Name	Mathematical constant	Value	Notation
One	$1$	1	M_ONE
Two	$2$	2	M_TWO
One half	$1/2$	0.5	M_ONE_HALF
Ludolph's number	$π$	3.1415926535898	M_PI
Tau	$τ=2\cdot π$	6.2831853071796	M_TAU
Euler's number	$e=\sum_{n=0}^{\infty}\cfrac{1}{n!}=1+\cfrac{1}{1}+\cfrac{1}{1\cdot 2}+\cfrac{1}{1\cdot 2\cdot 3}+\cdots$	2.718281828459	M_E
Euler's constant	$γ=\lim_{n\rightarrow\infty}\left(-\log{n}+\sum_{k=1}^{n}\right)\cfrac{1}{k}$	0.57721566490153	M_EULER
Apéry's constant	$ζ(3)=\sum_{n=1}^{\infty}\cfrac{1}{n^3}=1+\cfrac{1}{2^3}+\cfrac{1}{3^3}+\cfrac{1}{4^3}+\cfrac{1}{5^3}+\cdots$	1.2020569031596	M_APERY
Catalan's constant	$G=\sum_{n=0}^{\infty}\cfrac{\left(-1\right)^n}{\left(2n+1\right)^2}=\cfrac{1}{1^2}-\cfrac{1}{3^2}+\cfrac{1}{5^2}-\cfrac{1}{7^2}+\cfrac{1}{9^2}-\cdots$	0.91596559417722	M_CATALAN
Feigenbaum constant α	$α$	2.5029078750959	M_FEIGENBAUM_ALPHA
Feigenbaum constant δ	$δ$	4.669201609103	M_FEIGENBAUM_DELTA
Lemniscate constant	$ϖ=2\int_{0}^{1}\cfrac{\text{d}t}{\sqrt{1-t^4}}$	2.6220575542921	M_LEMNISCATE
Glaisher–Kinkelin constant	$A$	1.2824271291006	M_GLAISHER
Khinchin's constant	$K_0=\lim_{n\rightarrow\infty}\left(a_1a_2\ldots a_n\right)^{1/n}$	2.6854520010653	M_KHINCHIN
Golden Ratio	$φ=\cfrac{1+\sqrt{5}}{2}$	1.6180339887499	M_GOLDEN_RATIO
Silver Ratio	$δ_S=\sqrt{2}+1$	2.4142135623731	M_SILVER_RATIO
Supergolden Ratio	$ψ=\cfrac{1+\sqrt[3]{\cfrac{29+3\cdot\sqrt{93}}{2}}+\sqrt[3]{\cfrac{29-3\cdot\sqrt{93}}{2}}}{3}$	1.4655712318768	M_SUPERGOLDEN_RATIO
Connective constant	$μ=\sqrt{2+\sqrt{2}}$	1.8477590650226	M_CONNECTIVE
Kepler–Bouwkamp constant	$K'=\prod_{n=3}^{\infty}\cos\left(\cfrac{π}{n}\right)=\cos\left(\cfrac{π}{3}\right)\cos\left(\cfrac{π}{4}\right)\cos\left(\cfrac{π}{5}\right)\cdots$	0.1149420448533	M_KEPLER_BOUWKAMP
Erdős–Borwein constant	$E=\sum_{n=1}^{\infty}\cfrac{1}{2^n-1}=\cfrac{1}{1}+\cfrac{1}{3}+\cfrac{1}{7}+\cfrac{1}{15}\cdots$	1.6066951524153	M_ERDOS_BORWEIN
Omega constant	$Ω=\cfrac{1}{π}\int_{0}^{π}\log\left(1+\cfrac{\sin t}{t}e^{t \cot t}\right)dt$	0.56714329040978	M_OMEGA
Gauss's constant	$G=\cfrac{1}{\text{agm}\left(1, \sqrt{2}\right)}=\cfrac{1}{4π}\sqrt{\cfrac{2}{π}}Γ\left(\cfrac{1}{4}\right)^2=\cfrac{ϖ}{π}$	0.83462684167407	M_GAUSS
Second Hermite constant	$γ_2=\cfrac{2}{\sqrt{3}}$	1.1547005383793	M_SECOND_HERMITE
Liouville's constant	$L=\sum_{n=1}^{\infty}\cfrac{1}{10^{n!}}=\cfrac{1}{10^{1!}}+\cfrac{1}{10^{2!}}+\cfrac{1}{10^{3!}}+\cfrac{1}{10^{4!}}+\cdots$	0.110001	M_LIOUVILLE
Ramanujan's constant	${e}^{π\cdot\sqrt{163}}$	2.6253741264077E+17	M_RAMANUJAN
Dottie number	$D$	0.73908513321516	M_DOTTIE
Meissel-Mertens constant	$M=\lim_{n\rightarrow\infty}\left(\sum_{p\le n}\cfrac{1}{p}-\ln\left(\ln n\right)\right)=γ+\sum_{p}\left(\ln\left(1-\cfrac{1}{p}\right)+\cfrac{1}{p}\right)$	0.26149721284764	M_MEISSEL_MERTENS
Universal parabolic constant	$\ln{\left(1+\sqrt{2}\right)}+\sqrt{2}$	2.2955871493926	M_UNIVERSAL_PARABOLIC
Cahen's constant	$C=\sum_{k=1}^{\infty}\cfrac{\left(-1\right)^k}{s_k-1}=\cfrac{1}{1}-\cfrac{1}{2}+\cfrac{1}{6}-\cfrac{1}{42}+\cfrac{1}{1806}\pm\cdots$	0.64341054628834	M_CAHEN
Gelfond's constant	${e}^π$	23.140692632779	M_GELFOND
Gelfond-Schneider constant	$2^{\sqrt{2}}$	2.6651441426902	M_GELFOND_SCHNEIDER
Second Favard constant	$K_2=\cfrac{π^2}{8}$	1.2337005501362	M_SECOND_FAVARD
Golden angle	$g=π\cdot\left(3-\sqrt{5}\right)$	2.3999632297287	M_GOLDEN_ANGLE
Sierpiński's constant	$K=π\left(2γ+\ln\cfrac{4π^3}{Γ\left(\cfrac{1}{4}\right)^4}\right)=π\left(2γ+4\ln Γ\left(\cfrac{3}{4}\right)-\ln π\right)=π\left(2\ln2+3\ln π+2γ-4\ln Γ\left(\cfrac{1}{4}\right)\right)$	2.5849817595793	M_SIERPINSKI
Landau-Ramanujan constant	$b=\cfrac{1}{\sqrt{2}}\prod_{p\equiv 3\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{-\cfrac{1}{2}}=\cfrac{π}{4}\prod_{p\equiv 1\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{\cfrac{1}{2}}$	0.76422365358922	M_LANDAU_RAMANUJAN
First Nielsen-Ramanujan constant	$a_1=\cfrac{π^2}{12}$	0.82246703342411	M_FIRST_NIELSEN_RAMANUJAN
Gieseking constant	$G=\cfrac{3\sqrt{3}}{4}\left(1-\sum_{n=0}^{\infty}\cfrac{1}{\left(3n+2\right)^2}+\sum_{n=1}^{\infty}\cfrac{1}{\left(3n+1\right)^2}\right)=\cfrac{\sqrt{3}}{4}\left(\cfrac{ψ_1\left(1/3\right)}{2}-\cfrac{π^2}{3}\right)$	1.0149416064097	M_GIESEKING
Bernstein's constant	$β=\lim_{n\rightarrow\infty}2nE_{2n}\left(f\right)$	0.28016949902387	M_BERNSTEIN
Tribonacci constant	$\cfrac{1+\sqrt[3]{19+3\cdot\sqrt{33}}+\sqrt[3]{19-3\cdot\sqrt{33}}}{3}$	1.8392867552142	M_TRIBONACCI
Brun's constant	$B_2=\sum_{p}\left(\cfrac{1}{p}+\cfrac{1}{p+2}\right)=\left(\cfrac{1}{3}+\cfrac{1}{5}\right)+\left(\cfrac{1}{5}+\cfrac{1}{7}\right)+\left(\cfrac{1}{11}+\cfrac{1}{13}\right)\cdots$	1.902160583104	M_BRUN
Twin primes constant	$C_2=\prod_{p\ \text{prime,}\ p\geq3}\left(1-\cfrac{1}{\left(p-1\right)^2}\right)$	0.66016181584687	M_TWIN_PRIMES
Plastic Ratio	$ρ=\sqrt[3]{\cfrac{1}{2}+\cfrac{\sqrt{69}}{18}}+\sqrt[3]{\cfrac{1}{2}-\cfrac{\sqrt{69}}{18}}$	1.3247179572447	M_PLASTIC_RATIO
Prouhet-Thue-Morse constant	$τ=\sum_{n=0}^{\infty}\cfrac{t_n}{2^{n+1}}=\cfrac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\cfrac{1}{2^{2^n}}\right)\right]$	0.41245403364011	M_PROUHET_THUE_MORSE
Golomb-Dickman constant	$λ=\int_0^1e^{Li\left(t\right)}dt=\int_0^{\infty}\cfrac{ρ\left(t\right)}{t+2}dt$	0.62432998854355	M_GOLOMB_DICKMAN
Lebesgue constant	$c=\lim_{n\rightarrow\infty}\left(L_n-\cfrac{4}{π^2}\ln\left(2n+1\right)\right)$	0.98943127383115	M_LEBESGUE
Feller-Tornier constant	$C_{FT}=\cfrac{1}{2}\prod_{p\ \text{prime}}\left(1-\cfrac{2}{p^2}\right)+\cfrac{1}{2}$	0.66131704946962	M_FELLER_TORNIER
Champernowne constant	$C_{10}=0.1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\cdots$	0.12345678910111	M_CHAMPERNOWNE
Salem constant	$σ_{10}=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$	1.1762808182599	M_SALEM
Lévy's constant	$β=\cfrac{π^2}{12\cdot\ln{2}}$	1.1865691104156	M_LEVY
Copeland-Erdős constant	$C_{CE}=0.2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\ 23\ 29\ 31\ 37\cdots$	0.23571113171923	M_COPELAND_ERDOS
Mills' constant	$\left\lfloor A^{3^n}\right\rfloor$	1.3063778838631	M_MILLS
Gompertz constant	$δ=\int_0^{\infty}\cfrac{e^{-x}}{1+x}dx$	0.59634736232319	M_GOMPERTZ
Van der Pauw constant	$\cfrac{π}{\ln{2}}$	4.5323601418272	M_VAN_DER_PAUW
Magic angle	$θ_m=\tan^{-1}{\sqrt{2}}$	0.95531661812451	M_MAGIC_ANGLE
Artin's constant	$C_{Artin}=\prod_{p\ \text{prime}}\left(1-\cfrac{1}{p\left(p-1\right)}\right)$	0.3739558136192	M_ARTIN
Porter's constant	$C=\cfrac{6\ln 2}{π^2}\left(3\ln 2+4γ-\cfrac{24}{π^2}ζ'\left(2\right)-2\right)-\cfrac{1}{2}$	1.467078079434	M_PORTER
Lochs constant	$L=\cfrac{6\ln 2\ln 10}{π^2}$	0.97027011439203	M_LOCHS
Lieb's square ice constant	$\left(\cfrac{4}{3}\right)^{\cfrac{3}{2}}$	1.539600717839	M_LIEB_SQUARE_ICE
Niven's constant	$C=1+\sum_{n=2}^{\infty}\left(1-\cfrac{1}{ζ\left(n\right)}\right)$	1.7052111401054	M_NIVEN
Stephens' constant	$C_S=\prod_{p\ \text{prime}}\left(1-\cfrac{p}{p^3-1}\right)$	0.57595996889295	M_STEPHENS
Zero	$0$	0	M_ZERO
Negative one	$-1$	-1	M_NEGATIVE_ONE
Square Root of 2	$\sqrt{2}$	1.4142135623731	M_SQRT2
Square Root of 3	$\sqrt{3}$	1.7320508075689	M_SQRT3
Square Root of 5	$\sqrt{5}$	2.2360679774998	M_SQRT5
Cube Root of 2	$\sqrt[3]{2}$	1.2599210498949	M_CURT2
Cube Root of 3	$\sqrt[3]{3}$	1.4422495703074	M_CURT3
Twelfth Root of 2	$\sqrt[12]{2}$	1.0594630943593	M_TWRT2
Natural Log of 2	$\ln(2)$	0.69314718055995	M_LN2
Natural Log of 10	$\ln(10)$	2.302585092994	M_LN10
Natural Log of Pi	$\ln(π)$	1.1447298858494	M_LNPI
Base 10 Log of e	$\log10(e)$	0.43429448190325	M_LOG10E
Base 2 Log of e	$\log2(e)$	1.442695040889	M_LOG2E
Half of Pi	$π/2$	1.5707963267949	M_PI_2
Quarter of Pi	$π/4$	0.78539816339745	M_PI_4
Inverse of Pi	$1/π$	0.31830988618379	M_1_PI
Two over Pi	$2/π$	0.63661977236758	M_2_PI
Square Root of Pi	$\sqrt{π}$	1.7724538509055	M_SQRTPI
Two over Square Root of Pi	$2/\sqrt{π}$	1.1283791670955	M_2_SQRTPI
Inverse of Square Root of 2	$1/\sqrt{2}$	0.70710678118655	M_SQRT1_2

