# JME¶

JME expressions are used by students to enter answers to algebraic questions, and by question authors to define variables. JME syntax is similar to what you’d type on a calculator.

## Syntax¶

Expressions are strings of text, made of variable names, literal values, grouped terms, function applications, operators, collections and indices.

Whitespace (space characters, tabs, newlines, etc.) can be used to separate literal values, for example a 2 is not interpreted the same as a2. For all other purposes, whitespace is ignored.

### Variable names¶

The first character of a variable name must be an alphabet letter; after that, any combination of letters, numbers and underscores is allowed, with any number of ' on the end.

Examples:
• x

• x_1

• time_between_trials

• var1

• row1val2

• y''

This screencast describes which variable names are valid, and gives some advice on how you should pick names:

#### Variable name annotations¶

Names can be given annotations to change how they are displayed. The following annotations are built-in:

• verb – does nothing, but names like i, pi and e are not interpreted as the famous mathematical constants.

• op – denote the name as the name of an operator — wraps the name in the LaTeX \operatorname command when displayed

• v or vector – denote the name as representing a vector — the name is displayed in boldface

• unit – denote the name as representing a unit vector — places a hat above the name when displayed

• dot – places a dot above the name when displayed, for example when representing a derivative

• m or matrix – denote the name as representing a matrix — displayed using a non-italic font

• diff - denote the name as a differential - the name is prefixed with \mathrm{d} when displayed, e.g. $$\mathrm{d}x$$

• degrees - denote the name as representing an angle in degrees - the name has a suffix of ^{\circ} when displayed, e.g. $$x^{\circ}$$

Any other annotation is taken to be a LaTeX command. For example, a name vec:x is rendered in LaTeX as \vec{x}, which places an arrow above the name.

You can apply multiple annotations to a single variable. For example, v:dot:x produces a bold x with a dot on top: $$\boldsymbol{\dot{x}}$$.

Names with different annotations are considered to represent different values, for the purpose of simplification and evaluation. For example, in the expression dot:x + x, the two terms will not be collected together.

### Literal values¶

boolean, integer, number and string data types can be constructed with literal values.

Examples: true, 1, 4.3, "Numbas"

There are also some built-in constants which are interpreted as number values:

• pi or π - the ratio of a circle’s circumference to its diameter;

• e - the base of the natural logarithm;

• i - the square root of -1;

• infinity, infty or ∞ - infinity;

• nan - “Not a number”, sometimes returned by JavaScript functions.

You can disable, override or define new constants in the question editor’s Constants tab.

### Grouped terms¶

Use parentheses ( ) to group terms. For example:

(a+2)(a+1)


No other symbols can be used to group terms - square brackets [ ] and curly braces { } have other meanings in Numbas.

### Function application¶

A name followed immediately by a bracketed group is interpreted as a function application. Function names can be annotated similarly to variables.

A function takes one or more arguments, separated by commas. Each argument is a JME expression.

Examples:
• f(x)

• g(a,b)

### Operators¶

There are three kinds of operators:

Binary

A symbol written between two terms, for example a + b (“$$a$$ plus $$b$$”). Each binary operator has a precedence which determines when it is evaluated in relation to other operators.

Prefix

A symbol written to the left of a term, for example !a (“not $$a$$”).

Postfix

A symbol written to the right of a term, for example a! (“$$a$$ factorial”).

#### Relations¶

Some binary operators are marked as relations. Relations can be chained together, for example a<b<c is interpreted as a<b and b<c.

The relation operators are <, <=, >, >=, =, <> and in.

### Collections¶

There are two kinds of collection: lists and dictionaries.

Both are written as a series of terms written between square brackets and separated by commas.

For a dictionary, each term is a keypair: a variable name or string followed by a colon and an expression. For example, [a: 1, "first name": "Owen"] is a dictionary with keys "a" and "first name".

A collection made of any other kind of term is interpreted as a list.

An empty pair of brackets [] is interpreted as an empty list. To construct an empty dictionary, use dict().

Examples:
• [1,2,3]

• [a]

• []

### Indices¶

Many JME data types support indexing.

An index is a single term inside square brackets immediately following another term.

The first element in a list has index 0.

Negative indices read from the end: the index -1 corresponds to the last item.

Examples:
• [1,2,3][0] produces the first element in the list, 1.

• x[3..7] produces the fourth to the eighth elements of the list x.

• id(4)[1] produces the second row of a $$4 \times 4$$ identity matrix, vector(0,1,0,0).

• info["name"] returns the value corresponding to the key "name" in the dictionary info.

• "Numbas"[0] produces the first letter of the string "Numbas", "N".

### Implicit multiplication¶

JME supports implicit multiplication in some cases, allowing you to omit the multiplication symbol:

Examples:
• (a+1)2 = (a+1)*2

• (x+y)z = (x+y)*z

• 2x = 2*x

• x y = x*y

Warning

Note that a name followed by a bracket is not always interpreted as implicit multiplication. Instead, it’s interpreted as a function application.

To interpret such expressions as products, in a mathematical expression part turn off the Allow unknown function names? option, and when dealing with expression values, use expand_juxtapositions().

### Case-sensitivity¶

The default behaviour of variable and function names in JME expressions is that they are considered case-insensitively: two names are considered equivalent if they are exactly equal when converted to lower-case.

You can force the JME system to be case-sensitive using the scope_case_sensitive() function.

## Data types¶

JME expressions are composed of the following data types. Some extensions add new data types.

number

A real or complex floating-point number.

i, e, infinity and pi are reserved keywords for the imaginary unit, the base of the natural logarithm, ∞ and π, respectively.

Examples: 0.0, -1.0, 0.234, i, e, pi

Numbers of this type are represented using JavaScript’s built-in Number object, which is a 64-bit IEEE 754 floating-point number. This representation offers a very good compromise between precision and the range of values that can be stored, but can behave in unexpected ways. Accuracy is easily lost when dealing with very big or very small numbers, and on division.

See functions related to Arithmetic, Number operations, Trigonometry and Number theory.

Automatically converts to:
integer

An element of the set of integers, $$\mathbb{Z}$$.

Examples: 0, 1, -2, 431.

Automatically converts to:
rational

A fraction; an element of the set of rationals, $$\mathbb{Q}$$. The numerator and denominator are integers.

Instances of this data type may be top-heavy, with numerator bigger than the denominator, and are not required to be reduced.

Examples: 1/1, -34/2, 3/4.

Automatically converts to:
decimal

A number with a guaranteed level of precision, and arbitrary order of magnitude.

Numbers of this type are represented using the Decimal.js library. They’re guaranteed to be accurate to 40 significant figures. The order of magnitude is stored separately from the significant digits, so there’s no less of precision for very big or very small numbers.

Examples: dec(0), dec("1.23e-5"), 6.0221409*10^23

Automatically converts to:
boolean

Booleans represent either truth or falsity. The logical operations and, or and xor operate on and return booleans.

Examples: true, false

See functions related to Logic and Control flow.

string

Use strings to create non-mathematical text. Either ' or " can be used to delimit strings.

You can escape a character by placing a single backslash character before it. The following escape codes have special behaviour:

 \n New-line \{ \{ \} \}

If you want to write a string which contains a mixture of single and double quotes, you can delimit it with triple-double-quotes or triple-single-quotes, to save escaping too many characters.

Examples: "hello there", 'hello there', """ I said, "I'm Mike's friend" """

See functions related to Strings.

list

An ordered list of elements of any data type.

Examples: [0,1,2,3], [a,b,c], [true,false,true]

See functions related to Lists.

dict

A ‘dictionary’, mapping key strings to values of any data type.

A dictionary is created by enclosing one or more key-value pairs (a string followed by a colon and any JME expression) in square brackets, or with the dict function.

Key strings are case-sensitive.

Examples:

• ["a": 1, "b": 2]

• ["name": "Tess Tuser", "age": 106, "hobbies": ["reading","writing","arithmetic"] ]

• dict("key1": 0.1, "key2": 1..3)

• dict([["key1",1], ["key2",2]])

Warning

Because lists and dicts use similar syntax, [] produces an empty list, not an empty dictionary. To create an empty dictionary, use dict().

See functions related to Dictionaries and JSON.

range

A range a..b#c represents (roughly) the set of numbers $$\{a+nc \: | \: 0 \leq n \leq \frac{b-a}{c} \}$$. If the step size is zero, then the range is the continuous interval $$[a,b]$$.

Examples: 1..3, 1..3#0.1, 1..3#0

See functions related to Ranges.

Automatically converts to:
set

An unordered set of elements of any data type. The elements are pairwise distinct - if you create a set from a list with duplicate elements, the resulting set will not contain the duplicates.

Examples: set(a,b,c), set([1,2,3,4]), set(1..5)

See functions related to Sets.

