematical Notation

Under Construction: Consider this content a preview of the real thing which is coming soon (hopefully).

graph LR
classDef currentPage stroke:#333,stroke-width:4px

ALG(["fas:fa-trophy Algorithmis fas:fa-trophy "])

ASY_ANA(["fas:fa-check Asymptotic Analysis#160;"])
click ASY_ANA "./math-asymptotic-analysis"

MAT_NOT(["fas:fa-check Mathematical Notation#160;"])
click MAT_NOT "./math-notation"

class MAT_NOT currentPage

POL(["fas:fa-check Polynomials #160;"])
click POL "./math-polynomials"

MAT_FUN(["fas:fa-check Math Functions#160;"])
click MAT_FUN "./math-functions"

LOG(["fas:fa-check Logarithms#160;"])
click LOG "./math-logarithms"

COM(["fas:fa-check Combinatorics#160;"])
click COM "./math-combinatorics"

SET_NOT(["fas:fa-check Set Notation#160;"])
click SET_NOT "./math-set-notation"

GRA(["fas:fa-check Graphing#160;"])
click GRA "./math-graphing"

BW(["fas:fa-check Bitwise Logic#160;"])
click BW "./math-bitwise"

ASY_ANA-->ALG
BW-->ALG
COM & GRA & SET_NOT-->ASY_ANA
MAT_NOT--> SET_NOT
POL & LOG--> MAT_FUN
MAT_FUN--> GRA


It’s important to become fluent in written math for two reasons. The first is that reading and writing proofs is requisite for algorithm mastery. The second is that it is an efficient medium for communicating dense concepts. The concise symbols that comprise the language of math is far more expressive and precise than natural languages such as English, Spanish, etc…

Those in the software field often struggle with mathematical notation because written math, while far more precise than natural language, is not nearly as Draconian as typical programming languages. Miswritten syntax in math has no impact (assuming it’s still comprehensible). A syntax error while programming results in a compiler error. Remember that mathematical notation is a matter of convention and authors tend to interpret the convention loosely. There are several correct ways to write the same thing. For instance, $A^\prime$, $A^\complement$, and $A^\sim$ are all equivalent. What’s worse, it’s also common to see the same symbol with a different meaning depending on the context. For instance, the symbol $\Sigma$ denotes a summation; however, it’s not uncommon to define it to be an arbitrary variable. Deriving meaning from mathematical notation takes practice. Do not be discouraged if it isn’t painfully obvious at first. Keeping all this in mind, understand that every attempt is made to showcase the most common representations; however, it’s highly likely that you will encounter variations.

PRO TIP: Bookmark this page in the event that you encounter a symbol you don’t recognize.

Things that should be covered

• May not be logical sequential order
• floor & ceiling $\lfloor x \rfloor \lceil x \rceil$
• minimum and maximum $\min, \max$
• bar $\bar{x}$
• prime $x^{\prime}, x^{\prime\prime}$
• grave $\grave{x}$
• hat $\hat{x}$
• tilde $\tilde{x}$
• vector $\vec{x}$

Conflicting meanings $\vert x \vert$ can mean

• The absolute or positive value of x.
• The length of the vector x.
• The cardinality of the set x.

Greek alphabet

Name lower UPPER Name lower UPPER
Alpha $\alpha$ $A$ Nu $\nu$ $N$
Beta $\beta$ $B$ Omicron $\omicron$ $O$
Gamma $\gamma$ $\Gamma$ Pi $\pi$ $\Pi$
Delta $\delta$ $\Delta$ Rho $\rho$ $R$
Epsilon $\epsilon$ $E$ Sigma $\sigma$ $\Sigma$
Zeta $\zeta$ $Z$ Tau $\tau$ $T$
Eta $\eta$ $E$ Upsilon $\upsilon$ $\Upsilon$
Theta $\theta$ $\Theta$ Phi $\phi$ $\Phi$
Iota $\iota$ $I$ Chi $\chi$ $X$
Kappa $\kappa$ $K$ Psi $\psi$ $\Psi$
Lambda $\lambda$ $\Lambda$ Omega $\omega$ $\Omega$
Mu $\mu$ $M$

https://xkcd.com/2343/