Name

Mathematical constant

Value

Notation

One

$1$

M_ONE

Two

$2$

M_TWO

One half

$1/2$

0.5

M_ONE_HALF

Ludolph's number

$π$

3.1415926535898

M_PI

Tau

$τ=2\cdot π$

6.2831853071796

M_TAU

Euler's number

$e=\sum_{n=0}^{\infty}\cfrac{1}{n!}=1+\cfrac{1}{1}+\cfrac{1}{1\cdot 2}+\cfrac{1}{1\cdot 2\cdot 3}+\cdots$

2.718281828459

M_E

Euler's constant

$γ=\lim_{n\rightarrow\infty}\left(-\log{n}+\sum_{k=1}^{n}\right)\cfrac{1}{k}$

0.57721566490153

M_EULER

Apéry's constant

$ζ(3)=\sum_{n=1}^{\infty}\cfrac{1}{n^3}=1+\cfrac{1}{2^3}+\cfrac{1}{3^3}+\cfrac{1}{4^3}+\cfrac{1}{5^3}+\cdots$

1.2020569031596

M_APERY

Catalan's constant

$G=\sum_{n=0}^{\infty}\cfrac{\left(-1\right)^n}{\left(2n+1\right)^2}=\cfrac{1}{1^2}-\cfrac{1}{3^2}+\cfrac{1}{5^2}-\cfrac{1}{7^2}+\cfrac{1}{9^2}-\cdots$

0.91596559417722

M_CATALAN

Feigenbaum constant α

$α$

2.5029078750959

M_FEIGENBAUM_ALPHA

Feigenbaum constant δ

$δ$

4.669201609103

M_FEIGENBAUM_DELTA

Lemniscate constant

$ϖ=2\int_{0}^{1}\cfrac{\text{d}t}{\sqrt{1-t^4}}$

2.6220575542921

M_LEMNISCATE

Glaisher–Kinkelin constant

$A$

1.2824271291006

M_GLAISHER

Khinchin's constant

$K_0=\lim_{n\rightarrow\infty}\left(a_1a_2\ldots a_n\right)^{1/n}$

2.6854520010653

M_KHINCHIN

Golden Ratio

$φ=\cfrac{1+\sqrt{5}}{2}$

1.6180339887499

M_GOLDEN_RATIO

Silver Ratio

$δ_S=\sqrt{2}+1$

2.4142135623731

M_SILVER_RATIO

Supergolden Ratio

$ψ=\cfrac{1+\sqrt[3]{\cfrac{29+3\cdot\sqrt{93}}{2}}+\sqrt[3]{\cfrac{29-3\cdot\sqrt{93}}{2}}}{3}$

1.4655712318768

M_SUPERGOLDEN_RATIO

Connective constant

$μ=\sqrt{2+\sqrt{2}}$

1.8477590650226

M_CONNECTIVE

Kepler–Bouwkamp constant

$K'=\prod_{n=3}^{\infty}\cos\left(\cfrac{π}{n}\right)=\cos\left(\cfrac{π}{3}\right)\cos\left(\cfrac{π}{4}\right)\cos\left(\cfrac{π}{5}\right)\cdots$

0.1149420448533

M_KEPLER_BOUWKAMP

Erdős–Borwein constant

$E=\sum_{n=1}^{\infty}\cfrac{1}{2^n-1}=\cfrac{1}{1}+\cfrac{1}{3}+\cfrac{1}{7}+\cfrac{1}{15}\cdots$

1.6066951524153

M_ERDOS_BORWEIN

Omega constant

$Ω=\cfrac{1}{π}\int_{0}^{π}\log\left(1+\cfrac{\sin t}{t}e^{t \cot t}\right)dt$

0.56714329040978

M_OMEGA

Gauss's constant

$G=\cfrac{1}{\text{agm}\left(1, \sqrt{2}\right)}=\cfrac{1}{4π}\sqrt{\cfrac{2}{π}}Γ\left(\cfrac{1}{4}\right)^2=\cfrac{ϖ}{π}$

0.83462684167407

M_GAUSS

Second Hermite constant

$γ_2=\cfrac{2}{\sqrt{3}}$

1.1547005383793

M_SECOND_HERMITE

Liouville's constant

$L=\sum_{n=1}^{\infty}\cfrac{1}{10^{n!}}=\cfrac{1}{10^{1!}}+\cfrac{1}{10^{2!}}+\cfrac{1}{10^{3!}}+\cfrac{1}{10^{4!}}+\cdots$

0.110001

M_LIOUVILLE

Ramanujan's constant

${e}^{π\cdot\sqrt{163}}$

2.6253741264077E+17

M_RAMANUJAN

Dottie number

$D$

0.73908513321516

M_DOTTIE

Meissel-Mertens constant

$M=\lim_{n\rightarrow\infty}\left(\sum_{p\le n}\cfrac{1}{p}-\ln\left(\ln n\right)\right)=γ+\sum_{p}\left(\ln\left(1-\cfrac{1}{p}\right)+\cfrac{1}{p}\right)$

0.26149721284764

M_MEISSEL_MERTENS

Universal parabolic constant

$\ln{\left(1+\sqrt{2}\right)}+\sqrt{2}$

2.2955871493926

M_UNIVERSAL_PARABOLIC

Cahen's constant

$C=\sum_{k=1}^{\infty}\cfrac{\left(-1\right)^k}{s_k-1}=\cfrac{1}{1}-\cfrac{1}{2}+\cfrac{1}{6}-\cfrac{1}{42}+\cfrac{1}{1806}\pm\cdots$

0.64341054628834

M_CAHEN

Gelfond's constant

${e}^π$

23.140692632779

M_GELFOND

Gelfond-Schneider constant

$2^{\sqrt{2}}$

2.6651441426902

M_GELFOND_SCHNEIDER

Second Favard constant

$K_2=\cfrac{π^2}{8}$

1.2337005501362

M_SECOND_FAVARD

Golden angle

$g=π\cdot\left(3-\sqrt{5}\right)$

2.3999632297287

M_GOLDEN_ANGLE

Sierpiński's constant

$K=π\left(2γ+\ln\cfrac{4π^3}{Γ\left(\cfrac{1}{4}\right)^4}\right)=π\left(2γ+4\ln Γ\left(\cfrac{3}{4}\right)-\ln π\right)=π\left(2\ln2+3\ln π+2γ-4\ln Γ\left(\cfrac{1}{4}\right)\right)$

2.5849817595793

M_SIERPINSKI

Landau-Ramanujan constant

$b=\cfrac{1}{\sqrt{2}}\prod_{p\equiv 3\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{-\cfrac{1}{2}}=\cfrac{π}{4}\prod_{p\equiv 1\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{\cfrac{1}{2}}$

0.76422365358922

M_LANDAU_RAMANUJAN

First Nielsen-Ramanujan constant

$a_1=\cfrac{π^2}{12}$

0.82246703342411

M_FIRST_NIELSEN_RAMANUJAN

Gieseking constant

$G=\cfrac{3\sqrt{3}}{4}\left(1-\sum_{n=0}^{\infty}\cfrac{1}{\left(3n+2\right)^2}+\sum_{n=1}^{\infty}\cfrac{1}{\left(3n+1\right)^2}\right)=\cfrac{\sqrt{3}}{4}\left(\cfrac{ψ_1\left(1/3\right)}{2}-\cfrac{π^2}{3}\right)$