Automatically converts to:
vector

The components of a vector must be numbers.

When combining vectors of different dimensions, the smaller vector is padded with zeros to make up the difference.

Examples: vector(1,2), vector([1,2,3,4])

See functions related to Vector and matrix arithmetic.

Automatically converts to:
matrix

Matrices are constructed from lists of numbers, representing the rows.

When combining matrices of different dimensions, the smaller matrix is padded with zeros to make up the difference.

To retrieve the n th row of a matrix m, write m[n]. The row is returned as a vector value. To retrieve the cell in row a and column b, write m[a][b]. Rows and columns are both numbered starting from zero.

Examples: matrix([1,2,3],[4,5,6]), matrix(row1,row2,row3)

See functions related to Vector and matrix arithmetic.

Automatically converts to:
name

A variable name. When an expression is evaluated, all variable names are replaced withe their corresponding value in the current scope.

function

An application of a function.

Examples: f(x), sin(x)

op

An infix binary operation, or a pre-/post-fix unary operation.

Examples: x+y, n!, a and b

html

An HTML DOM node.

Examples: html("<div>things</div>")

See functions related to HTML.

expression

A JME sub-expression. Sub-expressions can be simplified, rearranged, pattern-matched, or evaluated using given values for their free variables.

See functions related to Sub-expressions.

### Automatic data type conversion¶

Some data types can be automatically converted to others when required. For example, the number-like types such as integer and decimal can be automatically converted to number values.

The data types of the arguments to a function application determine which version of the function is used. Ideally, this will do what you expect without you having to think about it. For reference, the process for deciding on what conversions to perform is as follows:

• If there is a version of the function which takes exactly the given data types, that is used.

• Otherwise, each definition of the function is compared by looking at each of the arguments, working from left to right.

• A definition which does not convert an argument is preferred to one that does.

• If both definitions being compared need to convert an argument, the type that occurs first in the input type’s list of automatic conversion methods is used. (This follows the order of the types under the “Automatically converts to” headers above)

The following examples illustrate how this works.

 Expression Type of result Explanation 1+3.3 number The 1 is converted to a number, and then added to 3.3. 1+1/2 rational integer prefers to convert to rational over number. 1.23+dec("1.2") decimal decimal values are preferred to number because they’re more precise. 1/2+0.5 number rational can convert to number, but not the other way round, so number addition is used. set(1,2,3,4) except [2] list except() is only defined on list values, so the set is converted to a list automatically.

## Function reference¶

### Arithmetic¶

x+y

Definitions:
Examples:
• 1+23

• vector(1,2)+vector(3,4)vector(4,6)

• matrix([1,2],[3,4])+matrix([5,6],[7,8])matrix([6,8],[10,12])

• [1,2,3]+4[1,2,3,4]

• [1,2,3]+[4,5,6][1,2,3,4,5,6]

• "hi "+"there""hi there"

x-y

Subtraction.

Definitions:
Examples:
• 1-2-1

• vector(3,2)-vector(1,4)vector(2,-2)

• matrix([5,6],[3,4])-matrix([1,2],[7,8])matrix([4,4],[-4,-4])

x*y

Multiplication.

Definitions:
Examples:
• 1*22

• 2*vector(1,2,3)vector(2,4,6)

• matrix([1,2],[3,4])*2matrix([2,4],[6,8])

• matrix([1,2],[3,4])*vector(1,2)vector(5,11)

x/y

Division. Only defined for numbers.

Definitions:
Example:
• 1.2/30.4.

x^y

Exponentiation.

exp(x) is a synonym for e^x.

Definitions:
Examples:
• 3^29

• e^(pi * i)-1

exp(x)

$$e^x$$ - a synonym for e^x.

Definitions:

### Number operations¶

decimal(n)
decimal(x)
dec(x)

Construct a decimal value. Any string accepted by Decimal.js is accepted.

Definitions:
rational(n)

Convert n to a rational number, taking an approximation when necessary.

Definition:
Example:
• rational(pi)355/113

int(n)

Convert n to an integer, rounding to the nearest integer.

Definitions:
Example:
• int(3.0)3

abs(x)
len(x)
length(x)

Absolute value, length, or modulus.

Definitions:
Examples:
• abs(-8)8

• abs(3-4i)5

• abs("Hello")5

• abs([1,2,3])3

• len([1,2,3])3

• len(set([1,2,2]))2

• length(vector(3,4))5

• abs(vector(3,4,12))13

• len(["a": 1, "b": 2, "c": 1])3

arg(z)

Argument of a complex number.

Definitions:
Example:
• arg(-1)pi

re(z)

Real part of a complex number.

Definitions:
Example:
• re(1+2i)1

im(z)

Imaginary part of a complex number.

Definitions:
Example:
• im(1+2i)2

conj(z)

Complex conjugate.

Definitions:
Example:
• conj(1+i)1-i

isint(x)

Returns true if x is an integer - that is, it is real and has no fractional part.

Definitions:
Example:
• isint(4.0)true

iszero(n)

Returns true when n is exactly 0.

Definitions:
sqrt(x)
sqr(x)

Square root of a number.

Examples:
• sqrt(4)2

• sqrt(-1)i

root(x, n)

nth root of x.

Definitions:
Example:
• root(8,3)2.

ln(x)

Natural logarithm.

Definitions:
Example:
• ln(e)1

log(x, b)

Logarithm with base b, or base 10 if b is not given.

Definitions:
Example:
• log(100)2.

• log(343,7)3.

degrees(x)

Convert radians to degrees.

Definitions:
Example:
• degrees(pi/2)90

Convert degrees to radians.

Example:
• radians(180)pi

sign(x)
sgn(x)

Sign of a number. Equivalent to $$\frac{x}{|x|}$$, or 0 when x is 0.

Definitions:
Examples:
• sign(3)1

• sign(-3)-1

max(a, b)

Greatest of the given numbers.

Definitions:
Example:
• max(46,2)46

• max([1,2,3])3

min(a, b)
min(numbers)

Least of the given numbers.

Definitions:
Example:
• min(3,2)2

• min([1,2,3])1

• min(1/2, 2/3)1/2

clamp(x, a, b)

Return the point nearest to x in the interval $$[a,b]$$.

Equivalent to max(a,min(x,b)).

Definitions:
Example:
• clamp(1,0,2)1

• clamp(-1,0,2)0

• clamp(3,0,2)2

precround(n, d)

Round n to d decimal places. On matrices and vectors, this rounds each element independently.

Definitions:
Examples:
• precround(pi,5)3.14159

• precround(matrix([[0.123,4.56],[54,98.765]]),2)matrix([0.12,4.56],[54,98.77])

• precround(vector(1/3,2/3),1)vector(0.3,0.7)

siground(n, f)

Round n to f significant figures. On matrices and vectors, this rounds each element independently.

Definitions:
Examples:
• siground(pi,3)3.14

• siground(matrix([[0.123,4.56],[54,98.765]]),2)matrix([0.12,4.6],[54,99])

• siground(vector(10/3,20/3),2)vector(3.3,6.7)

withintolerance(a, b, t)

Returns true if $$b-t \leq a \leq b+t$$.

Definitions:
Example:
• withintolerance(pi,22/7,0.1)true

dpformat(n, d[, style])

Round n to d decimal places and return a string, padding with zeros if necessary.

If style is given, the number is rendered using the given notation style. See the page on Number notation for more on notation styles.

Definitions:
Example:
• dpformat(1.2,4)"1.2000"

countdp(n)

Assuming n is a string representing a number, return the number of decimal places used. The string is passed through cleannumber() first.

Definitions:
Example:
• countdp("1.0")1

• countdp("1")0

• countdp("not a number")0

sigformat(n, d[, style])

Round n to d significant figures and return a string, padding with zeros if necessary.

Definitions:
Example:
• sigformat(4,3)"4.00"

countsigfigs(n)

Assuming n is a string representing a number, return the number of significant figures. The string is passed through cleannumber() first.

Definitions:
Example:
• countsigfigs("1")1

• countsigfigs("100")1

• countsigfigs("1.0")2

• countsigfigs("not a number")0

togivenprecision(str, precisionType, precision, strict)

Returns true if str is a string representing a number given to the desired number of decimal places or significant figures.

precisionType is either "dp", for decimal places, or "sigfig", for significant figures.

If strict is true, then trailing zeroes must be included.

Definitions:
Examples:
• togivenprecision("1","dp",1,true)false

• togivenprecision("1","dp",1,false)true

• togivenprecision("1.0","dp",1,true)true

• togivenprecision("100","sigfig",1,true)true

• togivenprecision("100","sigfig",3,true)true

tonearest(a, b)

Round a to the nearest multiple of b.

Definitions:
Example:
• tonearest(1.234,0.1)1.2

formatnumber(n, style)

Render the number n using the given number notation style.

