Built-in functions

One entry per function, in the order autosuggest lists them. Matrix and vector functions, the symbolic operations and the table functions have pages of their own: Matrix functions, Symbolic functions and Table functions. The guides cover units, angles and how results are shown.

abs

abs(x)

The absolute value of a number or quantity. On a matrix it applies to every element. The bars |x| are a different reading, the length of a vector or the determinant of a square matrix. See Matrix functions.

Parameters

Parameter	Meaning
x	A number, a quantity with units, or a matrix.

Returns

The magnitude, keeping the units; for a matrix, a matrix of magnitudes.

Errors

Message	When
Too few arguments in function abs / Too many arguments in function abs	Not exactly one argument
Unexpected type of argument in function abs	The argument isn’t numeric

Example

$$\begin{array}{rl}\mathrm{abs}(-3\mathrm{m})& =3\mathrm{m}\\ \mathrm{abs}\left(\left[\begin{array}{ccc}-1& 2& -3\end{array}\right]\right)& =\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\end{array}$$

acos

acos(x)

The inverse cosine. Returns an angle in radians as a plain number; set the block’s result unit to deg to show it in degrees.

Parameters

Parameter	Meaning
x	A number without units. Outside $-1\dots 1$ the result is complex.

Returns

The angle in radians, as a number.

Errors

Message	When
Too few arguments in function acos / Too many arguments in function acos	Not exactly one argument
Unexpected type of argument in function acos	The argument carries units or isn’t numeric

Example

$$\mathrm{acos}(0.5)=1.0472$$

acosh

acosh(x)

The inverse hyperbolic cosine.

Parameters

Parameter	Meaning
x	A number without units. Below $1$ the result is complex.

Returns

A number.

Errors

Message	When
Too few arguments in function acosh / Too many arguments in function acosh	Not exactly one argument
Unexpected type of argument in function acosh	The argument carries units or isn’t numeric

Example

$$\mathrm{acosh}(1)=0$$

asin

asin(x)

The inverse sine. Returns an angle in radians as a plain number; set the block’s result unit to deg to show it in degrees.

Parameters

Parameter	Meaning
x	A number without units. Outside $-1\dots 1$ the result is complex.

Returns

The angle in radians, as a number.

Errors

Message	When
Too few arguments in function asin / Too many arguments in function asin	Not exactly one argument
Unexpected type of argument in function asin	The argument carries units or isn’t numeric

Example

$$\mathrm{asin}(0.5)=0.5236$$

With the block’s result unit set to deg, the same value shows as $30°$.

asinh

asinh(x)

The inverse hyperbolic sine.

Parameters

Parameter	Meaning
x	A number without units.

Returns

A number.

Errors

Message	When
Too few arguments in function asinh / Too many arguments in function asinh	Not exactly one argument
Unexpected type of argument in function asinh	The argument carries units or isn’t numeric

Example

$$\mathrm{asinh}(1)=0.8814$$

atan

atan(x)

The inverse tangent. Returns an angle in radians as a plain number; set the block’s result unit to deg to show it in degrees. For a full-circle angle from two coordinates, use atan2.

Parameters

Parameter	Meaning
x	A number without units.

Returns

The angle in radians, between $-\pi /2$ and $\pi /2$.

Errors

Message	When
Too few arguments in function atan / Too many arguments in function atan	Not exactly one argument
Unexpected type of argument in function atan	The argument carries units or isn’t numeric

Example

$$\mathrm{atan}(1)=0.7854$$

atan2

atan2(y, x)

The angle of the point $(x,y)$ from the positive $x$-axis, in the correct quadrant. Note the argument order: $y$ first.

Parameters

Parameter	Meaning
y	A number without units.
x	A number without units.

Returns

The angle in radians, between $-\pi$ and $\pi$, as a number; set the block’s result unit to deg to show it in degrees.