1.0149416064097

M_GIESEKING

Bernstein's constant

$β=\lim_{n\rightarrow\infty}2nE_{2n}\left(f\right)$

0.28016949902387

M_BERNSTEIN

Tribonacci constant

$\cfrac{1+\sqrt[3]{19+3\cdot\sqrt{33}}+\sqrt[3]{19-3\cdot\sqrt{33}}}{3}$

1.8392867552142

M_TRIBONACCI

Brun's constant

$B_2=\sum_{p}\left(\cfrac{1}{p}+\cfrac{1}{p+2}\right)=\left(\cfrac{1}{3}+\cfrac{1}{5}\right)+\left(\cfrac{1}{5}+\cfrac{1}{7}\right)+\left(\cfrac{1}{11}+\cfrac{1}{13}\right)\cdots$

1.902160583104

M_BRUN

Twin primes constant

$C_2=\prod_{p\ \text{prime,}\ p\geq3}\left(1-\cfrac{1}{\left(p-1\right)^2}\right)$

0.66016181584687

M_TWIN_PRIMES

Plastic Ratio

$ρ=\sqrt[3]{\cfrac{1}{2}+\cfrac{\sqrt{69}}{18}}+\sqrt[3]{\cfrac{1}{2}-\cfrac{\sqrt{69}}{18}}$

1.3247179572447

M_PLASTIC_RATIO

Prouhet-Thue-Morse constant

$τ=\sum_{n=0}^{\infty}\cfrac{t_n}{2^{n+1}}=\cfrac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\cfrac{1}{2^{2^n}}\right)\right]$

0.41245403364011

M_PROUHET_THUE_MORSE

Golomb-Dickman constant

$λ=\int_0^1e^{Li\left(t\right)}dt=\int_0^{\infty}\cfrac{ρ\left(t\right)}{t+2}dt$

0.62432998854355

M_GOLOMB_DICKMAN

Lebesgue constant

$c=\lim_{n\rightarrow\infty}\left(L_n-\cfrac{4}{π^2}\ln\left(2n+1\right)\right)$

0.98943127383115

M_LEBESGUE

Feller-Tornier constant

$C_{FT}=\cfrac{1}{2}\prod_{p\ \text{prime}}\left(1-\cfrac{2}{p^2}\right)+\cfrac{1}{2}$

0.66131704946962

M_FELLER_TORNIER

Champernowne constant

$C_{10}=0.1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\cdots$

0.12345678910111

M_CHAMPERNOWNE

Salem constant

$σ_{10}=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$

1.1762808182599

M_SALEM

Lévy's constant

$β=\cfrac{π^2}{12\cdot\ln{2}}$

1.1865691104156

M_LEVY

Copeland-Erdős constant

$C_{CE}=0.2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\ 23\ 29\ 31\ 37\cdots$

0.23571113171923

M_COPELAND_ERDOS

Mills' constant

$\left\lfloor A^{3^n}\right\rfloor$

1.3063778838631

M_MILLS

Gompertz constant

$δ=\int_0^{\infty}\cfrac{e^{-x}}{1+x}dx$

0.59634736232319

M_GOMPERTZ

Van der Pauw constant

$\cfrac{π}{\ln{2}}$

4.5323601418272

M_VAN_DER_PAUW

Magic angle

$θ_m=\tan^{-1}{\sqrt{2}}$

0.95531661812451

M_MAGIC_ANGLE

Artin's constant

$C_{Artin}=\prod_{p\ \text{prime}}\left(1-\cfrac{1}{p\left(p-1\right)}\right)$

0.3739558136192

M_ARTIN

Porter's constant

$C=\cfrac{6\ln 2}{π^2}\left(3\ln 2+4γ-\cfrac{24}{π^2}ζ'\left(2\right)-2\right)-\cfrac{1}{2}$