See the page on Number notation for more on notation styles.

Definitions:
Example:
• formatnumber(1234.567,"eu")"1.234,567"

scientificnumberlatex(n)

Return a LaTeX string representing the given number in scientific notation, $$a \times 10^b$$.

This function exists because scientific notation may use superscripts, which aren’t easily typeset in plain text.

Definitions:
Example:
• scientificnumberlatex(123)"1.23 \\times 10^{2}"

scientificnumberhtml(n)

Return an HTML element representing the given number in scientific notation, $$a \times 10^b$$.

This function exists because scientific notation may use superscripts, which aren’t easily typeset in plain text.

Definitions:
Example:
• scientificnumberhtml(123)html("<span>1.23 × 10<sup>2</sup></span>")

cleannumber(str, styles)

Clean a string potentially representing a number. Remove space, and then try to identify a notation style, and rewrite to the plain-en style.

styles is a list of notation styles. If styles is given, str will be tested against the given styles. If it matches, the string will be rewritten using the matched integer and decimal parts, with punctuation removed and the decimal point changed to a dot.

Definitions:
Examples:
• cleannumber("100 000,02",["si-fr"])"100000.02"

• cleannumber(" 1 ")"1"

• cleannumber("1.0")"1.0"

matchnumber(str, styles)

Try to match a string representing a number in any of the given styles at the start of the given string, and return both the matched text and a corresponding number value.

Definitions:
Examples:
• matchnumber("1.234",["plain","eu"])[ "1.234", 1.234 ]

• matchnumber("1,234",["plain","eu"])[ "1,234", 1.234 ]

• matchnumber("5 000 things",["plain","si-en"])[ "5 000", 5000 ]

• matchnumber("apple",["plain"])[ "", NaN ]

parsenumber(string, style)

Parse a string representing a number written in the given style.

If a list of styles is given, the first that accepts the given string is used.

See the page on Number notation for more on notation styles.

Examples:
• parsenumber("1 234,567","si-fr")1234.567

• parsenumber("1.001",["si-fr","eu"])1001

parsenumber_or_fraction(string, style)

Works the same as parsenumber(), but also accepts strings of the form number/number, which it interprets as fractions.

Example:
• parsenumber_or_fraction("1/2")0.5

parsedecimal(string, style)

Parse a string representing a number written in the given style, and return a decimal value.

If a list of styles is given, the first that accepts the given string is used.

See the page on Number notation for more on notation styles.

Definitions:
Examples:
• parsedecimal("1 234,567","si-fr")1234.567

• parsedecimal("1.001",["si-fr","eu"])1001

parsedecimal_or_fraction(string, style)

Works the same as parsedecimal(), but also accepts strings of the form number/number, which it interprets as fractions.

Definitions:
Example:
• parsedecimal_or_fraction("1/2")0.5

tobinary(n)

Write the given number in binary: base 2.

Definitions:
Example:
• tobinary(13)"1101"

tooctal(n)

Write the given number in octal: base 8.

Definitions:
Example:
• tooctal(13)"15"

Write the given number in hexadecimal: base 16.

Definitions:
Example:
• tohexadecimal(44)"2c"

tobase(n, base)

Write the given number in the given base. base can be any integer between 2 and 36.

Definitions:
Example:
• tobase(13,4)"31"

• tobase(13,5)"23"

• tobase(50,20)"2a"

frombinary(s)

Convert a string representing a number written in binary (base 2) to a integer value.

Definitions:
Example:
• frombinary("1010")10

fromoctal(s)

Convert a string representing a number written in octal (base 8) to a integer value.

Definitions:
Example:
• fromoctal("54")44

Convert a string representing a number written in hexadecimal (base 16) to a integer value.

Definitions:
Example:
• fromhexadecimal("b4")180

frombase(s, base)

Convert a string representing a number written in the given base to a integer value. base can be any integer between 2 and 36.

Definitions:
Example:
• frombase("b4",20)224

• frombase("321",5)86

• frombase("621",5)NaN

isnan(n)

Is n the “not a number” value, NaN?

Definitions:
Examples:
• isnan(1)false

• isnan(parsenumber("a","en"))true

### Trigonometry¶

Trigonometric functions all work in radians, and have as their domain the complex numbers.

sin(x)

Sine.

Definitions:
cos(x)

Cosine.

Definitions:
tan(x)

Tangent: $$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

Definitions:
cosec(x)

Cosecant: $$\csc(x) = \frac{1}{sin(x)}$$

Definitions:
sec(x)

Secant: $$\sec(x) = \frac{1}{cos(x)}$$

Definitions:
cot(x)

Cotangent: $$\cot(x) = \frac{1}{\tan(x)}$$

Definitions:
arcsin(x)

Inverse of sin(). When $$x \in [-1,1]$$, arcsin(x) returns a value in $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$.

Definitions:
arccos(x)

Inverse of cos(). When $$x \in [-1,1]$$, arccos(x) returns a value in $$[0, \frac{\pi}]$$.

Definitions:
arctan(x)

Inverse of tan(). When $$x$$ is non-complex, arctan(x) returns a value in $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$.

Definitions:
atan2(y, x)

The angle in radians between the positive $$x$$-axis and the line through the origin and $$(x,y)$$. This is often equivalent to arctan(y/x), except when $$x \lt 0$$, when $$pi$$ is either added or subtracted from the result.

Definitions:
Examples:
• atan2(0,1)0

• atan2(sin(1),cos(1))1

• atan2(sin(pi/4), cos(pi/4)) / pi0.25

• atan2(sin(pi/4), -cos(pi/4)) / pi0.75

sinh(x)

Hyperbolic sine: $$\sinh(x) = \frac{1}{2} \left( \mathrm{e}^x - \mathrm{e}^{-x} \right)$$

Definitions:
cosh(x)

Hyperbolic cosine: $$\cosh(x) = \frac{1}{2} \left( \mathrm{e}^x + \mathrm{e}^{-x} \right)$$

Definitions:
tanh(x)

Hyperbolic tangent: $$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$$

Definitions:
cosech(x)

Hyperbolic cosecant: $$\operatorname{cosech}(x) = \frac{1}{\sinh(x)}$$

Definitions:
sech(x)

Hyperbolic secant: $$\operatorname{sech}(x) = \frac{1}{\cosh(x)}$$

Definitions:
coth(x)

Hyperbolic cotangent: $$\coth(x) = \frac{1}{\tanh(x)}$$

Definitions:
arcsinh(x)

Inverse of sinh().

Definitions:
arccosh(x)

Inverse of cosh().

Definitions:
arctanh(x)

Inverse of tanh().

Definitions:

### Number theory¶

x!
fact(x)

Factorial. When x is not an integer, $$\Gamma(x+1)$$ is used instead.

fact(x) is a synonym for x!.

Definitions:
Examples:
• fact(3)6

• 3!6

• fact(5.5)287.885277815

factorise(n)

Factorise n. Returns the exponents of the prime factorisation of n as a list.

Definitions:
Examples
• factorise(18)[1,2]

• factorise(70)[1,0,1,1]

gamma(x)

Gamma function.

Definitions:
Examples:
• gamma(3)2

• gamma(1+i)0.4980156681 - 0.1549498283i

ceil(x)

Round up to the nearest integer. When x is complex, each component is rounded separately.

Definitions:
Examples:
• ceil(3.2)4

• ceil(-1.3+5.4i)-1+6i

floor(x)

Round down to the nearest integer. When x is complex, each component is rounded separately.

Definitions:
Example:
• floor(3.5)3

round(x)

Round to the nearest integer. 0.5 is rounded up.

Definitions:
Examples:
• round(0.1)0

• round(0.9)1

• round(4.5)5

• round(-0.5)0

trunc(x)

If x is positive, round down to the nearest integer; if it is negative, round up to the nearest integer.

Definitions:
Example:
• trunc(3.3)3

• trunc(-3.3)-3

fract(x)

Fractional part of a number. Equivalent to x-trunc(x).

Definitions:
Example:
• fract(4.3)0.3

rational_approximation(n[, accuracy])

Compute a rational approximation to the given number by computing terms of its continued fraction, returning the numerator and denominator separately. The approximation will be within $$e^{-\text{accuracy}}$$ of the true value; the default value for accuracy is 15.

Definitions:
Examples:
• rational_approximation(pi)[355,113]

• rational_approximation(pi,3)[22,7]

mod(a, b)

Modulo; remainder after integral division, i.e. $$a \bmod b$$.

Definitions:
Example:
• mod(5,3)2

perm(n, k)

Count permutations, i.e. $$^n \kern-2pt P_k = \frac{n!}{(n-k)!}$$.