Errors

Message	When
Too few arguments in function atan2 / Too many arguments in function atan2	Not exactly two arguments
Unexpected type of argument in function atan2	An argument carries units or isn’t numeric

Example

$$\begin{array}{rl}atan2(1,1)& =0.7854\\ atan2(-1,-1)& =-2.3562\end{array}$$

atanh

atanh(x)

The inverse hyperbolic tangent.

Parameters

Parameter	Meaning
x	A number without units. Outside $-1\dots 1$ the result is complex.

Returns

A number.

Errors

Message	When
Too few arguments in function atanh / Too many arguments in function atanh	Not exactly one argument
Unexpected type of argument in function atanh	The argument carries units or isn’t numeric

Example

$$\mathrm{atanh}(0.5)=0.5493$$

ceil

ceil(x)
ceil(x, unit)

Rounds up, towards $+\infty$. A quantity with units must say which unit to round in.

Parameters

Parameter	Meaning
x	A number, or a quantity with units.
unit	Required when x has units: the unit to round in, as a bare unit such as cm.

Returns

The rounded value; for a quantity, in the unit named.

Errors

Message	When
Too few arguments in function ceil	A quantity with units but no unit to round in
Too many arguments in function ceil	Too many arguments
Unexpected type of argument in function ceil	The argument isn’t numeric

Example

$$\begin{array}{rl}\mathrm{ceil}(2.1)& =3\\ \mathrm{ceil}(-2.1)& =-2\\ \mathrm{ceil}(3.241\mathrm{cm},\mathrm{cm})& =4\mathrm{cm}\end{array}$$

cos

cos(x)

The cosine. A plain number is taken as radians; write the angle with a unit, 60.deg, to work in degrees.

Parameters

Parameter	Meaning
x	An angle: a plain number in radians, or a quantity in deg or rad.

Returns

A number.

Errors

Message	When
Unit in function cot is no angle	The argument carries a unit that isn’t an angle (the message names cot whichever trig function was used)
Too few arguments in function cos / Too many arguments in function cos	Not exactly one argument

Example

$$\begin{array}{rl}\cos (60°)& =0.5\\ \cos (\pi )& =-1\end{array}$$

cosh

cosh(x)

The hyperbolic cosine.

Parameters

Parameter	Meaning
x	A number without units.

Returns

A number.

Errors

Message	When
Too few arguments in function cosh / Too many arguments in function cosh	Not exactly one argument
Unexpected type of argument in function cosh	The argument carries units or isn’t numeric

Example

$$\cosh (0)=1$$

cot

cot(x)

The cotangent, $1/\tan x$. A plain number is taken as radians.

Parameters

Parameter	Meaning
x	An angle: a plain number in radians, or a quantity in deg or rad.

Returns

A number.

Errors

Message	When
Unit in function cot is no angle	The argument carries a unit that isn’t an angle
Too few arguments in function cot / Too many arguments in function cot	Not exactly one argument

Example

$$\cot (45°)=1$$

csc

csc(x)

The cosecant, $1/\sin x$. A plain number is taken as radians.

Parameters

Parameter	Meaning
x	An angle: a plain number in radians, or a quantity in deg or rad.

Returns

A number.

Errors

Message	When
Unit in function cot is no angle	The argument carries a unit that isn’t an angle (the message names cot whichever trig function was used)
Too few arguments in function csc / Too many arguments in function csc	Not exactly one argument

Example

$$\csc (90°)=1$$

exp

exp(x)

$e$ raised to the power $x$. Writing e^x is the same thing.

Parameters

Parameter	Meaning
x	A number without units.

Returns

A number.

Errors

Message	When
Unexpected type of argument in function exp	The argument carries units
Cannot convert “…” to a number	The argument is text
Too few arguments in function exp / Too many arguments in function exp	Not exactly one argument

Example

$$\begin{array}{rl}\exp (1)& =2.7183\\ \exp (0)& =1\end{array}$$

floor

floor(x)
floor(x, unit)

Rounds down, towards $-\infty$. A quantity with units must say which unit to round in.

Parameters

Parameter	Meaning
x	A number, or a quantity with units.
unit	Required when x has units: the unit to round in, as a bare unit such as m.

