Arithmetic

Constants

Symbol	Value
ExponentialE	\approx 2.7182818284\ldots	Euler's number
MachineEpsilon	2^{−52}	The difference between 1 and the next larger floating point number. See Machine Epsilon on Wikipedia
CatalanConstant	\approx 0.9159655941\ldots	\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} See Catalan's Constant on Wikipedia
GoldenRatio	\approx 1.6180339887\ldots	\frac{1+\sqrt{5}}{2} See Golden Ratio on Wikipedia
EulerGamma	\approx 0.5772156649\ldots	See Euler-Mascheroni Constant on Wikipedia

See also Trigonometry for Pi and related constants

See also Complex for ImaginaryUnit

Functions

Arithmetic Functions

Function	Notation
Add	a + b	Addition
Subtract	a - b	Subtraction
Negate	-a	Additive inverse
Multiply	a\times b	Multiplication
Divide	\frac{a}{b}	Divide
Power	a^b	Exponentiation
Root	\sqrt[n]{x}=x^{\frac1n}	nth root
Sqrt	\sqrt{x}=x^{\frac12}	Square root
Square	x^2

Sums and Products

Sum(xs: collection)

Evaluate to a sum of all the elements in collection. If all the elements are numbers, the result is a number. Otherwise it is an ["Add"] expression.

\sum x_{i}

$$$\sum x_{i}$$

["Sum", ["List", 5, 7, 11]]
// ➔ 23

["Sum", ["List", 5, 7, "x" , "y"]]
// ➔ ["Add", 12, "x", "y"]

Note this is equivalent to:

["Reduce", ["List", 5, 7, 11], "Add"]

Sum(body: function, ...bounds: tuple) -> number

Evaluate to the sum of body for each value in bounds.

\sum{i=1}^{10} i+1

$$$\sum{i=1}^{10} i+1$$

["Sum", ["Add", "i", 1], ["Tuple", "i", 1, 10]]
// ➔ 65

Product(xs: collection)

Evaluate to a product of all the elements in collection.

If all the elements are numbers, the result is a number. Otherwise it is a ["Multiply"] expression.

\prod x_{i}

$$$\prod x_{i}$$

["Product", ["List", 5, 7, 11]]
// ➔ 385

["Product", ["List", 5, "x", 11]]
// ➔ ["Multiply", 55, "x"]

Note this is equivalent to:

["Reduce", ["List", 5, 7, 11], "Multiply"]

Product(f: function, bounds:tuple)

Return the product of bodyfor each value in bounds.

\prod_{i=1}^{n} f(i)

$$$\prod_{i=1}^{n} f(i)$$

["Product", ["Add", "x", 1], ["Tuple", "x", 1, 10]]
// ➔ 39916800

Transcendental Functions

Function	Notation
Exp	\exponentialE^{x}	Exponential function
Ln	\ln(x)	Logarithm function, the natural logarithm, the inverse of Exp
Log	\log_b(x)	["Log", <v>, <b>] Logarithm of base b, default 10
Lb	\log_2(x)	Binary logarithm function, the base-2 logarithm
Lg	\log_{10}(x)	Common logarithm, the base-10 logarithm
LogOnePlus	\ln(x+1)

See also Trigonometry for trigonometric functions

See also Complex for complex functions

See also Statistics for statistics functions and functions on lists

Rounding

Function	Notation
Abs	\|x\|	Absolute value, magnitude
Ceil	\lceil x \rceil	Rounds a number up to the next largest integer
Floor	\lfloor x \rfloor	Round a number to the greatest integer less than the input value
Chop		Replace real numbers that are very close to 0 (less than 10^{-10}) with 0
Round

Other Relational Operators

Congruent(a, b, modulus)

Evaluate to True if a is congruent to b modulo modulus.

["Congruent", 26, 11, 5]
// ➔ True

26 \equiv 11 \pmod{5}

$$$26 \equiv 11 \pmod{5}$$

Other Functions

Clamp(value)

Clamp(value, lower, upper)

• If value is less than lower, evaluate to lower
• If value is greater than upper, evaluate to upper
• Otherwise, evaluate to value

If lowerand upperare not provided, they take the default values of -1 and +1.

["Clamp", 0.42]
// ➔ 1
["Clamp", 4.2]
// ➔ 1
["Clamp", -5, 0, "+Infinity"]
// ➔ 0
["Clamp", 100, 0, 11]
// ➔ 11

Max(x1, x2, ...)

Max(list)

If all the arguments are real numbers, excluding NaN, evaluate to the largest of the arguments.

Otherwise, simplify the expression by removing values that are smaller than or equal to the largest real number.

["Max", 5, 2, -1]
// ➔ 5
["Max", 0, 7.1, "NaN", "x", 3]
// ➔ ["Max", 7.1, "NaN", "x"]

Max(x1, x2, ...)

Max(list)

If all the arguments are real numbers, excluding NaN, evaluate to the smallest of the arguments.

Otherwise, simplify the expression by removing values that are greater than or equal to the smallest real number.

\min(0, 7.1, 3) = 0

$$$\min(0, 7.1, 3) = 0$$

["Min", 5, 2, -1]
// ➔ -1
["Min", 0, 7.1, "x", 3]
// ➔ ["Min", 0, "x"]

Mod(a, b)

Evaluate to the Euclidian division (modulus) of a by b.

When a and b are positive integers, this is equivalent to the % operator in JavaScript, and returns the remainder of the division of a by b.

However, when a and b are not positive integers, the result is different.

The result is always the same sign as b, or 0.

["Mod", 7, 5]
// ➔ 2

["Mod", -7, 5]
// ➔ 3

["Mod", 7, -5]
// ➔ -3

["Mod", -7, -5]
// ➔ -2

Rational(n)

Evaluate to a rational approximating the value of the number n.

["Rational", 0.42]
// ➔ ["Rational", 21, 50]

Rational(numerator, denominator)

Represent a rational number equal to numeratorover denominator.

Numerator(expr)

Return the numerator of expr.

Note that expr may be a non-canonical form.

["Numerator", ["Rational", 4, 5]]
// ➔ 4

Denominator(expr)

Return the denominator of expr.

Note that expr may be a non-canonical form.

["Denominator", ["Rational", 4, 5]]
// ➔ 5

NumeratorDenominator(expr)

Return the numerator and denominator of expr as a sequence.

Note that expr may be a non-canonical form.

["NumeratorDenominator", ["Rational", 4, 5]]
// ➔ ["Sequence", 4, 5]

The sequence can be used with another function, for example GCD to check if the fraction is in its canonical form:

["GCD", ["NumeratorDenominator", ["Rational", 4, 5]]]
// ➔ 1

["GCD", ["NumeratorDenominator", ["Rational", 8, 10]]]
// ➔ 2

Relational Operators

Function	Notation
Equal	x = y	Mathematical relationship asserting that two quantities have the same value
NotEqual	x \ne y
Greater	x \gt y
GreaterEqual	x \geq y
Less	x \lt y
LessEqual	x \leq y

See below for additonal relational operators: Congruent, etc...