Definitions:
Example:
• perm(5,2)20

comb(n, k)

Count combinations, i.e. $$^n \kern-2pt C_k = \frac{n!}{k!(n-k)!}$$.

Definitions:
Example:
• comb(5,2)10.

gcd(a, b)
gcf(a, b)

Greatest common divisor of integers a and b. Can also write gcf(a,b).

Definitions:
Example:
• gcd(12,16)4

gcd_without_pi_or_i(a, b)

Take out factors of $$\pi$$ or $$i$$ from a and b before computing their greatest common denominator.

Definitions:
Example:
• gcd_without_pi_or_i(6*pi, 9)3

coprime(a, b)

Are a and b coprime? True if their gcd() is $$1$$, or if either of a or b is not an integer.

Definitions:
Examples:
• coprime(12,16)false

• coprime(2,3)true

• coprime(1,3)true

• coprime(1,1)true

lcm(a, b)

Lowest common multiple of integers a and b. Can be used with any number of arguments; it returns the lowest common multiple of all the arguments.

Definitions:
Examples
• lcm(8,12)24

• lcm(8,12,5)120

x|y

x divides y.

Definitions:
Example:
• 4|8true

### Vector and matrix arithmetic¶

vector(a1, a2, ..., aN)

Create a vector with given components. Alternately, you can create a vector from a single list of numbers.

Definitions:
Examples:
• vector(1,2,3)

• vector([1,2,3])

matrix(row1, row2, ..., rowN)

Create a matrix with given rows, which should be either vectors or lists of numbers. Or, you can pass in a single list of lists of numbers.

Definitions:
Examples:
• matrix([1,2],[3,4])

• matrix([[1,2],[3,4]])

id(n)

Identity matrix with $$n$$ rows and columns.

Definitions:
Example:
• id(3)matrix([1,0,0],[0,1,0],[0,0,1])

numrows(matrix)

The number of rows in the given matrix

Definitions:
Example:
• numrows(matrix([1,2],[3,4],[5,6]))3

numcolumns(matrix)

The number of columns in the given matrix

Definitions:
Example:
• numcolumns(matrix([1,2],[3,4],[5,6]))2

rowvector(a1, a2, ..., aN)

Create a row vector ($$1 \times n$$ matrix) with the given components. Alternately, you can create a row vector from a single list of numbers.

Definitions:
Examples:
• rowvector(1,2)matrix([1,2])

• rowvector([1,2])matrix([1,2])

dot(x, y)

Dot (scalar) product. Inputs can be vectors or column matrices.

Definitions:
Examples:
• dot(vector(1,2,3),vector(4,5,6))32

• dot(matrix([1],[2]), matrix([3],[4]))11

cross(x, y)

Cross product. Inputs can be vectors or column matrices.

Definitions:
Examples:
• cross(vector(1,2,3),vector(4,5,6))vector(-3,6,-3)

• cross(matrix([1],[2],[3]), matrix([4],[5],[6]))vector(-3,6,-3)

angle(a, b)

Angle between vectors a and b, in radians. Returns 0 if either a or b has length 0.

Definitions:
Example:
• angle(vector(1,0),vector(0,1))1.5707963268

is_zero(x)

Returns true if every component of the vector x is zero.

Definitions:
Example:
• is_zero(vector(0,0,0))true

det(x)

Determinant of a matrix. Throws an error if used on anything larger than a 3×3 matrix.

Definitions:
Examples:
• det(matrix([1,2],[3,4]))-2

• det(matrix([1,2,3],[4,5,6],[7,8,9]))0

transpose(x)

Matrix transpose.

Definitions:
Examples:
• transpose(matrix([1,2],[3,4]))matrix([1,3],[2,4])

• transpose(vector(1,2,3))matrix([1,2,3])

sum_cells(m)

Calculate the sum of all the cells in a matrix.

Definitions:
Example:
• sum_cells(matrix([1,2],[3,4]))10

### Strings¶

x[n]

Get the Nth character of the string x. Indices start at 0.

Example:
• "hello"[1]"e"

x[a..b]

Slice the string x - get the substring between the given indices. Note that indices start at 0, and the final index is not included.

Example:
• "hello"[1..4]"ell"

substring in string

Test if substring occurs anywhere in string. This is case-sensitive.

Example:
• "plain" in "explains"true

string(x)

Convert x to a string.

When converting a expression value to a string, you can give a list of display options as a second argument, either as a comma-separated string or a list of strings.

Definitions:
Example:
• string(123)"123"

• string(x)"x"

• string(expression("0.5"))"0.5"

• string(expression("0.5"),"fractionNumbers")"1/2"

latex(x)

Mark string x as containing raw LaTeX, so when it’s included in a mathmode environment it doesn’t get wrapped in a \textrm environment.

If x is a expression value, it’s rendered to LaTeX.

Note that backslashes must be double up, because the backslash is an escape character in JME strings.

Definitions:
Example:
• latex('\\frac{1}{2}').

• latex(expression("x^2 + 3/4"))"x^{2} + \\frac{3}{4}"

safe(x)

Mark string x as safe: don’t substitute variable values into it when this expression is evaluated.

Use this function to preserve curly braces in string literals.

Definitions:
Example:
• safe('From { to }')

render(x, values)

Substitute variable values into the string x, even if it’s marked as safe (see safe()).

The optional dictionary values overrides any previously-defined values of variables.

Definitions:
Example:
• render(safe("I have {num_apples} apples."), ["num_apples": 5])"I have 5 apples."

• render(safe("Let $x = \\var{x}$"), ["x": 2])"Let $x = {2}$"

Note

The variable dependency checker can’t establish which variables will be used in the string until render is evaluated, so you may encounter errors if using render in the definition of a question variable. You can ensure a variable has been evaluated by including it in the values argument, e.g.:

render("a is {}",["a": a])


This function is intended for use primarily in content areas.

capitalise(x)

Capitalise the first letter of a string.

Definitions:
Example:
• capitalise('hello there').

pluralise(n, singular, plural)

Return singular if n is 1, otherwise return plural.

Definitions:
Example:
• pluralise(num_things,"thing","things")

upper(x)

Convert string to upper-case.

Definitions:
Example:
• upper('Hello there').

lower(x)

Convert string to lower-case.

Definitions:
Example:
• lower('CLAUS, Santa').

join(strings, delimiter)

Join a list of strings with the given delimiter.

Definitions:
Example:
• join(['a','b','c'],',')"a,b,c"

split(string, delimiter)

Split a string at every occurrence of delimiter, returning a list of the the remaining pieces.

Definitions:
Example:
• split("a,b,c,d",",")["a","b","c","d"]

match_regex(pattern, str, flags)

If str matches the regular expression pattern, returns a list of matched groups, otherwise returns an empty list.

This function uses JavaScript regular expression syntax.

flags is an optional string listing the options flags to use. If it’s not given, the default value of "u" is used.

Definitions:
Examples:
• match_regex("\\d+","01234")["01234"]

• match_regex("a(b+)","abbbb")["abbbb","bbbb"]

• match_regex("a(b+)","ABBBB")[]

• match_regex("a(b+)","ABBBB","i")["ABBBB","BBBB"]

split_regex(string, pattern, flags)

Split a string at every occurrence of a substring matching the given regular expression pattern, returning a list of the the remaining pieces.

flags is an optional string listing the options flags to use. If it’s not given, the default value of "u" is used.

Definitions:
Example:
• split_regex("a, b,c, d ",", *")["a","b","c","d"]

• split_regex("this and that AND THIS"," and ","i")["this","that","THIS"]

replace_regex(pattern, replacement, string, flags)

Replace a substring of string matching the given regular expression pattern with the string replacement.

flags is an optional string listing the options flags to use. If it’s not given, the default value of "u" is used.

Remember that backslashes must be doubled up inside JME strings, and curly braces are normally used to substitute in variables. You can use the safe() function to avoid this behaviour.

To replace all occurrences of the pattern, add the flag "g".

Definitions:
Example:
• replace_regex("day","DAY","Monday Tuesday Wednesday")"MonDAY Tuesday Wednesday"

• replace_regex("day","DAY","Monday Tuesday Wednesday","g")"MonDAY TuesDAY WednesDAY"

• replace_regex("a","o","Aardvark")"Aordvark"

• replace_regex("a","o","Aardvark","i")"oardvark"

• replace_regex("a","o","Aardvark","ig")"oordvork"

• replace_regex(safe("(\\d+)x(\\d+)"),"$1 by$2","32x24")"32 by 24"

• replace_regex(safe("a{2}"),"c","a aa aaa")"a c aaa"

trim(str)

Remove whitespace from the start and end of str.

Definitions:
Example:
• trim(" a string  ")"a string"

currency(n, prefix, suffix)

Write a currency amount, with the given prefix or suffix characters.