Returns

The rounded value; for a quantity, in the unit named.

Errors

Message	When
Too few arguments in function floor	A quantity with units but no unit to round in
Too many arguments in function floor	Too many arguments
Unexpected type of argument in function floor	The argument isn’t numeric

Example

$$\begin{array}{rl}\mathrm{floor}(2.9)& =2\\ \mathrm{floor}(-2.9)& =-3\\ \mathrm{floor}(3.7\mathrm{m},\mathrm{m})& =3\mathrm{m}\end{array}$$

if

if(condition, then, else)

A conditional value. Only the branch that is taken is evaluated, so an error in the other branch does no harm: if(x > 0, 1/x, 0) is safe at $x=0$. Comparisons of text compare the text; a comparison of two matrices is true when every element matches.

Parameters

Parameter	Meaning
condition	Anything that is true or false: a comparison such as a > b, comparisons joined with and, or, not, or a number, where zero is false and anything else true.
then	The value when the condition holds.
else	The value otherwise.

Returns

Whichever of then or else applies, evaluated.

Errors

Message	When
if expects 3 arguments: if(condition, then, else)	Not exactly three arguments

Example

$$\begin{array}{rl}\mathrm{if}(3>2,10,20)& =10\\ \mathrm{if}(3<2,10,20)& =20\end{array}$$

integrate

integrate(expression, x)
integrate(expression, x, a, b)
∫ expression dx
∫_a^b expression dx
expression → integrate, x

With limits, the definite integral of an expression over x from a to b, worked out numerically by adaptive Gauss–Kronrod quadrature to a relative accuracy of about $10^{-10}$. x is the integral’s own variable: a value of x defined on the sheet is ignored inside the call and left unchanged outside it. The limits are evaluated where the call is written, so a function of its upper limit, $Q(s):=\mathrm{integrate}(q(x),x,0,s)$, is a running integral. The limits themselves are never sampled, so an expression that is infinite at a limit but integrable there, such as $1/\sqrt{x}$ at 0, still converges. The limits must be finite.

Without limits, the antiderivative as a formula, with no constant of integration. It is a symbolic operation: names defined above are substituted except x, and the formula evaluates to a number once every name in it is defined, so $F(x):=\mathrm{integrate}(f(x),x)$ is a callable function and $F(b)-F(a)$ an exact definite integral. Every formula is checked by differentiating it numerically against the expression before it is shown.

The integral sign is the same call written as typeset maths, and nests for a double or triple integral; the arrow is the formula’s Mathcad spelling.

Parameters

Parameter	Meaning
expression	What to integrate, written in terms of x. It may call functions and use values defined above. With limits, it must give a single real number at every point of the range, in the same units throughout.
x	The variable of integration: a plain name such as x or t.
a	The lower limit. A finite number, or a quantity. Leave out a and b for the formula.
b	The upper limit, in the same units as a. Below a, the integral comes out negative.

Returns

With limits, the integral, as a number or as a quantity whose units are the expression’s units times the limits’ units: a load in kN/m integrated over metres gives kN.

Without limits, the antiderivative as an expression, or its value once every name in it is defined. Coefficients that come back as decimals are shown to six significant figures.