1.467078079434

M_PORTER

Lochs constant

$L=\cfrac{6\ln 2\ln 10}{π^2}$

0.97027011439203

M_LOCHS

Lieb's square ice constant

$\left(\cfrac{4}{3}\right)^{\cfrac{3}{2}}$

1.539600717839

M_LIEB_SQUARE_ICE

Niven's constant

$C=1+\sum_{n=2}^{\infty}\left(1-\cfrac{1}{ζ\left(n\right)}\right)$

1.7052111401054

M_NIVEN

Stephens' constant

$C_S=\prod_{p\ \text{prime}}\left(1-\cfrac{p}{p^3-1}\right)$

0.57595996889295

M_STEPHENS

Zero

$0$

M_ZERO

Negative one

$-1$

-1

M_NEGATIVE_ONE

Square Root of 2

$\sqrt{2}$

1.4142135623731

M_SQRT2

Square Root of 3

$\sqrt{3}$

1.7320508075689

M_SQRT3

Square Root of 5

$\sqrt{5}$

2.2360679774998

M_SQRT5

Cube Root of 2

$\sqrt[3]{2}$

1.2599210498949

M_CURT2

Cube Root of 3

$\sqrt[3]{3}$

1.4422495703074

M_CURT3

Twelfth Root of 2

$\sqrt[12]{2}$

1.0594630943593

M_TWRT2

Natural Log of 2

$\ln(2)$

0.69314718055995

M_LN2

Natural Log of 10

$\ln(10)$

2.302585092994

M_LN10

Natural Log of Pi

$\ln(π)$

1.1447298858494

M_LNPI

Base 10 Log of e

$\log10(e)$

0.43429448190325

M_LOG10E

Base 2 Log of e

$\log2(e)$

1.442695040889

M_LOG2E

Half of Pi

$π/2$

1.5707963267949

M_PI_2

Quarter of Pi

$π/4$

0.78539816339745

M_PI_4

Inverse of Pi

$1/π$

0.31830988618379

M_1_PI

Two over Pi

$2/π$

0.63661977236758

M_2_PI

Square Root of Pi

$\sqrt{π}$

1.7724538509055

M_SQRTPI

Two over Square Root of Pi

$2/\sqrt{π}$

1.1283791670955

M_2_SQRTPI

Inverse of Square Root of 2

$1/\sqrt{2}$

0.70710678118655

M_SQRT1_2

Function	Description	Sample example
ABS	Returns the absolute value of a number. The absolute value of a number is the same number without a sign.	ABS(-9)=9
ACCRINT	Returns the total interest on securities that pay interest on regular dates.	ACCRINT(39508, 39691, 39569, 0.1, 1000, 2, 0)=16.666666666667
ACCRINTM	Returns the accrued interest on a security that pays interest on the maturity date.	ACCRINTM(39539, 39614, 0.1, 1000, 3)=20.547945205479
ACOS	Returns the arccosine (the inverse of the cosine function) of the specified number. The arccosine is the angle whose cosine is the specified number. The resulting angle is given in radians in the range zero to pi.	ACOS(0.5)=1.0471975511966
ACOSH	Returns the inverse hyperbolic cosine of a number. The number must be greater than or equal to 1. The inverse hyperbolic cosine is a value whose hyperbolic cosine is a number, so ACOSH(COSH(number)) equals the number.	ACOSH(2)=1.3169578969248
ACOT	Returns the value of the arccotangent (inverse cotangent) of the specified number.	ACOT(3.14)=0.30831566219543
ACOTH	Returns the inverse hyperbolic cotangent of the specified number.	ACOTH(3.14)=0.32994497940173
AMORDEGRC	Returns the depreciation for one accounting period. This function is adapted to the French accounting model. If the asset was acquired during the accounting period, the depreciation is calculated on a straight-line basis. The AMORDEGRC function is similar to the AMORLINC function, but differs in that the depreciation coefficient depends on the useful life of the asset.	AMORDEGRC(2400, 39679, 39813, 300, 1, 0.15, 1)=776
AMORLINC	Returns the depreciation for one accounting period. This function is adapted to the French accounting model. If the asset was acquired during the accounting period, proportional depreciation is considered.	AMORLINC(2400, 39679, 39813, 300, 1, 0.15, 1)=360
ARABIC	Converts Roman numerals to Arabic numerals.	ARABIC(V)=5
ASIN	Returns the arcsine (the inverse of the sine) of the specified number. The arcsine is the angle whose sine is the specified number. The resulting angle is given in radians in the interval from pi/2 to pi/2.	ASIN(1)=1.5707963267949
ASINH	Returns the hyperbolic arcsine of the specified number. The hyperbolic arcsine is a value whose hyperbolic sine is the specified number, so ARCSINH(SINH(number)) is equal to the specified number.	ASINH(1)=0.88137358701954
ATAN	Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.	ATAN(2)=1.1071487177941
ATAN2	Returns the arctangent (the inverse of tangent) of the specified x and y coordinates. The arctangent is the angle between the x axis and the line containing the origin (0;0) and the point at coordinates (x_number, y_number). This angle is given in radians in the interval -pi to pi, except for the value -pi.	ATAN2(2, 3)=0.98279372324733
ATANH	Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.	ATANH(0.5)=0.54930614433405
AVEDEV	Returns the average of the absolute deviations of data points from their mean. STANDARD DEVIATION is a measure of the variability of a data set.	AVEDEV(10, 30, 2, 3)=9.375
AVERAGEA	Calculates the average (arithmetic mean) of the values in the argument list.	AVERAGEA(10, 20, 30)=20
BESSELI	Returns a modified Bessel function that is equivalent to the Bessel function calculated for purely imaginary arguments.	BESSELI(1, 2)=0.1357476665807
BESSELJ	Returns the Bessel function.	BESSELJ(1, 2)=0.1149034849319
BESSELK	Returns a modified Bessel function that is equivalent to the Bessel function calculated for purely imaginary arguments.	BESSELK(1, 2)=1.6248388844172
BESSELY	Returns the Bessel function, also known as the Weber or Neumann function.	BESSELY(1, 2)=-1.6506826133039
BIN2DEC	Converts a binary number to decimal.	BIN2DEC(1101)=13
CEILING	Rounds a number up to the nearest integer or to the nearest multiple of significance.	CEILING(1.2)=2
CEILING.MATH	The CEILING.MATH function rounds a number up to the nearest integer or, optionally, to the nearest multiple of significance.	CEILING.MATH(1.2, 1)=2
CEILING.PRECISE	Returns a number that is rounded up to the nearest integer or to the nearest multiple of significance. Regardless of the sign of the number, the number is rounded up. However, if the number or the significance is zero, zero is returned.	CEILING.PRECISE(1.2, 1)=2
CHIDIST	Returns the right-tailed probability of a chi-square distribution. The χ2 distribution is related to the χ2 test. The χ2 test compares observed and expected values. For example, in a genetic experiment, you might predict that the next generation of plants will have flowers of a certain color. By comparing the observed results with the expected results, you can determine whether the original prediction is valid.	CHIDIST(15, 15)=0.45141721122575
CHIINV	Returns the inverse of the right-tailed probability distribution function of the chi-square distribution. If the probability is equal to CHIDIST(x;...), the value of CHIINV(probability;...) is equal to x. Use this function to compare observed results with expected results to determine whether the original assumption is valid.	CHIINV(0.05, 10)=18.307038053334
CODE	Returns the numeric code of the first character of a given text string.	CODE(1)=49
COMBIN	Returns the number of combinations for a specified number of elements. Use the COMBIN function to determine the total number of possible groups for a specified number of elements.	COMBIN(15, 9)=5005
COMBINA	Returns the number of combinations (with repetition) for a given number of items.	COMBINA(15, 6)=38760
COS	Returns the cosine of the specified angle.	COS(1)=0.54030230586814
COSH	Returns the hyperbolic cosine of the specified number.	COSH(1)=1.5430806348152
COT	Returns the cotangent of an angle specified in radians.	COT(1)=0.64209261593433