Definitions:
Example:
• currency(123.321,"£","")"£123.32"

separateThousands(n, separator)

Write a number, with the given separator character between every 3 digits

To write a number using notation appropriate to a particular culture or context, see formatnumber().

Definitions:
Example:
• separateThousands(1234567.1234,",")"1,234,567.1234"

unpercent(str)

Get rid of the % on the end of a percentage and parse as a number, then divide by 100.

Definitions:
Example:
• unpercent("2%")0.02

Add copies of prefix to the start of str until the result is at least n characters long.

Definitions:
Example:
• lpad("3", 2, "0")"03"

Add copies of suffix to the end of str until the result is at least n characters long.

Definitions:
Example:
• rpad("3", 2, "0")"30"

formatstring(str, values)

For each occurrence of %s in str, replace it with the corresponding entry in the list values.

Definitions:
Example:
• formatstring("Their name is %s",["Hortense"])"Their name is Hortense"

• formatstring("You should %s the %s",["simplify","denominator"])"You should simplify the denominator"

letterordinal(n)

Get the $$n$$th element of the sequence a, b, c, ..., aa, ab, ....

Note that the numbering starts from 0.

Definitions:
Examples:
• letterordinal(0)"a"

• letterordinal(1)"b"

• letterordinal(26)"aa"

translate(str, arguments)

Translate the given string, if it’s in the localisation file.

Look at the default localisation file for strings which can be translated. This function takes a key representing a string to be translated, and returns the corresponding value from the current localisation file.

arguments is a dictionary of named substitutions to make in the string.

Definitions:
Examples:
• translate("question.header",["number": 2])"Question 2" (when the en-GB locale is in use)

• translate("question.header",["number": 2]) "Pregunta 2" (when the es-ES locale is in use)

isbool(str)

After converting to lower case, is str any of the strings "true", "false", "yes" or "no"?

Definitions:
Examples:
• isbool("true")true

• isbool("YES")true

• isbool("no")true

• isbool("y")false

### Logic¶

x<y

Returns true if x is less than y.

Definitions:
Example:
• 4<5

x>y

Returns true if x is greater than y.

Definitions:
Example:
• 5>4

x<=y

Returns true if x is less than or equal to y.

Definitions:
Example:
• 4<=4

x>=y

Returns true if x is greater than or equal to y.

Definitions:
Example:
• 4>=4

x<>y

Returns true if x is not equal to y. Returns true if x and y are not of the same data type.

Definitions:
Examples:
• 'this string' <> 'that string'

• 1<>2

• '1' <> 1

x=y

Returns true if x is equal to y. Returns false if x and y are not of the same data type.

Definitions:
Examples:
• vector(1,2)=vector(1,2,0)

• 4.0=4

isclose(x, y[, rel_tol][, abs_tol])

Returns true if x is close to y.

The arguments rel_tol and abs_tol are optional, with default values of $$10^{-15}$$.

Equivalent to the following expression:

abs(x-y) <= max( rel_tol*max(abs(a),abs(b)), abs_tol )

Definitions:
resultsequal(a, b, checkingFunction, accuracy)

Returns true if a and b are both of the same data type, and “close enough” according to the given checking function.

Vectors, matrices, and lists are considered equal only if every pair of corresponding elements in a and b is “close enough”.

checkingFunction is the name of a checking function to use. These are documented in the Numbas runtime documentation.

Definitions:
Examples:
• resultsequal(22/7,pi,"absdiff",0.001)false

• resultsequal(22/7,pi,"reldiff",0.001)true

x and y
x && y
x & y

Logical AND. Returns true if both x and y are true, otherwise returns false.

Definitions:
Examples:
• true and true

• true && true

• true & true

not x

Logical NOT.

Definitions:
Examples:
• not true

• !true

x or y

Logical OR. Returns true when at least one of x and y is true. Returns false when both x and y are false.

Definitions:
Examples:
• true or false

• true || false

x xor y

Logical XOR. Returns true when at either x or y is true but not both. Returns false when x and y are the same expression.

Definitions:
Example:
• true XOR false.

x implies y

Logical implication. If x is true and y is false, then the implication is false. Otherwise, the implication is true.

Definitions:
Example:
• false implies true.

### Collections¶

x[y]

Get the yth element of the collection x.

For matrices, the yth row is returned.

For dictionaries, the value corresponding to the key y is returned. If the key is not present in the dictionary, an error will be thrown.

Definitions:
Examples:
• [0,1,2,3][1]1

• vector(0,1,2)[2]2

• matrix([0,1,2],[3,4,5],[6,7,8])[0]vector(0,1,2)

• ["a": 1, "b": 2]["a"]1

x[a..b]
x[a..b#c]

Slice the collection x - return elements with indices in the given range. Note that list indices start at 0, and the final index is not included.

Definitions:
Example:
• [0,1,2,3,4,5][1..3][1,2]

• [0,1,2,3,4,5][1..6#2][1,3,5]

x in collection

Is element x in collection?

Definitions:
Examples:
• 3 in [1,2,3,4]true

• 3 in (set(1,2,3,4) and set(2,4,6,8))false

• "a" in ["a": 1]true

• 1 in ["a": 1] throws an error because dictionary keys must be strings.

### Ranges¶

a .. b

Define a range. Includes all integers between and including a and b.

Definitions:
Examples:
• 1..5

• -6..6

range # step

Set the step size for a range. Default is 1. When step is 0, the range includes all real numbers between the limits.

Definitions:
Examples:
• 0..1 # 0.1

• 2..10 # 2

• 0..1#0

a except b

Exclude a number, range, or list of items from a list or range.

Definitions:
Examples:
• -9..9 except 0

• -9..9 except [-1,1]

• 3..8 except 4..6

• [1,2,3,4,5] except [2,3]

### Lists¶

repeat(expression, n)

Evaluate expression n times, and return the results in a list.

Definitions:
Example:
• repeat(random(1..4),5) [2, 4, 1, 3, 4]

all(list)

Returns true if every element of list is true.

Definitions:
Examples:
• all([true,true])true

• all([true,false])false

• all([])true

some(list)

Returns true if at least one element of list is true.

Definitions:
Examples:
• some([false,true,false])true

• some([false,false,false])false

• some([])false

map(expression,name[s],d)

Evaluate expression for each item in list, range, vector or matrix d, replacing variable name with the element from d each time.

You can also give a list of names if each element of d is a list of values. The Nth element of the list will be mapped to the Nth name.

Definitions:
Examples:
• map(x+1,x,1..3)[2,3,4]

• map(capitalise(s),s,["jim","bob"])["Jim","Bob"]

• map(sqrt(x^2+y^2),[x,y],[ [3,4], [5,12] ])[5,13]

• map(x+1,x,id(2))matrix([2,1],[1,2])

• map(sqrt(x),x,vector(1,4,9))vector(1,2,3)

filter(expression, name, d)

Filter each item in list or range d, replacing variable name with the element from d each time, returning only the elements for which expression evaluates to true.

Definitions:
Example:
• filter(x>5,x,[1,3,5,7,9])[7,9]

foldl(expression, accumulator_name, item_name, first_value, d)

Accumulate a value by iterating over a collection. This can be used as an abstraction of routines such as “sum of a list of numbers”, or “maximum value in a list”.

Evaluate expression for each item in the list, range, vector or matrix d, accumulating a single value which is returned.

At each iteration, the variable item_name is replaced with the corresponding value from d. The variable accumulator_name is replaced with first_value for the first iteration, and the result of expression from the previous iteration subsequently.

Definitions:
Examples:
• foldl(total + x, total, x, 0, [1,2,3])6

• foldl(if(len(x)>len(longest),x,longest), longest, x, "", ["banana","pineapple","plum"])"pineapple"

iterate(expression, name, initial, times)

Iterate an expression on the given initial value the given number of times, returning a list containing the values produced at each step.

You can also give a list of names. The Nth element of the value will be mapped to the Nth name.

Definitions:
Example:
• iterate(x+1, x, 0, 3)[0,1,2,3]

• iterate([b,a+b], [a,b], [1,1], 3)[ [1,1], [1,2], [2,3], [3,5] ]

• iterate(l[1..len(l)]+[l[0]], l, ["a","b","c"], 3)[ ["a","b","c"], ["b","c","a"], ["c","a","b"], ["a","b","c"] ]

iterate_until(expression, name, initial, condition, max_iterations)

Iterate an expression on the given initial value until the condition is satisfied, returning a list containing the values produced at each step.

You can also give a list of names. The Nth element of the value will be mapped to the Nth name.

max_iterations is an optional parameter specifying the maximum number of iterations that may be performed. If not given, the default value of 100 is used. This parameter prevents the function from running indefinitely, when the condition is never met.