Errors

Message	When
integrate expects 2 arguments for a formula, integrate(expression, variable), or 4 for a number, integrate(expression, variable, a, b)	Neither two nor four arguments
integrate needs the variable to integrate over: write integrate(x^2, x), or end an integral with d and the variable, as in ∫ x² dx	Without limits: the second argument isn’t a plain name, or an integral sign has no dx
integrate needs the variable to integrate over: write integrate(x^2, x, 0, 2), or end an integral with d and the variable, as in ∫ x² dx	With limits: the same
integrate found no formula for this integral in x — for its value over a range, give the limits: integrate(f, x, a, b)	Without limits, when no formula exists in ordinary functions, such as for $x^x$
integrate’s formula for this needs the sine integral Si, which Maths Explore can’t evaluate — for its value over a range, give the limits: integrate(f, x, a, b)	Without limits, when the formula needs a special function; the message names it (the sine or cosine integral, erfi, a Fresnel integral, li, Ei and others)
integrate gave up looking for a formula for this integral in x after 1 s — for its value over a range, give the limits: integrate(f, x, a, b)	Without limits, when the search takes more than a second
integrate couldn’t find a formula for this integral in x that checks out — for its value over a range, give the limits: integrate(f, x, a, b)	Without limits, when the formula found fails the numerical check
integrate can’t work out a formula through if(…) — for its value over a range, give the limits: integrate(f, x, a, b)	Without limits, when the expression applies a function with no formula form, such as if, to x
integrate expects a and b to share the same units	One limit has units and the other doesn’t, or they differ in kind
integrate expects a to be a finite real number / integrate expects b to be a finite real number	A limit isn’t numeric, or is infinite
integrate expects the expression to give a real number, but at x = … it is complex	The expression goes complex inside the range, such as $\sqrt{x}$ below zero
integrate expects the expression to give a single value, not a matrix or vector	The expression is a matrix or vector
integrate expects the expression to give a single real number at x = …	The expression is text, or true or false
integrate expects the expression’s units to stay the same across the range; they change at x = …	The expression changes dimension partway, such as an if with differently dimensioned branches
integrate: the expression is not a finite number at x = …	The expression is infinite or undefined inside the range
integrate did not converge within 20000 evaluations: the integral may not exist over this range, or the expression changes too sharply	The integral diverges, such as $1/x$ from 0 to 1, or can’t be pinned down to the accuracy
integrate stopped: these nested integrals need too many evaluations (more than 100000 between them)	A double or triple integral would cost too much

Example

$$\begin{array}{rl}\mathrm{integrate}(x^2,x,0,2)& =2.6667\\ {\int }_0^{\pi }\sin (t)dt& =2\\ {\int }_0^2{\int }_0^{1}x\cdot ydydx& =1\end{array}$$ $$\begin{array}{rl}w& :=3kN/m\\ {\int }_{0\mathrm{m}}^{2\mathrm{m}}wdx& =6\mathrm{kN}\end{array}$$ $$\begin{array}{rl}\mathrm{integrate}(3x^2+2x+1,x)& =x^3+x^2+x\\ x\cdot e^x\to \mathrm{integrate},x& =x\cdot e^x-e^x\\ F(x)& :=\int x\cdot e^{x}dx\\ F(1)-F(0)& =1\end{array}$$

ln

ln(x)

The natural logarithm, base $e$. The \ln command and the typed letters are the same function. For base 10 use log.

Parameters

Parameter	Meaning
x	A number without units. Zero gives $-\infty$; a negative number gives a complex result.

Returns

A number.

Errors

Message	When
Too few arguments in function log	The argument carries units (the message names log, which ln is built on)
Too few arguments in function ln / Too many arguments in function ln	Not exactly one argument

Example

$$\begin{array}{rl}\ln (10)& =2.3026\\ \ln (e)& =1\end{array}$$

log

log(x)
log(x, base)

The logarithm. With one argument it is base 10, as on paper and in Mathcad, not the natural log. With two arguments it is the log of x to base. The \log command is the same function; for the natural log use ln.

Parameters

Parameter	Meaning
x	A number without units. Zero gives $-\infty$; a negative number gives a complex result.
base	Optional. The base; 10 when omitted.

Returns

A number.

Errors

Message	When
Too few arguments in function log	The argument carries units
Too many arguments in function log	More than two arguments

Example

$$\begin{array}{rl}\log (100)& =2\\ \log (8,2)& =3\end{array}$$

log10

log10(x)

The base-10 logarithm, written out. Identical to one-argument log.

Parameters

Parameter	Meaning
x	A number without units. Zero gives $-\infty$; a negative number gives a complex result.

Returns

A number.