COTH	Returns the hyperbolic cotangent of a hyperbolic angle.	COTH(1)=1.3130352854993
COUNT	Returns the number of items in the argument list.	COUNT(1, 2)=2
CSC	Returns the cosecant of an angle specified in radians.	CSC(1)=1.1883951057781
CSCH	Returns the hyperbolic cosecant of an angle specified in radians.	CSCH(1)=0.85091812823932
DEGREES	Converts radians to degrees.	DEGREES(PI())=180
DELTA	Tests the equality of two numbers. Returns 1 if number1 = number2; otherwise, returns 0. This function can be used to filter a set of values. For example, when adding several DELTA functions, you can find the number of identical pairs. This function is also known as the Kronecker Delta function.	DELTA(2, 2)=1
DEVSQ	Returns the sum of the squares of the deviations of data points from their sample mean.	DEVSQ(10, 3, 2)=38
EXP	Returns e raised to the power of the argument number. The constant e is equal to 2.718281828459, the base of natural logarithms.	EXP(1)=2.718281828459
FACT	Returns the factorial of a number. The factorial of a number is equal to 123... number.	FACT(9.5)=362880
FACTDOUBLE	Returns the double factorial of the specified number.	FACTDOUBLE(12)=46080
FISHER	Returns the Fisher transformation value of x. This transformation produces a function with a normal distribution rather than a skewed distribution. You can use this function to test the hypothesis about the correlation coefficient.	FISHER(0.9)=1.4722194895832
FISHERINV	Returns the inverse of the Fisher transform. You can use this transform to analyze correlations between regions or matrices of data. If y = FISHER(x), FISHERINV(y) = x.	FISHERINV(10)=0.99999999587769
FLOOR	Rounds a number down to the nearest whole number or to the nearest multiple of the specified value.	FLOOR(1.2)=1
GAMMA	Returns the value of the gamma function.	GAMMA(12)=39916800.000056
GAMMALN	Returns the natural logarithm of the gamma function.	GAMMALN(12)=17.502307845875
GAUSS	Calculates the probability that a member of a standard population falls within the interval between the mean standard deviation and the standard deviation z from the mean.	GAUSS(12)=0.5
GCD	Returns the greatest common divisor of two or more integers.	GCD(12, 15, 1.5)=1
GEOMEAN	Returns the geometric mean of an array or range of positive data. For example, you can use the GEOMEAN function to calculate the average growth rate of a given compound interest with a variable interest rate.	GEOMEAN(5, 2)=3.1622776601684
GESTEP	It returns a value of 1 if the number ≥ the specified value, otherwise it returns 0 (zero).	GESTEP(5, 2)=1
HARMEAN	Returns the harmonic mean of a data set. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.	HARMEAN(10, 3)=4.6153846153846
INDEX	Returns the value of the element in the array at the selected index.	INDEX((15, 20), 1)=15
INT	Rounds a number down to the nearest integer.	INT(1.2)=1
LCM	Returns the least common multiple of integers. The least common multiple is the smallest positive integer that is a multiple of all integer arguments number1, number2, and so on. Use LCM to add fractions with different denominators.	LCM(10, 6, 3)=30
LN	Returns the natural logarithm of a number. The base of the natural logarithm is the constant e (2.71828182845904).	LN(10)=2.302585092994
LOG	Returns the logarithm of a number to a given base.	LOG(10, 3)=2.0959032742894
LOG10	Returns the base 10 logarithm of a number.	LOG10(10)=1
MAX	Returns the maximum value in a set of values.	MAX(10, -10)=10
MAXA	Returns the maximum value in a set of values.	MAXA(10, -10, TRUE)=10
MEDIAN	Returns the median of the specified numbers. The median is the number that lies in the middle of a set of numbers.	MEDIAN(10, 20, 15, 1)=12.5
MIN	Returns the minimum value in a set of values.	MIN(10, -10)=-10
MOD	Returns the remainder after dividing a number by a divisor.	MOD(20, 3)=2
MODE	Returns the most frequently occurring or recurring value in a range of data.	MODE(20, 3, 20, 3, 26, 3)=3
MROUND	Returns a number rounded to the desired multiple.	MROUND(20, 2)=20
MULTINOMIAL	Returns the factorial of the sum of values divided by the product of their factorials.	MULTINOMIAL(10, 20, 30)=3.553261127085E+24
PERMUT	Returns the number of permutations for a given number of objects that can be selected from number objects. A permutation is any set or subset of objects or events where internal order is significant. Permutations are different from combinations, for which the internal order is not significant. Use this function for lottery-style probability calculations.	PERMUT(10, 2)=90
PERMUTATIONA	Returns the number of permutations for a given number of objects (with repetitions) that can be selected from the total objects.	PERMUTATIONA(10, 2)=100
PHI	Returns the value of the density function for the standard deviation.	PHI(2)=0.053990966513188
PI	Returns the number 3.1415926535898, the mathematical constant pi.	PI()=3.1415926535898
POWER	Returns the power of a number.	POWER(3, 2)=9
QUOTIENT	Returns the whole part of a division. This function is used to remove the remainder after division.	QUOTIENT(3, 2)=1
RADIANS	Converts degrees to radians.	RADIANS(180)=3.1415926535898
RAND	RAND returns uniformly distributed random real numbers greater than or equal to 0 and less than 1. A new random real number is returned each time the worksheet is recalculated.	RAND()=0.42277311925905
RANDBETWEEN	Returns a random integer from the specified interval. A new random integer is returned each time the worksheet is recalculated.	RANDBETWEEN(180, 220)=201
ROUNDDOWN	Rounds a number down towards zero.	ROUNDDOWN(180.66669, 3)=180.666
ROUNDUP	Rounds a number up, away from zero.	ROUNDUP(180.66669, 3)=180.667
SEC	Returns the secant of an angle.	SEC(0.5)=1.1394939273245
SECH	Returns the hyperbolic secant of an angle.	SECH(1)=0.64805427366389
SIGN	Returns the sign of the argument. Returns 1 if the number is positive, 0 for 0, and -1 if the number is negative.	SIGN(1)=1
SIN	Returns the sine of a given angle.	SIN(PI()/6)=0.5
SINH	Returns the hyperbolic sine of a number.	SINH(PI()/6)=0.54785347388804
SKEW	Returns the skewness of the distribution of a random variable. Skewness indicates the degree of asymmetry of the distribution of the variable around the mean. Positive skewness indicates a distribution with an asymmetric side that is skewed toward more positive values. Negative skewness indicates a distribution with an asymmetric side that is skewed toward more negative values.	SKEW(10, 3, 4, 5)=1.5970779829308
SKEW.P	Returns the skewness coefficient of a distribution based on a set of values: a characteristic of the degree of skewness of the distribution around the mean value.	SKEW.P(12, 5, 6)=0.65201211704405
SMALL	Returns the kth smallest value in a data set. This function is used to determine the value that has a specific relative position in a data set.	SMALL((12, 15), 2)=15
SQRT	Returns the positive square root.	SQRT(100)=10
STDEVP	Calculates standard deviation based on the entire population given as arguments. The standard deviation is a measure of how widely values are dispersed from the average value (the mean).	STDEVP(10, 20)=5
SUM	Returns the sum of all numbers.	SUM(-2, 20)=18
TAN	Returns the tangent of the specified angle.	TAN(1)=1.5574077246549
TANH	Returns the hyperbolic tangent of the argument.	TANH(1)=0.76159415595576

