Mathematical function reference

x: a real or complex value; returns its magnitude.

Result: 3

x: nonnegative for real output; complex inputs use the supported principal branch.

Result: 3

x: a finite exponent; very large results can exceed numerical range.

Result: 1

x: positive for real logarithms.

Result: 1

x: positive; base-10 logarithm. Use ln for the natural logarithm.

Result: 2

x: angle in the selected mode. Examples use radians. Reciprocal functions are undefined where their denominator is zero.

Result: 0

x: angle in the selected mode. Examples use radians. Reciprocal functions are undefined where their denominator is zero.

Result: 1

x: angle in the selected mode. Examples use radians. Reciprocal functions are undefined where their denominator is zero.

Result: 0

x: angle in the selected mode. Examples use radians. Reciprocal functions are undefined where their denominator is zero.

Result: 1

x: angle in the selected mode. Examples use radians. Reciprocal functions are undefined where their denominator is zero.

Result: 1

x: angle in the selected mode. Examples use radians. Reciprocal functions are undefined where their denominator is zero.

Result: 1

Domain: −1 ≤ x ≤ 1. The result follows the selected angle mode.

Result: 0

Domain: −1 ≤ x ≤ 1. The result follows the selected angle mode.

Result: 0

Domain: any finite real x. The result follows the selected angle mode.

Result: 0

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: 0

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: 1

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: 0

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: 1

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: approximately 0.850918

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: approximately 1.313035

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: 0

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: 0

Hyperbolic functions use real numeric arguments, independent of angle mode. csch/coth exclude zero; arccosh needs x ≥ 1; arctanh needs −1 < x < 1.

Result: 0

x: a finite real number. floor rounds down, ceil up, round to the nearest integer; sign gives −1, 0 or 1.

Result: 2

x: a finite real number. floor rounds down, ceil up, round to the nearest integer; sign gives −1, 0 or 1.

Result: 3

x: a finite real number. floor rounds down, ceil up, round to the nearest integer; sign gives −1, 0 or 1.

Result: 3

x: a finite real number. floor rounds down, ceil up, round to the nearest integer; sign gives −1, 0 or 1.

Result: −1

a: dividend; b: nonzero divisor. The result uses the divisor-sign modulo convention.

Result: 1

For integer factorials use nonnegative n. Values beyond the finite numerical range are undefined.

Result: 120

Integer arguments; greatest common divisor.

Result: 6

Integer arguments; least common multiple.

Result: 12

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 4

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 4

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 12

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 3

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 4

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 8/3

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 2

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: approximately 1.632993

L: a finite numeric list. variance/stdev use sample normalization (at least two values); varp/stdevp use population normalization. mad is the mean absolute deviation from the mean.

Result: 4/3

L: nonempty finite numeric list; p in [0,1]. Uses type-7 linear interpolation.

Result: 2.5

Like quantile, but p is a percentage in [0,100].

Result: 2.5

L: finite numeric list; k is an integer from 0 to 4 (minimum through maximum).

Result: 2

L: nonempty finite numeric list; returns minimum, Q1, median, Q3, maximum.

Result: [0,1,2,3,4]

X and Y: paired finite lists of equal length. Sample covariance needs two pairs; correlations need nonzero spread. Spearman correlates ranks.

Result: 2

X and Y: paired finite lists of equal length. Sample covariance needs two pairs; correlations need nonzero spread. Spearman correlates ranks.

Result: 1

X and Y: paired finite lists of equal length. Sample covariance needs two pairs; correlations need nonzero spread. Spearman correlates ranks.

Result: 1

X and Y: paired finite lists of equal length. Sample covariance needs two pairs; correlations need nonzero spread. Spearman correlates ranks.

Result: 1

count: nonnegative integer; seed: optional finite number. Returns values in [0,1). random() returns one scalar; seeded calls repeat exactly.

Result: A reproducible three-value list

L: list; optional finite seed. Reorders entries without removing any.

Result: A reproducible permutation of [1,2,3]

L: nonempty list; count: nonnegative integer. Samples with replacement; seed is optional.

Result: Three entries, each 1 or 2

L and counts: matching lists; each count is a nonnegative integer.

Result: [2,2,4]

a and b: real parts of a + bi.

Result: 1 + 2i

z: complex value. arg returns its angle in the selected angle mode; examples use radians.

Result: 1

z: complex value. arg returns its angle in the selected angle mode; examples use radians.

Result: 2

z: complex value. arg returns its angle in the selected angle mode; examples use radians.

Result: 1 − 2i

z: complex value. arg returns its angle in the selected angle mode; examples use radians.

Result: approximately 1.107149 radians

rows: a rectangular nested numeric list; at most 16 by 16.

Result: A 2 × 2 explicit matrix

A: explicit matrix. det/inverse/trace need a square matrix; inverse also needs a nonsingular matrix.

Result: −2

A: explicit matrix. det/inverse/trace need a square matrix; inverse also needs a nonsingular matrix.

Result: [[-2,1],[1.5,-0.5]]

A: explicit matrix. det/inverse/trace need a square matrix; inverse also needs a nonsingular matrix.

Result: [[1,3],[2,4]]

A: explicit matrix. det/inverse/trace need a square matrix; inverse also needs a nonsingular matrix.

Result: [[1,0],[0,1]]

A: explicit matrix. det/inverse/trace need a square matrix; inverse also needs a nonsingular matrix.

Result: 5

n: positive integer at most 16; creates an n × n identity matrix.

Result: [[1,0],[0,1]]

A and B: matrices whose inner dimensions agree; this is a matrix product.

Result: [[1,2],[3,4]]

sd > 0. Parameters are optional.

Result: A distribution object; use its methods below.

upper > lower. Parameters are optional.

Result: A distribution object; use its methods below.

Positive degrees of freedom, up to 1,000,000.

Result: A distribution object; use its methods below.

Positive degrees of freedom, up to 1,000,000.

Result: A distribution object; use its methods below.

Integer trials from 0 to 1,000,000; probability p in [0,1].

Result: A distribution object; use its methods below.

Mean from 0 to 1,000,000.

Result: A distribution object; use its methods below.

Positive rate up to 1,000,000.

Result: A distribution object; use its methods below.

0 < p ≤ 1; counts trials until the first success, starting at 1.

Result: A distribution object; use its methods below.

d: distribution; x: value. Returns density for continuous distributions or probability mass for discrete ones.

Result: Then d.pdf(0) ≈ 0.398942

Cumulative probability through x, or interval probability between the supplied bounds.

Result: Then d.cdf(0.5) = 0.5

p: probability in [0,1]. Infinite endpoint quantiles may be undefined in finite output.

Result: Then d.inversecdf(0.5) = 0

count: nonnegative integer; seed: optional finite number. Keep the seed to reproduce the sample.

Result: Then d.random(3,42) gives three integers between 0 and 10

L: finite values; width: optional positive bin width. At most 100 bins.

Result: Two occupied bins with two observations each

L: finite values; offset: vertical center (default 1); height: positive box height (default 1). Uses type-7 quartiles and min–max whiskers.

Result: Median 2, Q1 1, Q3 3; box centered at y = 2 with height 1

L: up to 2,000 finite values; width: optional positive grouping width.

Result: Two stacked dots at 1 and one at 2