Errors

Message	When
Unexpected type of argument in function log10	The argument carries units or isn’t numeric
Too few arguments in function log10 / Too many arguments in function log10	Not exactly one argument

Example

$$log10(1000)=3$$

max

max(a, b, c, …)
max(A)
max(A, dimension)

The largest of several values, or of all the values in a matrix. Quantities are compared after converting units, and the winner keeps the unit it was written in. On a matrix, dimension gives the maximum of each column (1) or of each row (2) instead.

Parameters

Parameter	Meaning
a, b, c, …	Two or more values to compare. All numbers, or all quantities of one kind.
A	A list or matrix of values.
dimension	Optional. 1 for the maximum down each column, 2 along each row.

Returns

The largest value, or a vector of them when dimension is given.

Errors

Message	When
Cannot calculate max, unexpected type of argument	A plain number is mixed with a quantity, or the values aren’t numeric
Cannot calculate max of an empty array	An empty list

Example

$$\begin{array}{rl}\max (1,5,3)& =5\\ \max (2\mathrm{m},300\mathrm{cm})& =300\mathrm{cm}\\ \max \left(\left[\begin{array}{cc}1& 5\\ 3& 2\end{array}\right],1\right)& =\left[\begin{array}{cc}3& 5\end{array}\right]\end{array}$$

mean

mean(a, b, c, …)
mean(A)
mean(A, dimension)

The arithmetic mean. Quantities are converted to one unit first. On a matrix, dimension gives the mean of each column (1) or of each row (2) instead.

Parameters

Parameter	Meaning
a, b, c, …	The values. All numbers, or all quantities of one kind.
A	A list or matrix of values.
dimension	Optional. 1 for the mean down each column, 2 along each row.

Returns

The mean, in the unit of the first value.

Errors

Message	When
Cannot calculate the mean of an empty array	An empty list
Cannot calculate mean, unexpected type of argument	A plain number is mixed with a quantity, or the values aren’t numeric

Example

$$\begin{array}{rl}\mathrm{mean}(1,2,3,4)& =2.5\\ \mathrm{mean}\left(\left[\begin{array}{cc}2\mathrm{m}& 4\mathrm{m}\end{array}\right]\right)& =3\mathrm{m}\end{array}$$

median

median(a, b, c, …)
median(A)

The middle value once the values are sorted; the average of the two middle values when there is an even number of them.

Parameters

Parameter	Meaning
a, b, c, …	The values. All numbers, or all quantities of one kind.
A	A list or matrix of values. Every element counts.

Returns

The median.

Errors

Message	When
Cannot calculate median of an empty array	An empty list
Cannot calculate median, unexpected type of argument	A plain number is mixed with a quantity, or the values aren’t numeric

Example

$$\begin{array}{rl}\mathrm{median}(3,1,2)& =2\\ \mathrm{median}\left(\left[\begin{array}{cccc}4& 1& 3& 2\end{array}\right]\right)& =2.5\end{array}$$

min

min(a, b, c, …)
min(A)
min(A, dimension)

The smallest of several values, or of all the values in a matrix. Quantities are compared after converting units, and the winner keeps the unit it was written in. On a matrix, dimension gives the minimum of each column (1) or of each row (2) instead.

Parameters

Parameter	Meaning
a, b, c, …	Two or more values to compare. All numbers, or all quantities of one kind.
A	A list or matrix of values.
dimension	Optional. 1 for the minimum down each column, 2 along each row.

Returns

The smallest value, or a vector of them when dimension is given.

Errors

Message	When
Cannot calculate min, unexpected type of argument	A plain number is mixed with a quantity, or the values aren’t numeric
Cannot calculate min of an empty array	An empty list

Example

$$\begin{array}{rl}\min (1,5,3)& =1\\ \min \left(\left[\begin{array}{cc}4\mathrm{kg}& 500\mathrm{g}\end{array}\right]\right)& =500\mathrm{g}\end{array}$$

mod

mod(x, y)

The remainder of $x/y$. The result takes the sign of y, so mod(-7, 3) is $2$, not $-1$. A divisor of zero returns x unchanged.