Definitions:
Example:
• iterate_until(if(mod(x,2)=0,x/2,3x+1), x, 5, x=1)[ 5, 16, 8, 4, 2, 1 ]

• iterate_until([b,mod(a,b)], [a,b], [37,32], b=0)[ [ 37, 32 ], [ 32, 5 ], [ 5, 2 ], [ 2, 1 ], [ 1, 0 ] ]

take(n, expression, name, d)

Take the first n elements from list or range d, replacing variable name with the element from d each time, returning only the elements for which expression evaluates to true.

This operation is lazy - once n elements satisfying the expression have been found, execution stops. You can use this to filter a few elements from a large list, where the condition might take a long time to calculate.

Definitions:
Example:
• take(3,gcd(x,6)=1,x,10..30)[11,13,17]

flatten(lists)

“Flatten” a list of lists, returning a single list containing the concatenation of all the entries in lists.

Definitions:
Example:
• flatten([ [1,2], [3,4] ])[1,2,3,4]

let(name, definition, ..., expression)
let(definitions, expression)

Evaluate expression, temporarily defining variables with the given names. Use this to cut down on repetition. You can define any number of variables - in the first calling pattern, follow a variable name with its definition. Or you can give a dictionary mapping variable names to their values. The last argument is the expression to be evaluated.

Definitions:
Examples:
• let([a,b,c],[1,5,6],d,sqrt(b^2-4*a*c), [(-b+d)/2, (-b-d)/2])[-2,-3] (when [a,b,c] = [1,5,6])

• let(x,1, y,2, x+y)3

• let(["x": 1, "y": 2], x+y)3

sort(x)

Sort list x.

Definitions:
Example:
• sort([4,2,1,3])[1,2,3,4]

sort_destinations(x)

Return a list giving the index that each entry in the list will occupy after sorting.

Definitions:
Example:
• sort_destinations([4,2,1,3])[3,1,0,2]

• sort_destinations([1,2,3,4])[0,1,2,3]

sort_by(key, list)

Sort the given list of either list or dict values by their entries corresponding to the given key. When sorting a list of lists, the key is a number representing the index of each list to look at. When sorting a list of dictionaries, the key is a string.

Definitions:
Examples:
• sort_by(0, [[5,0], [3,2], [4,4]])[[3,2], [4,4], [5,0]]

• sort_by("width", [["label": "M", "width": 20], ["label": "L", "width": 30], ["label": "S", "width": 10]])[["label": "S", "width": 10], ["label": "M", "width": 20], ["label": "L", "width": 30]]

group_by(key, list)

Group the entries in the given list of either list or dict values by their entries corresponding to the given key. The returned value is a list of lists of the form [key, group], where key is the value all elements of the list group have in common.

When grouping a list of lists, the key argument is a number representing the index of each list to look at. When grouping a list of dictionaries, the key argument is a string.

Definitions:
Examples:
• group_by(0, [[0,0], [3,2], [0,4]])[[0, [[0,0], [0,4]]], [3, [[3,2]]]]

• group_by("a", [["a": 1, "b": "M"], ["a": 2, "b": "S"], ["a": 1, "b": "XL"]])[[1,[["a": 1, "b": "M"], ["a": 1, "b": "XL"]]], [2, [["a": 2, "b": "S"]]]]

reverse(x)

Reverse list x.

Definitions:
Example:
• reverse([1,2,3])[3,2,1]

indices(list, value)

Find the indices at which value occurs in list.

Definitions:
Examples:
• indices([1,0,1,0],1)[0,2]

• indices([2,4,6],4)[1]

• indices([1,2,3],5)[]

distinct(x)

Return a copy of the list x with duplicates removed.

Definitions:
Example:
• distinct([1,2,3,1,4,3])[1,2,3,4]

list(x)

Convert a value to a list of its components (or rows, for a matrix).

Definitions:
Examples:
• list(set(1,2,3))[1,2,3] (note that you can’t depend on the elements of sets being in any order)

• list(vector(1,2))[1,2]

• list(matrix([1,2],[3,4]))[[1,2], [3,4]]

make_variables(definitions)

Evaluate a dictionary of variable definitions and return a dictionary containing the generated values.

definitions is a dictionary mapping variable names to expression values corresponding to definitions.

The definitions can refer to other variables to be evaluated, or variables already defined in the current scope. Variables named in the dictionary which have already been defined will be removed before evaluation begins.

Definitions:
Example:
• make_variables(["a": expression("random(1..5)"), "b": expression("a^2")]) ["a": 3, "b": 9]

satisfy(names, definitions, conditions, maxRuns)

Each variable name in names should have a corresponding definition expression in definitions. conditions is a list of expressions which you want to evaluate to true. The definitions will be evaluated repeatedly until all the conditions are satisfied, or the number of attempts is greater than maxRuns. If maxRuns isn’t given, it defaults to 100 attempts.

Note

This function is deprecated, and retained only for backwards compatibility. Use make_variables() instead.

Definitions:
Example:
• satisfy([a,b,c],[random(1..10),random(1..10),random(1..10)],[b^2-4*a*c>0])

sum(numbers)

Add up a list of numbers

Definitions:
Example:
• sum([1,2,3])6

• sum(vector(4,5,6))15

prod(list)

Multiply a list of numbers together

Definitions:
Example:
• prod([2,3,4])24

product(list1, list2, ..., listN)

Cartesian product of lists. In other words, every possible combination of choices of one value from each given list.

If one list and a number are given, then the n-th Cartesian power of the list is returned: the Cartesian product of n copies of the list.

Definitions:
Example:
• product([1,2],[a,b])[ [1,a], [1,b], [2,a], [2,b] ]

• product([1,2],2)[ [1,1], [1,2], [2,1], [2,2] ]

zip(list1, list2, ..., listN)

Combine two (or more) lists into one - the Nth element of the output is a list containing the Nth elements of each of the input lists.

Definitions:
Example:
• zip([1,2,3],[4,5,6])[ [1,4], [2,5], [3,6] ]

combinations(collection, r)

All ordered choices of r elements from collection, without replacement.

Definitions:
Example:
• combinations([1,2,3],2)[ [1,2], [1,3], [2,3] ]

combinations_with_replacement(collection, r)

All ordered choices of r elements from collection, with replacement.

Definitions:
Example:
• combinations_with_replacement([1,2,3],2)[ [1,1], [1,2], [1,3], [2,2], [2,3], [3,3] ]

permutations(collection, r)

All choices of r elements from collection, in any order, without replacement.

Definitions:
Example:
• permutations([1,2,3],2)[ [1,2], [1,3], [2,1], [2,3], [3,1], [3,2] ]

frequencies(collection)

Count the number of times each distinct element of collection appears.

Returns a list of pairs [value, frequency], where value is a value from the list, and frequency is the number of times it appeared.

Definitions:
Examples:
• frequencies([1,2,3,2,2,1])[ [1,2], [2,3], [3,1] ]

• frequencies(["a","a","c","b","c","a"])[ ["a",3], ["c",2], ["b",1] ]

enumerate(collection)

Enumerate the elements of collection: this function returns a list containing, for each element v of collection, a new list of the form [i,v], where i is the index of the element in collection.

Definitions:
Example:
• enumerate(["A","B","C"])[ [0,"A"], [1,"B"], [2,"C"] ]

### Dictionaries¶

dict[key]

Get the value corresponding to the given key string in the dictionary d.

If the key is not present in the dictionary, an error will be thrown.

Definitions:
Example:
• ["a": 1, "b": 2]["a"]1

get(dict, key, default)

Get the value corresponding to the given key string in the dictionary.

If the key is not present in the dictionary, the default value will be returned.

Definitions:
Examples:
• get(["a":1], "a", 0)1

• get(["a":1], "b", 0)0

dict(a:b, c:d, ...)
dict(pairs)

Create a dictionary with the given key-value pairs. Equivalent to [ .. ], except when no key-value pairs are given: [] creates an empty list instead.

You can alternately pass a list of pairs of the form [key, value], to transform a list into a dictionary.

Definitions:
Examples:
• dict()

• dict("a": 1, "b": 2)

• dict([ ["a",1], ["b",2] ])

keys(dict)

A list of all of the given dictionary’s keys.

Definitions:
Example:
• keys(["a": 1, "b": 2, "c": 1])["a","b","c"]

values(dict[, keys])

A list of the values corresponding to each of the given dictionary’s keys.

If a list of keys is given, only the values corresponding to those keys are returned, in the same order.

Definitions:
Examples:
• values(["a": 1, "b": 2, "c": 1])[1,2,1]

• values(["a": 1, "b": 2, "c": 3], ["b","a"])[2,1]

items(dict)

A list of all of the [key,value] pairs in the given dictionary.

Definitions:
Example:
• items(["a": 1, "b": 2, "c": 1])[ ["a",1], ["b",2], ["c",1] ]

### Sets¶

set(elements)

Create a set with the given elements. Either pass the elements as individual arguments, or as a list.