Function

Description

Sample example

ABS

Returns the absolute value of a number. The absolute value of a number is the same number without a sign.

ABS(-9)=9

ACCRINT

Returns the total interest on securities that pay interest on regular dates.

ACCRINT(39508, 39691, 39569, 0.1, 1000, 2, 0)=16.666666666667

ACCRINTM

Returns the accrued interest on a security that pays interest on the maturity date.

ACCRINTM(39539, 39614, 0.1, 1000, 3)=20.547945205479

ACOS

Returns the arccosine (the inverse of the cosine function) of the specified number. The arccosine is the angle whose cosine is the specified number. The resulting angle is given in radians in the range zero to pi.

ACOS(0.5)=1.0471975511966

ACOSH

Returns the inverse hyperbolic cosine of a number. The number must be greater than or equal to 1. The inverse hyperbolic cosine is a value whose hyperbolic cosine is a number, so ACOSH(COSH(number)) equals the number.

ACOSH(2)=1.3169578969248

ACOT

Returns the value of the arccotangent (inverse cotangent) of the specified number.

ACOT(3.14)=0.30831566219543

ACOTH

Returns the inverse hyperbolic cotangent of the specified number.

ACOTH(3.14)=0.32994497940173

AMORDEGRC

Returns the depreciation for one accounting period. This function is adapted to the French accounting model. If the asset was acquired during the accounting period, the depreciation is calculated on a straight-line basis. The AMORDEGRC function is similar to the AMORLINC function, but differs in that the depreciation coefficient depends on the useful life of the asset.

AMORDEGRC(2400, 39679, 39813, 300, 1, 0.15, 1)=776

AMORLINC

Returns the depreciation for one accounting period. This function is adapted to the French accounting model. If the asset was acquired during the accounting period, proportional depreciation is considered.

AMORLINC(2400, 39679, 39813, 300, 1, 0.15, 1)=360

ARABIC

Converts Roman numerals to Arabic numerals.

ARABIC(V)=5

ASIN

Returns the arcsine (the inverse of the sine) of the specified number. The arcsine is the angle whose sine is the specified number. The resulting angle is given in radians in the interval from pi/2 to pi/2.

ASIN(1)=1.5707963267949

ASINH

Returns the hyperbolic arcsine of the specified number. The hyperbolic arcsine is a value whose hyperbolic sine is the specified number, so ARCSINH(SINH(number)) is equal to the specified number.

ASINH(1)=0.88137358701954

ATAN

Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.

ATAN(2)=1.1071487177941

ATAN2

Returns the arctangent (the inverse of tangent) of the specified x and y coordinates. The arctangent is the angle between the x axis and the line containing the origin (0;0) and the point at coordinates (x_number, y_number). This angle is given in radians in the interval -pi to pi, except for the value -pi.

ATAN2(2, 3)=0.98279372324733

ATANH

ATANH(0.5)=0.54930614433405

AVEDEV

Returns the average of the absolute deviations of data points from their mean. STANDARD DEVIATION is a measure of the variability of a data set.

AVEDEV(10, 30, 2, 3)=9.375

AVERAGEA

Calculates the average (arithmetic mean) of the values in the argument list.

AVERAGEA(10, 20, 30)=20

BESSELI

Returns a modified Bessel function that is equivalent to the Bessel function calculated for purely imaginary arguments.

BESSELI(1, 2)=0.1357476665807

BESSELJ

Returns the Bessel function.

BESSELJ(1, 2)=0.1149034849319

BESSELK

Returns a modified Bessel function that is equivalent to the Bessel function calculated for purely imaginary arguments.

BESSELK(1, 2)=1.6248388844172

BESSELY

Returns the Bessel function, also known as the Weber or Neumann function.

BESSELY(1, 2)=-1.6506826133039

BIN2DEC

Converts a binary number to decimal.

BIN2DEC(1101)=13

CEILING

Rounds a number up to the nearest integer or to the nearest multiple of significance.

CEILING(1.2)=2

CEILING.MATH

The CEILING.MATH function rounds a number up to the nearest integer or, optionally, to the nearest multiple of significance.

CEILING.MATH(1.2, 1)=2

CEILING.PRECISE

Returns a number that is rounded up to the nearest integer or to the nearest multiple of significance. Regardless of the sign of the number, the number is rounded up. However, if the number or the significance is zero, zero is returned.

CEILING.PRECISE(1.2, 1)=2

CHIDIST

Returns the right-tailed probability of a chi-square distribution. The χ2 distribution is related to the χ2 test. The χ2 test compares observed and expected values. For example, in a genetic experiment, you might predict that the next generation of plants will have flowers of a certain color. By comparing the observed results with the expected results, you can determine whether the original prediction is valid.

CHIDIST(15, 15)=0.45141721122575

CHIINV

Returns the inverse of the right-tailed probability distribution function of the chi-square distribution. If the probability is equal to CHIDIST(x;...), the value of CHIINV(probability;...) is equal to x. Use this function to compare observed results with expected results to determine whether the original assumption is valid.

CHIINV(0.05, 10)=18.307038053334

CODE

Returns the numeric code of the first character of a given text string.

CODE(1)=49

COMBIN

Returns the number of combinations for a specified number of elements. Use the COMBIN function to determine the total number of possible groups for a specified number of elements.

COMBIN(15, 9)=5005

COMBINA

Returns the number of combinations (with repetition) for a given number of items.

COMBINA(15, 6)=38760

COS

Returns the cosine of the specified angle.

COS(1)=0.54030230586814

COSH

Returns the hyperbolic cosine of the specified number.

COSH(1)=1.5430806348152

COT

Returns the cotangent of an angle specified in radians.

COT(1)=0.64209261593433

COTH

Returns the hyperbolic cotangent of a hyperbolic angle.

COTH(1)=1.3130352854993

COUNT

Returns the number of items in the argument list.

COUNT(1, 2)=2

CSC

Returns the cosecant of an angle specified in radians.

CSC(1)=1.1883951057781

CSCH

Returns the hyperbolic cosecant of an angle specified in radians.

CSCH(1)=0.85091812823932

DEGREES

Converts radians to degrees.

DEGREES(PI())=180

DELTA

Tests the equality of two numbers. Returns 1 if number1 = number2; otherwise, returns 0. This function can be used to filter a set of values. For example, when adding several DELTA functions, you can find the number of identical pairs. This function is also known as the Kronecker Delta function.

DELTA(2, 2)=1

DEVSQ

Returns the sum of the squares of the deviations of data points from their sample mean.

DEVSQ(10, 3, 2)=38

EXP

Returns e raised to the power of the argument number. The constant e is equal to 2.718281828459, the base of natural logarithms.

EXP(1)=2.718281828459

FACT

Returns the factorial of a number. The factorial of a number is equal to 1*2*3*...* number.

FACT(9.5)=362880

FACTDOUBLE

Returns the double factorial of the specified number.

FACTDOUBLE(12)=46080

FISHER

Returns the Fisher transformation value of x. This transformation produces a function with a normal distribution rather than a skewed distribution. You can use this function to test the hypothesis about the correlation coefficient.

FISHER(0.9)=1.4722194895832

FISHERINV

Returns the inverse of the Fisher transform. You can use this transform to analyze correlations between regions or matrices of data. If y = FISHER(x), FISHERINV(y) = x.

FISHERINV(10)=0.99999999587769

FLOOR

Rounds a number down to the nearest whole number or to the nearest multiple of the specified value.

FLOOR(1.2)=1

GAMMA

Returns the value of the gamma function.

GAMMA(12)=39916800.000056

GAMMALN

Returns the natural logarithm of the gamma function.

GAMMALN(12)=17.502307845875

GAUSS

Calculates the probability that a member of a standard population falls within the interval between the mean standard deviation and the standard deviation z from the mean.

GAUSS(12)=0.5

GCD

Returns the greatest common divisor of two or more integers.

GCD(12, 15, 1.5)=1

GEOMEAN

Returns the geometric mean of an array or range of positive data. For example, you can use the GEOMEAN function to calculate the average growth rate of a given compound interest with a variable interest rate.

GEOMEAN(5, 2)=3.1622776601684

GESTEP

It returns a value of 1 if the number ≥ the specified value, otherwise it returns 0 (zero).

GESTEP(5, 2)=1

HARMEAN

Returns the harmonic mean of a data set. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

HARMEAN(10, 3)=4.6153846153846

INDEX

Returns the value of the element in the array at the selected index.

INDEX((15, 20), 1)=15

INT

Rounds a number down to the nearest integer.

INT(1.2)=1

LCM

Returns the least common multiple of integers. The least common multiple is the smallest positive integer that is a multiple of all integer arguments number1, number2, and so on. Use LCM to add fractions with different denominators.

LCM(10, 6, 3)=30

Returns the natural logarithm of a number. The base of the natural logarithm is the constant e (2.71828182845904).