Parameters

Parameter	Meaning
x	A number without units.
y	A number without units.

Returns

A number with the sign of y.

Errors

Message	When
Unexpected type of argument in function mod	An argument carries units or isn’t numeric
Too few arguments in function mod / Too many arguments in function mod	Not exactly two arguments

Example

$$\begin{array}{rl}\mathrm{mod}(7,3)& =1\\ \mathrm{mod}(-7,3)& =2\end{array}$$

mode

mode(a, b, c, …)
mode(A)

The most frequent value. The result is always a vector, because there may be several, and when every value occurs equally often all of them are returned.

Parameters

Parameter	Meaning
a, b, c, …	The values.
A	A list or matrix of values.

Returns

A vector of the most frequent values.

Errors

Message	When
Too few arguments in function mode	No arguments

Example

$$\begin{array}{rl}\mathrm{mode}\left(\left[\begin{array}{cccc}1& 2& 2& 3\end{array}\right]\right)& =\left[\begin{array}{c}2\end{array}\right]\\ \mathrm{mode}\left(\left[\begin{array}{cccc}1& 1& 2& 2\end{array}\right]\right)& =\left[\begin{array}{cc}1& 2\end{array}\right]\end{array}$$

prod

prod(a, b, c, …)
prod(A)

The product of all the values. Units multiply through.

Parameters

Parameter	Meaning
a, b, c, …	The values.
A	A list or matrix of values. Every element counts.

Returns

The product, with the product of the units.

Errors

Message	When
Cannot calculate prod of an empty array	An empty list
Too few arguments in function prod	No arguments

Example

$$\begin{array}{rl}\mathrm{prod}(2,3,4)& =24\\ \mathrm{prod}\left(\left[\begin{array}{cc}2\mathrm{m}& 3\mathrm{m}\end{array}\right]\right)& =6{\mathrm{m}}^2\end{array}$$

random

random()
random(max)
random(min, max)
random(size)
random(size, max)
random(size, min, max)

A random number, uniformly distributed. The sheet re-evaluates whenever anything changes, so the value changes too. It is not fixed once drawn.

Parameters

Parameter	Meaning
max	Optional. Upper limit, excluded; 1 when omitted.
min	Optional. Lower limit, included; 0 when omitted.
size	Optional. A list such as [2, 3] giving the rows and columns of a matrix to fill.

Returns

A number $\ge$ min and $<$ max, or a matrix of them.

Errors

Message	When
Too many arguments in function random	Three arguments where the first isn’t a size

Example

$$\mathrm{random}(5,6)=5.4713\text{(a different value each evaluation, always }5\le x<6\text{)}$$

root

root(f, a, b)

A numerical root of $f(x)=0$ between a and b, found by Brent’s method. The bracket must straddle the root, with $f(a)$ and $f(b)$ of opposite sign, and the root is returned in the bracket’s units. This is the fallback solve points you at when an equation has no closed form.

Parameters

Parameter	Meaning
f	A function of one variable, defined above as f(x) := … and passed by name. Its value only matters by sign, so it may carry units.
a	One end of the bracket. A number, or a quantity.
b	The other end, in the same units as a.

Returns

The root, as a number or in the bracket’s units. If an end of the bracket is itself a root, that end is returned.

Errors

Message	When
root(f, a, b) needs a sign change: f(a) and f(b) have the same sign, so no root is bracketed	The bracket doesn’t straddle a root
root expects its first argument to be a function, as in root(f, a, b)	f isn’t a function
root expects a and b to share the same units	One end has units and the other doesn’t, or they differ in kind
root expects a to be a real number / root expects b to be a real number	An end isn’t numeric
root expects f(x) to return a finite real value across the bracket	f gives infinity or an error somewhere in the bracket
root(f, a, b) did not converge within 128 iterations	No root found within the iteration cap

Example

$$\begin{array}{rl}f(x)& :=x^2-2\\ \mathrm{root}(f,0,2)& =1.4142\end{array}$$ $$\begin{array}{rl}f(x)& :=\cos (x)-x\\ \mathrm{root}(f,0,1)& =0.7391\end{array}$$ $$\begin{array}{rl}g(d)& :=d-3\mathrm{mm}\\ \mathrm{root}(g,1\mathrm{mm},5\mathrm{mm})& =3\mathrm{mm}\end{array}$$

round

round(x)
round(x, n)
round(x, unit)
round(x, n, unit)

Rounds to the nearest whole number, or to n decimal places. Halves round away from zero. A quantity with units must say which unit to round in.

Parameters

Parameter	Meaning
x	A number, or a quantity with units.
n	Optional. Decimal places, a whole number from 0 to 15.
unit	Required when x has units: the unit to round in, as a bare unit such as cm.

Returns

The rounded value; for a quantity, in the unit named.

Errors

Message	When
Number of decimals in function round must be an integer from 0 to 15 inclusive	n is negative, fractional or too large
Too few arguments in function round	A quantity with units but no unit to round in
Too many arguments in function round	Too many arguments

Example

$$\begin{array}{rl}\mathrm{round}(2.567,2)& =2.57\\ \mathrm{round}(2.5)& =3\\ \mathrm{round}(2.5\mathrm{m},1,\mathrm{cm})& =250\mathrm{cm}\end{array}$$

sec

sec(x)

The secant, $1/\cos x$. A plain number is taken as radians.

Parameters

Parameter	Meaning
x	An angle: a plain number in radians, or a quantity in deg or rad.

Returns

A number.

Errors

Message	When
Unit in function cot is no angle	The argument carries a unit that isn’t an angle (the message names cot whichever trig function was used)
Too few arguments in function sec / Too many arguments in function sec	Not exactly one argument

Example

$$\sec (0)=1$$

sin

sin(x)

The sine. A plain number is taken as radians; write the angle with a unit, 30.deg, to work in degrees.

Parameters

Parameter	Meaning
x	An angle: a plain number in radians, or a quantity in deg or rad.

Returns

A number.

Errors

Message	When
Unit in function cot is no angle	The argument carries a unit that isn’t an angle (the message names cot whichever trig function was used)
Too few arguments in function sin / Too many arguments in function sin	Not exactly one argument

Example

$$\begin{array}{rl}\sin (30°)& =0.5\\ \sin (\pi /6)& =0.5\\ \sin (90)& =0.8940\text{(90 radians, not degrees)}\end{array}$$

sinh

sinh(x)

The hyperbolic sine.

Parameters

Parameter	Meaning
x	A number without units.

Returns

A number.

Errors

Message	When
Too few arguments in function sinh / Too many arguments in function sinh	Not exactly one argument
Unexpected type of argument in function sinh	The argument carries units or isn’t numeric

Example

$$\sinh (1)=1.1752$$

sqrt

sqrt(x)

The square root. The toolbar’s radical and the \sqrt command are the same function. Units come through: $\sqrt{4{\mathrm{m}}^2}=2\mathrm{m}$.

Parameters

Parameter	Meaning
x	A number or a quantity. A negative number gives a complex result, not an error.

Returns

The principal square root, with the square root of the units.

Errors

Message	When
Too few arguments in function sqrt / Too many arguments in function sqrt	Not exactly one argument
Unexpected type of argument in function sqrt	The argument isn’t numeric

Example

$$\begin{array}{rl}\sqrt{16}& =4\\ \sqrt{4{\mathrm{m}}^2}& =2\mathrm{m}\\ \sqrt{-4}& =2i\end{array}$$

std

std(a, b, c, …)
std(A)
std(A, normalization)

The standard deviation. By default this is the sample standard deviation, dividing by $n-1$. Quantities keep their unit.

Parameters

Parameter	Meaning
a, b, c, …	The values.
A	A list or matrix of values. Every element counts.
normalization	Optional, as text: "unbiased" (default, divide by $n-1$), "uncorrected" (divide by $n$) or "biased" (divide by $n+1$).

Returns

The standard deviation, in the values’ unit.

Errors

Message	When
Cannot calculate variance of an empty array	An empty list
Cannot calculate std, unexpected type of argument	A plain number is mixed with a quantity, or the values aren’t numeric

Example

$$\begin{array}{rl}\mathrm{std}\left(\left[\begin{array}{cccccccc}2& 4& 4& 4& 5& 5& 7& 9\end{array}\right]\right)& =2.1381\\ \mathrm{std}\left(\left[\begin{array}{cccccccc}2& 4& 4& 4& 5& 5& 7& 9\end{array}\right],\text{"uncorrected"}\right)& =2\\ \mathrm{std}\left(\left[\begin{array}{cc}2\mathrm{m}& 4\mathrm{m}\end{array}\right]\right)& =1.4142\mathrm{m}\end{array}$$

sum

sum(a, b, c, …)
sum(A)
sum(A, dimension)

The total of all the values. Quantities are converted to one unit first. On a matrix, dimension gives the total of each column (1) or of each row (2) instead. An empty list sums to zero.

Parameters

Parameter	Meaning
a, b, c, …	The values. All numbers, or all quantities of one kind.
A	A list or matrix of values.
dimension	Optional. 1 for the total down each column, 2 along each row.

Returns

The total, in the unit of the first value.

Errors

Message	When
Cannot calculate sum, unexpected type of argument	A plain number is mixed with a quantity, or the values aren’t numeric

Example

$$\begin{array}{rl}\mathrm{sum}(1,2,3)& =6\\ \mathrm{sum}\left(\left[\begin{array}{ccc}1\mathrm{m}& 2\mathrm{m}& 50\mathrm{cm}\end{array}\right]\right)& =3.5\mathrm{m}\\ \mathrm{sum}\left(\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],2\right)& =\left[\begin{array}{cc}3& 7\end{array}\right]\end{array}$$

tan

tan(x)

The tangent. A plain number is taken as radians; write the angle with a unit, 45.deg, to work in degrees.

Parameters

Parameter	Meaning
x	An angle: a plain number in radians, or a quantity in deg or rad.

Returns

A number. At $90°$ the result is a very large number rather than an error.

Errors

Message	When
Unit in function cot is no angle	The argument carries a unit that isn’t an angle (the message names cot whichever trig function was used)
Too few arguments in function tan / Too many arguments in function tan	Not exactly one argument

Example

$$\tan (45°)=1$$

tanh

tanh(x)

The hyperbolic tangent.

Parameters

Parameter	Meaning
x	A number without units.

Returns

A number between $-1$ and $1$.

Errors

Message	When
Too few arguments in function tanh / Too many arguments in function tanh	Not exactly one argument
Unexpected type of argument in function tanh	The argument carries units or isn’t numeric

Example

$$\tanh (1)=0.7616$$

variance

variance(a, b, c, …)
variance(A)
variance(A, normalization)

The variance. By default this is the sample variance, dividing by $n-1$. For quantities the unit is squared.

Parameters

Parameter	Meaning
a, b, c, …	The values.
A	A list or matrix of values. Every element counts.
normalization	Optional, as text: "unbiased" (default, divide by $n-1$), "uncorrected" (divide by $n$) or "biased" (divide by $n+1$).

Returns

The variance, in the values’ unit squared.

Errors

Message	When
Cannot calculate variance of an empty array	An empty list
Cannot calculate variance, unexpected type of argument	A plain number is mixed with a quantity, or the values aren’t numeric

Example

$$\begin{array}{rl}\mathrm{variance}\left(\left[\begin{array}{cccc}1& 2& 3& 4\end{array}\right]\right)& =1.6667\\ \mathrm{variance}\left(\left[\begin{array}{cccc}1& 2& 3& 4\end{array}\right],\text{"uncorrected"}\right)& =1.25\\ \mathrm{variance}\left(\left[\begin{array}{cc}2\mathrm{m}& 4\mathrm{m}\end{array}\right]\right)& =2{\mathrm{m}}^2\end{array}$$