Definitions:
Examples:
• set(1,2,3)

• set([1,2,3])

union(a, b)

Union of sets a and b

Definitions:
Examples:
• union(set(1,2,3),set(2,4,6))set(1,2,3,4,6)

• set(1,2,3) or set(2,4,6)set(1,2,3,4,6)

intersection(a, b)

Intersection of sets a and b, i.e. elements which are in both sets.

Definitions:
Examples:
• intersection(set(1,2,3),set(2,4,6))set(2)

• set(1,2,3) and set(2,4,6)set(2)

a - b

Set minus - elements which are in a but not b

Example:
• set(1,2,3,4) - set(2,4,6)set(1,3)

### Randomisation¶

random(x)

Pick uniformly at random from a range, list, or from the given arguments.

Definitions:
Examples:
• random(1..5)

• random([1,2,4])

• random(1,2,3)

weighted_random(x)

Pick random from a weighted list of items. Each element in the input list is a pair of the form [item, probability], where probability is a number value.

Items with negative weight are ignored.

Definitions:
Examples:
• weighted_random([ ["a", 1], ["b", 2], ["c", 0.5] ])

deal(n)

Get a random shuffling of the integers $$[0 \dots n-1]$$

Definitions:
Example:
• deal(3) [2,0,1]

reorder(list, order)

Reorder a list given a permutation. The ith element of the output is the order[i]th element of list.

Definitions:
Examples:
• reorder([0,1,2,3],[3,2,0,1])[3,2,0,1]

• reorder(["a","b","c","d"],[3,2,0,1])["d","c","a","b"]

shuffle(x)

Random shuffling of list or range.

Definitions:
Examples:
• shuffle(["a","b","c"]) ["c","b","a"]

• shuffle(0..4) [2,3,0,4,1]

shuffle_together(lists)

Shuffle several lists together - each list has the same permutation of its elements applied. The lists must all be the same length, otherwise an error is thrown.

Definitions:
Example:
• shuffle_together([ ["a","b","c","d"], [0,1,2,3] ]) [ ["d","a","c","b"], [3,0,2,1] ]

### Control flow¶

award(a, b)

Return a if b is true, else return 0.

Definitions:
Example:
• award(5,true)5

if(p, a, b)

If p is true, return a, else return b. Only the returned value is evaluated.

Definitions:
Example:
• if(false,1,0)0

switch(p1, a1, p2, a2, ..., pn, an, d)

Select cases. Alternating boolean expressions with values to return, with the final argument representing the default case. Only the returned value is evaluated.

Definitions:
Examples:
• switch(true,1,false,0,3)1

• switch(false,1,true,0,3)0

• switch(false,1,false,0,3)3

assert(condition, value)

If condition is false, then return value, otherwise don’t evaluate value and return false. This is intended for use in marking scripts, to apply marking feedback only if a condition is met.

Definitions:
Example:
• assert(studentAnswer<=0, correct("Student answer is positive"))

try(expression, name, except)

Try to evaluate expression. If it is successfully evaluated, return the result. Otherwise, evaluate except, with the error message available as name.

Definitions:
Examples:
• try(eval(expression("x+")),err, "Error: "+err)"Error: Not enough arguments for operation <code>+</code>"

• try(1+2,err,0)3

### HTML¶

html(x)

Parse string x as HTML.

Definitions:
Example:
• html('<div>Text!</div>')

isnonemptyhtml(str)

Does str represent a string of HTML containing text? Returns false for the empty string, or HTML elements with no text content.

Definitions:
Examples:
• isnonemptyhtml("<p>Yes</p>")true

• isnonemptyhtml("<p></p>")false

table(data)

Create an HTML with cell contents defined by data, which should be a list of lists of data, and column headers defined by the list of strings headers.

Definitions:
Examples:
• table([[0,1],[1,0]], ["Column A","Column B"])

• table([[0,1],[1,0]])

image(url[, width][, height])

Create an HTML img element loading the image from the given URL. Images uploaded through the resources tab are stored in the relative URL resources/images/<filename>.png, where <filename> is the name of the original file.

The optional width and height parameters specify the size the image should be displayed at, in em units. If only the width is given, the height will be automatically set to maintain the image’s original proportions. 1 em is the width of the letter ‘m’.

If neither width nor height are given, the image is displayed at its original size.

Definitions:
Examples:
• image('resources/images/picture.png')

• image(chosenimage)

max_width(width, element)

Apply a CSS max-width attribute to the given element. You can use this to ensure that an image is not displayed too wide. The given width is in em units.

Definitions:
Example:

max_width(400,html("<p>Text</p>"))html("<p style=\"max-width: 400em;\">a</p>")

max_height(width, element)

Apply a CSS max-height attribute to the given element. You can use this to ensure that an image is not displayed too long. The given height is in em units.

Definitions:
Example:

max_height(400,html("<p>Text</p>"))html("<p style=\"max-height: 400em;\">a</p>")

### JSON¶

JSON is a lightweight data-interchange format. Many public data sets come in JSON format; it’s a good way of encoding data in a way that is easy for both humans and computers to read and write.

For an example of how you can use JSON data in a Numbas question, see the exam Working with JSON data.

json_decode(json)

Decode a JSON string into JME data types.

JSON is decoded into numbers, strings, booleans, lists, or dictionaries. To produce other data types, such as matrices or vectors, you will have to post-process the resulting data.

Warning

The JSON value null is silently converted to an empty string, because JME has no “null” data type. This may change in the future.

Definitions:
Example:
• json_decode(safe(' {"a": 1, "b": [2,true,"thing"]} '))["a": 1, "b": [2,true,"thing"]]

json_encode(data)

Convert the given object to a JSON string.

Numbers, strings, booleans, lists, and dictionaries are converted in a straightforward manner. Other data types may behave unexpectedly.

Definitions:
Example:
• json_encode([1,"a",true])"[1,\"a\",true]"

### Sub-expressions¶

The expression data type represents a JME expression. You can use it to store and manipulate expressions symbolically, substituting in other variables before evaluating or displaying to the student.

There are functions to construct sub-expressions from strings of code, or by joining other sub-expressions together as the arguments to operators or functions.

Once you’ve got a sub-expression, you can evaluate it to a normal JME data type, test if it matches a pattern, substitute values or other sub-expressions for free variables, rearrange using simplification rules, or test if it is equivalent to another sub-expression.

expression(string)
parse(string)

Parse a string as a JME expression. The expression can be substituted into other expressions, such as the answer to a mathematical expression part, or the \simplify LaTeX command.

parse(string) is a synonym for expression(string).

Warning

Note that the argument to expression is evaluated using the same rules as any other JME expression, so for example expression("2" + "x") is equivalent to expression("2x"), not expression("2 + x"). A good way to construct a randomised sub-expression is using substitute().

Definitions:
Example:
eval(expression, values)

Evaluate the given sub-expression.

If values is given, it should be a dictionary mapping names of variables to their values.

Definitions:
Example:
• eval(expression("1+2"))3

• eval(expression("x+1"), ["x":1])2

args(expression)

Returns the arguments of the top-level operation of expression, as a list of sub-expressions. If expression is a data type other than an operation or function, an empty list is returned.

Binary operations only ever have two arguments. For example, 1+2+3 is parsed as (1+2)+3.

Definitions:
Examples:
• args(expression("f(x)"))[expression("x")]

• args(expression("1+2+3"))[expression("1+2"), expression("3")]

• args(expression("1"))[]

type(expression)

Returns the name of the data type of the top token in the expression, as a string.

Definitions:
Examples:
• type(expression("x"))"name"

• type(expression("1"))"integer"

• type(expression("x+1"))"op"

• type(expression("sin(x)"))"function"

name(string)

Construct a name token with the given name.

Definitions:
Example:
• name("x")x

op(name)

Construct an operator with the given name.

Definitions:
Example:
• op("+") +

function(name)

Construct a function token with the given name.

Definitions:
Example:
• function("sin") sin

exec(op, arguments)

Returns a sub-expression representing the application of the given operation to the list of arguments.

Definitions:
Example:
• exec(op("+"), [2,1])expression("2+1")

• exec(op("-"), [2,name("x")])expression("2-x")

findvars(expression)

Return a list of all unbound variables used in the given expression. Effectively, this is all the variables that need to be given values in order for this expression to be evaluated.

Bound variables are those defined as part of operations which also assign values to those variables, such as map or let.

Definitions:
Examples:
• findvars(expression("x+1"))["x"]

• findvars(expression("x + x*y"))["x","y"]

• findvars(expression("map(x+2, x, [1,2,3])"))[]

substitute(variables, expression)

Substitute the given variable values into expression.

variables is a dictionary mapping variable names to values.