LN(10)=2.302585092994

LOG

Returns the logarithm of a number to a given base.

LOG(10, 3)=2.0959032742894

LOG10

Returns the base 10 logarithm of a number.

LOG10(10)=1

MAX

Returns the maximum value in a set of values.

MAX(10, -10)=10

MAXA

Returns the maximum value in a set of values.

MAXA(10, -10, TRUE)=10

MEDIAN

Returns the median of the specified numbers. The median is the number that lies in the middle of a set of numbers.

MEDIAN(10, 20, 15, 1)=12.5

MIN

Returns the minimum value in a set of values.

MIN(10, -10)=-10

MOD

Returns the remainder after dividing a number by a divisor.

MOD(20, 3)=2

MODE

Returns the most frequently occurring or recurring value in a range of data.

MODE(20, 3, 20, 3, 26, 3)=3

MROUND

Returns a number rounded to the desired multiple.

MROUND(20, 2)=20

MULTINOMIAL

Returns the factorial of the sum of values divided by the product of their factorials.

MULTINOMIAL(10, 20, 30)=3.553261127085E+24

PERMUT

Returns the number of permutations for a given number of objects that can be selected from number objects. A permutation is any set or subset of objects or events where internal order is significant. Permutations are different from combinations, for which the internal order is not significant. Use this function for lottery-style probability calculations.

PERMUT(10, 2)=90

PERMUTATIONA

Returns the number of permutations for a given number of objects (with repetitions) that can be selected from the total objects.

PERMUTATIONA(10, 2)=100

PHI

Returns the value of the density function for the standard deviation.

PHI(2)=0.053990966513188

Returns the number 3.1415926535898, the mathematical constant pi.

PI()=3.1415926535898

POWER

Returns the power of a number.

POWER(3, 2)=9

QUOTIENT

Returns the whole part of a division. This function is used to remove the remainder after division.

QUOTIENT(3, 2)=1

RADIANS

Converts degrees to radians.

RADIANS(180)=3.1415926535898

RAND

RAND returns uniformly distributed random real numbers greater than or equal to 0 and less than 1. A new random real number is returned each time the worksheet is recalculated.

RAND()=0.42277311925905

RANDBETWEEN

Returns a random integer from the specified interval. A new random integer is returned each time the worksheet is recalculated.

RANDBETWEEN(180, 220)=201

ROUNDDOWN

Rounds a number down towards zero.

ROUNDDOWN(180.66669, 3)=180.666

ROUNDUP

Rounds a number up, away from zero.

ROUNDUP(180.66669, 3)=180.667

SEC

Returns the secant of an angle.

SEC(0.5)=1.1394939273245

SECH

Returns the hyperbolic secant of an angle.

SECH(1)=0.64805427366389

SIGN

Returns the sign of the argument. Returns 1 if the number is positive, 0 for 0, and -1 if the number is negative.

SIGN(1)=1

SIN

Returns the sine of a given angle.

SIN(PI()/6)=0.5

SINH

Returns the hyperbolic sine of a number.

SINH(PI()/6)=0.54785347388804

SKEW

Returns the skewness of the distribution of a random variable. Skewness indicates the degree of asymmetry of the distribution of the variable around the mean. Positive skewness indicates a distribution with an asymmetric side that is skewed toward more positive values. Negative skewness indicates a distribution with an asymmetric side that is skewed toward more negative values.

SKEW(10, 3, 4, 5)=1.5970779829308

SKEW.P

Returns the skewness coefficient of a distribution based on a set of values: a characteristic of the degree of skewness of the distribution around the mean value.

SKEW.P(12, 5, 6)=0.65201211704405

SMALL

Returns the kth smallest value in a data set. This function is used to determine the value that has a specific relative position in a data set.

SMALL((12, 15), 2)=15

SQRT

Returns the positive square root.

SQRT(100)=10

STDEVP

Calculates standard deviation based on the entire population given as arguments. The standard deviation is a measure of how widely values are dispersed from the average value (the mean).

STDEVP(10, 20)=5

SUM

Returns the sum of all numbers.

SUM(-2, 20)=18

TAN

Returns the tangent of the specified angle.

TAN(1)=1.5574077246549

TANH

Returns the hyperbolic tangent of the argument.

TANH(1)=0.76159415595576

Speed of sound

Values for calculation

Derivative of

$γ$ with respect to the dimensionless pressure

$π$

$γ_π$

Inverse reduced temperature

$τ$

Cross derivative of

$γ$ with respect to

$π$ and temperature

$τ$

$γ_{πτ}$

Second partial derivative of

$γ$ with respect to

$τ$

$γ_{ττ}$

Second partial derivative of

$γ$ with respect to

$π$

$γ_{ππ}$

Specific gas constant of ordinary water

$R$

$\mathrm{J\cdot\ kg^{-1}\cdot\ K^{-1}}$

The water temperature

$T$

$\mathrm{°C}$

Reduced pressure

$π$

Derivative of

$γ^r$ with respect to the dimensionless pressure

$π$

$γ^r_π$

Second partial derivative of

$γ^r$ with respect to

$π$

$γ^r_{ππ}$

Cross derivative of

$γ^r$ with respect to

$π$ and temperature

$τ$

$γ^r_{πτ}$

Second partial derivative of

$γ^o$ with respect to

$τ$

$γ^o_{ττ}$

Second partial derivative of

$γ^r$ with respect to

$τ$

$γ^r_{ττ}$

Reduced density

$δ$

Derivative of

$φ$ with respect to the dimensionless density

$δ$

$φ_δ$

Second partial derivative of

$φ$ with respect to

$δ$

$φ_{δδ}$

Cross derivative of

$φ$ with respect to

$δ$ and temperature

$τ$

$φ_{δτ}$

Second partial derivative of

$φ$ with respect to

$τ$

$φ_{ττ}$

Region

$\text{Region}$

Calculation

Speed of sound

$\text{if }\ \text{Region}= 1$

$w=\sqrt{\left(\cfrac{γ_π^2}{\cfrac{\left(γ_π-τ\cdot γ_{πτ}\right)^2}{τ^2\cdot γ_{ττ}}-γ_{ππ}}\right)\cdot R\cdot\left(T+273.15\right)}$

$\text{else if }\ \text{Region}= 2$

$w=\sqrt{\cfrac{1+2\cdot π\cdot γ^r_π+π^2\cdot {γ^r_π}^2}{\left(1-π^2\cdot γ^r_{ππ}\right)+\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)}}\cdot R\cdot\left(T+273.15\right)}$

$\text{else if }\ \text{Region}= 3$

$w=\sqrt{\left(2\cdot δ\cdot φ_δ+δ^2\cdot φ_{δδ}-\cfrac{\left(δ\cdot φ_δ-δ\cdot τ\cdot φ_{δτ}\right)^2}{τ^2\cdot φ_{ττ}}\right)\cdot R\cdot\left(T+273.15\right)}$

$\text{else if }\ \text{Region}= 5$

$\text{else}$

$w=0$

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Speed of sound

Values for calculation

Calculation

Speed of sound