Definitions:
Examples:
• substitute(["x": 1], expression("x + y"))expression("1 + y")

• substitute(["x": 1, "y": expression("sqrt(z+2)")], expression("x + y"))expression("1 + sqrt(z + 2)")

simplify(expression, rules)

Apply the given simplification rules to expression, until no rules apply.

rules is a list of names of rules to apply, given either as a string containing a comma-separated list of names, or a list of strings.

Unlike the \\simplify command in content areas, the basic rule is not turned on by default.

See Substituting variables into displayed maths for a list of rules available.

Definitions:
Examples:
• simplify(expression("1*x+cos(pi)"),"unitfactor")expression("x+cos(pi)")

• simplify(expression("1*x+cos(pi)"),["basic","unitfactor","trig"])expression("x-1")

expand_juxtapositions(expression, options)

Expand juxtapositions in variable and function names for implicit multiplication of terms or composition of functions. This is to do with strings of letters with no spaces or operator symbols between them.

options is an optional dictionary of settings for the process. It can contain the following keys.

• singleLetterVariables - Insist that all variable names consist of a single letter, interpreting longer strings of characters as implicit multiplication.

Greek letters are considered to be one letter long.

• noUnknownFunctions - When a name appears before a bracket, but it’s not the name of a defined function, interpret it as a multiplication instead. This does not apply function applications with more than one argument.

• implicitFunctionComposition - When several function names are juxtaposed together to form a string that is not the name of a defined function, or several function names are joined with the multiplication symbol *, interpret it as implicity composition of functions.

If options is not given, all of these are turned on.

Variable name annotations, subscripts and primes do not count towards the number of letters in a name.

Definition:
Examples:
• expand_juxtapositions(expression("xy"))expression("x*y")

• expand_juxtapositions(expression("x'y"))expression("x\'*y")

• expand_juxtapositions(expression("pizza"))expression("pi*z*z*a")

• expand_juxtapositions(expression("hat:abc"))expression("hat:a*b*c")

• expand_juxtapositions(expression("xcos(x)"))expression("x*cos(x)")

• expand_juxtapositions(expression("lnabs(x)"))expression("ln(abs(x))")

• expand_juxtapositions(expression("ln*abs(x)"))expression("ln(abs(x))")

• expand_juxtapositions(expression("xy"),["singleLetterVariables": false])expression("xy")

• expand_juxtapositions(expression("x(x+1)"))expression("x*(x+1)")

• expand_juxtapositions(expression("x(x+1)"),["noUnknownFunctions": false])expression("x(x+1)")

• expand_juxtapositions(expression("ln*abs(x)"),["implicitFunctionComposition": false, "singleLetterVariables": true, "noUnknownFunctions": true])expression("l*n*abs(x)")

• expand_juxtapositions(expression("xy^z"))expression("x*y^z")

• expand_juxtapositions(expression("xy!"))expression("x*y!")

canonical_compare(expr1, expr2)

Compare expressions a and b using the “canonical” ordering. Returns -1 if a should go before b, 0 if they are considered “equal”, and 1 if a should go after b.

Expressions are examined in the following order:

• Names used: all variable names used in each expression are collected in a depth-first search and the resulting lists are compared lexicographically.

• Data type: if a and b are of different data types, op and function go first, and then they are compared using the names of their data types.

• Polynomials: terms of the form x^b or a*x^b, where a and b are numbers and x is a variable name, go before anything else.

• Function name: if a and b are both function applications, they are compared using the names of the functions. If the functions are the same, the arguments are compared. Powers, or multiples of powers, go after anything else.

• Number: if a and b are both numbers, the lowest number goes first. Complex numbers are compared by real part and then by imaginary part.

• Elements of other data types are considered to be equal to any other value of the same data type.

Definitions:
Examples:
• canonical_compare(a,b)-1

• canonical_compare(f(y),g(x))1

• canonical_compare(f(x),g(x))-1

• canonical_compare("a","b")0

numerical_compare(a, b)

Compare expression a and b by substituting random values in for the free variables.

Returns true if a and b have exactly the same free variables, and produce the same results when evaluated against the randomly chosen values.

For more control over the evaluation, see resultsequal().

Definitions:
Example:
• numerical_compare(expression("x^2"), expression("x*x"))true

• numerical_compare(expression("x^2"), expression("2x"))false

• numerical_compare(expression("x^2"), expression("y^2"))false

scope_case_sensitive(expression[, case_sensitive])

Set the evaluation scope to be case-sensitive or not, depending on the value of case_sensitive, and then evaluate expression.

If case_sensitive is not given, it defaults to true.

Case-sensitivity affects variable and function names. The names x and X are considered equivalent when not in case-sensitive mode, but are considered to be different when in case-sensitive mode.

Definitions:
Example:
• scope_case_sensitive(findvars(expression("x+X")))["X","x"]

• scope_case_sensitive(let(x,1,X,2,x+X), true)3

### Calculus¶

diff(expression, variable)

Differentiate the given expression with respect to the given variable name

Definitions:
Example:
• diff(expression("x^2 + 2x + 4"), "x")expression("2x + 2")

• diff(expression("x * y + 3x + 2y"), "x")expression("y + 3")

• diff(expression("cos(x^2)"), "x")expression("-2 * sin(x^2) * x")

### Pattern-matching sub-expressions¶

match(expr, pattern, options)

If expr matches pattern, return a dictionary of the form ["match": boolean, "groups": dict], where "groups" is a dictionary mapping names of matches to sub-expressions.

If you don’t need to use any parts of the matched expression, use matches() instead.

Definitions:
Examples:
• match(expression("x+1"),"?;a + ?;b")["match": true, "groups": ["a": expression("x"), "b": expression("1"), "_match": expression("x+1")]]

• match(expression("sin(x)"), "?;a + ?;b")["match": false, "groups": dict()]

• match(expression("x+1"),"1+?;a")["match": true, "groups": ["a": expression("x"), "_match": expression("x+1")]]

matches(expr, pattern, options)

Return true if expr matches pattern.

Use this if you’re not interested in capturing any parts of the matched expression.

Definitions:
Examples:
• matches(expression("x+1"),"?;a + ?;b")true

• matches(expression("sin(x)"), "?;a + ?;b")false

replace(pattern, replacement, expr)

Replace occurrences of pattern in expr with the expression created by substituting the matched items into replacement.

Definitions:
Examples:
• replace("?;x + ?;y", "x*y", expression("1+2"))expression("1*2")

• replace("?;x + ?;y", "f(x,y)", expression("1+2+3"))expression("f(f(1,2),3)")

• replace("0*?", "0", expression("0*sin(x) + x*0 + 2*cos(0*pi)"))expression("0 + 0 + 2*cos(0)")

### Identifying data types¶

type(x)

Returns the name of the data type of x.

Example:
• type(1)"integer"

Keywords

x isa type

Returns true if x is of the data type type.

Definitions:
Examples:
• 1 isa "number"true

• x isa "name"true (if x is not defined in this scope)

• x isa "number" true (if x has a numerical value in this scope)

x as type

Convert x to the given data type, if possible.

If x can not be automatically converted to type, an error is thrown.

Definitions:
Examples:
• dec(1.23) as "number"1.23

• set(1,2,3) as "list"[1,2,3]

infer_variable_types(expression)

Attempt to infer the types of free variables in the given expression.

There can be more than one valid assignment of types to the variables in an expression. For example, in the expression a+a, the variable a can be any type which has a defined addition operation.

Returns the first possible assignment of types to variables, as a dictionary mapping variable names to the name of its type. If a variable name is missing from the dictionary, the algorithm can’t establish any constraint on it.

Definitions:
Examples:
• infer_variable_types(expression("x^2"))["x": "number"]

• infer_variable_types(expression("union(a,b)"))["a": "set", "b": "set"]

• infer_variable_types(expression("k*det(a)"))[ "k": "number", "a": "matrix" ]

infer_type(expression)

Attempt to infer the type of the value produced by the given expression, which may contain free variables.

First, the types of any free variables are inferred. Then, definitions of an operations or functions in the function are chosen to match the types of their arguments.

Returns the name of the expression’s output type as a string, or "?" if the type can’t be determined.

Definitions:
Examples:
• infer_type(expression("x+2"))"number"

• infer_type(expression("id(n)"))"matrix"

• infer_type(expression("random(2,true)"))"?"

### Inspecting the evaluation scope¶

definedvariables()

Returns a list containing the names of every variable defined in the current scope, as strings.

Definitions:
isset(name)

Returns true if the variable with the given name has been defined in the current scope.

Definitions:
unset(names, expression)

Temporarily remove the named variables, functions and rulesets from the scope, and evaluate the given expression.

names is a dictionary of the form ["variables": list, "functions": list, "rulesets": list].

Definitions: