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

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

Evaluate to the sum of body for each value in an Element-based indexing set.

This form uses ["Element", index, collection] to specify that the index variable iterates over a finite collection (Set, List, or Range).

\sum_{n \in \{1,2,3\}} n

$$$\sum_{n \in \{1,2,3\}} n$$

["Sum", "n", ["Element", "n", ["Set", 1, 2, 3]]]
// ➔ 6

["Sum", ["Power", "n", 2], ["Element", "n", ["Set", 1, 2, 3]]]
// ➔ 14  (1² + 2² + 3² = 1 + 4 + 9)

["Sum", "n", ["Element", "n", ["Range", 1, 5]]]
// ➔ 15  (1 + 2 + 3 + 4 + 5)

The indexing set can be:

• Set: ["Set", 1, 2, 3] - explicit finite set of values
• List: ["List", 1, 2, 3] - ordered list of values
• List (2-element): ["List", 1, 5] - when a List has exactly 2 integer elements, it is treated as a Range. This allows bracket notation like \sum_{n \in [1,5]} n to iterate over all integers from 1 to 5 (evaluates to 15, not 6).
• Range: ["Range", 1, 10] - integer range from 1 to 10
• Interval: ["Interval", 1, 10] - enumerates integers in the interval. Supports Open and Closed boundary markers:
  • ["Interval", 1, 5] → iterates 1, 2, 3, 4, 5 (closed bounds)
  • ["Interval", ["Open", 0], 5] → iterates 1, 2, 3, 4, 5 (excludes 0)
  • ["Interval", 1, ["Open", 6]] → iterates 1, 2, 3, 4, 5 (excludes 6)

Note: Evaluation requires a finite, enumerable domain with at most 1000 elements. Sums over infinite sets (like \sum_{n \in \mathbb{N}}) remain symbolic.

Multiple Indexing Sets

Multiple Element expressions can be specified for multi-index sums:

\sum_{n \in A}\sum_{m \in B} n \cdot m

$$$\sum_{n \in A}\sum_{m \in B} n \cdot m$$

["Sum", ["Multiply", "n", "m"], ["Element", "n", ["Set", 1, 2]], ["Element", "m", ["Set", 3, 4]]]
// ➔ 21  (1·3 + 1·4 + 2·3 + 2·4)

Mixed indexing sets (Element + Limits) can be used together:

["Sum", ["Add", "n", "m"], ["Element", "n", ["Set", 1, 2]], ["Limits", "m", 1, 2]]
// ➔ 12  (n iterates {1,2}, m iterates 1 to 2)

Condition Filtering

A condition can be attached to an Element expression to filter values from the set. The condition is the optional 4th operand of the Element expression.

\sum_{n \in S, n > 0} n

$$$\sum_{n \in S, n > 0} n$$

// Sum only positive values from S
["Sum", "n", ["Element", "n", ["Set", 1, 2, 3, -1, -2], ["Greater", "n", 0]]]
// ➔ 6  (only 1 + 2 + 3)

// Sum values greater than or equal to 2
["Sum", "n", ["Element", "n", ["Set", 1, 2, 3, 4], ["GreaterEqual", "n", 2]]]
// ➔ 9  (only 2 + 3 + 4)

// Product of negative values
["Product", "k", ["Element", "k", ["Set", 1, -2, 3, -4], ["Less", "k", 0]]]
// ➔ 8  (only (-2) × (-4))

Supported condition operators: Greater, GreaterEqual, Less, LessEqual, NotEqual.

Simplification

When simplify() is called on a Sum expression with symbolic bounds, the following closed-form formulas are applied when applicable:

Pattern	Simplifies to	Name
\sum_{n=1}^{b} c	b \cdot c	Constant body
\sum_{n=a}^{b} n	\frac{b(b+1) - a(a-1)}{2}	Triangular number (general bounds)
\sum_{n=1}^{b} n^2	\frac{b(b+1)(2b+1)}{6}	Sum of squares
\sum_{n=1}^{b} n^3	\left[\frac{b(b+1)}{2}\right]^2	Sum of cubes
\sum_{n=0}^{b} r^n	\frac{1-r^{b+1}}{1-r}	Geometric series
\sum_{n=0}^{b} (-1)^n	\frac{1+(-1)^b}{2}	Alternating unit series
\sum_{n=0}^{b} (-1)^n \cdot n	(-1)^b \lfloor\frac{b+1}{2}\rfloor	Alternating linear series
\sum_{n=0}^{b} (a + dn)	(b+1)\left(a + \frac{db}{2}\right)	Arithmetic progression
\sum_{k=0}^{n} \binom{n}{k}	2^n	Sum of binomial coefficients
\sum_{k=0}^{n} (-1)^k \binom{n}{k}	0	Alternating binomial sum
\sum_{k=0}^{n} k \binom{n}{k}	n \cdot 2^{n-1}	Weighted binomial sum
\sum_{k=1}^{n} \frac{1}{k(k+1)}	\frac{n}{n+1}	Partial fractions (telescoping)
\sum_{k=2}^{n} \frac{1}{k(k-1)}	\frac{n-1}{n}	Partial fractions (telescoping)
\sum_{k=0}^{n} k^2 \binom{n}{k}	n(n+1) \cdot 2^{n-2}	Weighted squared binomial sum
\sum_{k=0}^{n} k^3 \binom{n}{k}	n^2(n+3) \cdot 2^{n-3}	Weighted cubed binomial sum
\sum_{k=0}^{n} (-1)^k k \binom{n}{k}	0	Alternating weighted binomial sum (n ≥ 2)
\sum_{k=0}^{n} \binom{n}{k}^2	\binom{2n}{n}	Sum of binomial squares
\sum_{k=1}^{n} k(k+1)	\frac{n(n+1)(n+2)}{3}	Sum of consecutive products
\sum_{n=m}^{b} (a + dn)	(b-m+1)\left(a + \frac{d(m+b)}{2}\right)	Arithmetic progression (general bounds)
\sum_{n=1}^{b} c \cdot f(n)	c \cdot \sum_{n=1}^{b} f(n)	Factor out constant

Edge cases:

• Empty range (upper < lower): returns 0
• Single iteration (upper = lower): substitutes the bound value and returns the body
• Nested sums: inner sums are simplified first, enabling cascading simplification

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 body for 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

Product(body: function, ...bounds: Element) -> number

Evaluate to the product of body for each value in an Element-based indexing set.

This form uses ["Element", index, collection] to specify that the index variable iterates over a finite collection (Set, List, or Range).

\prod_{k \in \{1,2,3,4\}} k

$$$\prod_{k \in \{1,2,3,4\}} k$$

["Product", "k", ["Element", "k", ["Set", 1, 2, 3, 4]]]
// ➔ 24  (4!)

["Product", ["Power", "k", 2], ["Element", "k", ["Set", 1, 2, 3]]]
// ➔ 36  (1² × 2² × 3² = 1 × 4 × 9)

See the Sum documentation above for details on supported collection types.

Simplification

When simplify() is called on a Product expression with symbolic bounds, the following closed-form formulas are applied when applicable:

Pattern	Simplifies to	Name
\prod_{n=1}^{b} c	c^b	Constant body
\prod_{n=1}^{b} n	b!	Factorial
\prod_{n=1}^{b} (n+c)	\frac{(b+c)!}{c!}	Shifted factorial
\prod_{n=1}^{b} (2n-1)	(2b-1)!!	Odd double factorial
\prod_{n=1}^{b} 2n	2^b \cdot b!	Even double factorial
\prod_{k=0}^{n-1} (x+k)	(x)_n	Rising factorial (Pochhammer)
\prod_{k=0}^{n-1} (x-k)	\frac{x!}{(x-n)!}	Falling factorial
\prod_{k=1}^{n} \frac{k+1}{k}	n+1	Telescoping product
\prod_{k=2}^{n} (1 - \frac{1}{k^2})	\frac{n+1}{2n}	Wallis-like product
\prod_{n=1}^{b} c \cdot f(n)	c^b \cdot \prod_{n=1}^{b} f(n)	Factor out constant

Edge cases:

• Empty range (upper < lower): returns 1
• Single iteration (upper = lower): substitutes the bound value and returns the body

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...

Polynomial Arithmetic

These functions operate on polynomial expressions.

Function	Description
Expand	Expand products and positive integer powers
ExpandAll	Recursively expand products and positive integer powers
Factor	Factor an expression into irreducible factors
Together	Combine rational expressions into a single fraction
Distribute	Distribute multiplication over addition
PolynomialDegree	Return the degree of a polynomial
CoefficientList	Return the list of coefficients of a polynomial
PolynomialQuotient	Return the quotient of polynomial division
PolynomialRemainder	Return the remainder of polynomial division
PolynomialGCD	Return the greatest common divisor of two polynomials
Cancel	Cancel common polynomial factors in a rational expression

Expand(expr)

Expand out products and positive integer powers in expr.

["Expand", ["Power", ["Add", "x", 1], 2]]
// ➔ ["Add", ["Power", "x", 2], ["Multiply", 2, "x"], 1]

ExpandAll(expr)

Recursively expand out products and positive integer powers in expr and all subexpressions.

Factor(expr)

Factor(expr, var)

Factor a polynomial expression into a product of irreducible factors.

Supports:

• Perfect square trinomials: a^2 \pm 2ab + b^2 \to (a \pm b)^2
• Difference of squares: a^2 - b^2 \to (a-b)(a+b)
• Quadratic factoring: ax^2 + bx + c (when roots are rational)
• Common factor extraction: 2x + 4 \to 2(x+2)

The optional var parameter specifies which variable to factor over.

Perfect square trinomial:

["Factor", ["Add", ["Power", "x", 2], ["Multiply", 2, "x"], 1]]
// ➔ ["Power", ["Add", "x", 1], 2]  // (x+1)²

Quadratic with rational roots:

["Factor", ["Add", ["Power", "x", 2], ["Multiply", 5, "x"], 6]]
// ➔ ["Multiply", ["Add", "x", 2], ["Add", "x", 3]]  // (x+2)(x+3)

Difference of squares:

["Factor", ["Add", ["Power", "x", 2], -4]]
// ➔ ["Multiply", ["Add", "x", -2], ["Add", "x", 2]]  // (x-2)(x+2)

With coefficients:

["Factor", ["Add", ["Multiply", 4, ["Power", "x", 2]], ["Multiply", 12, "x"], 9]]
// ➔ ["Power", ["Add", ["Multiply", 2, "x"], 3], 2]  // (2x+3)²

Automatic use in sqrt simplification:

["Sqrt", ["Add", ["Power", "x", 2], ["Multiply", 2, "x"], 1]]
// ➔ ["Abs", ["Add", "x", 1]]  // |x+1| (auto-factors before applying sqrt rule)

Together(expr)

Combine rational expressions into a single fraction with a common denominator.

["Together", ["Add", ["Divide", 1, "x"], ["Divide", 1, "y"]]]
// ➔ ["Divide", ["Add", "x", "y"], ["Multiply", "x", "y"]]

Distribute(expr)

Distribute multiplication over addition in expr.

["Distribute", ["Multiply", "a", ["Add", "b", "c"]]]
// ➔ ["Add", ["Multiply", "a", "b"], ["Multiply", "a", "c"]]

PolynomialDegree(poly, var)

Return the degree of the polynomial poly with respect to the variable var.

["PolynomialDegree", ["Add", ["Power", "x", 3], ["Multiply", 2, "x"], 1], "x"]
// ➔ 3

CoefficientList(poly, var)

Return the list of coefficients of the polynomial poly with respect to the variable var, ordered from lowest to highest degree.

["CoefficientList", ["Add", ["Power", "x", 3], ["Multiply", 2, "x"], 1], "x"]
// ➔ ["List", 1, 2, 0, 1]

The result represents the polynomial 1 + 2x + 0x^2 + 1x^3.

PolynomialQuotient(dividend, divisor, var)

Return the quotient of the polynomial division of dividend by divisor with respect to the variable var.

["PolynomialQuotient", ["Subtract", ["Power", "x", 3], 1], ["Subtract", "x", 1], "x"]
// ➔ ["Add", ["Power", "x", 2], "x", 1]

This represents \frac{x^3 - 1}{x - 1} = x^2 + x + 1.

PolynomialRemainder(dividend, divisor, var)

Return the remainder of the polynomial division of dividend by divisor with respect to the variable var.

["PolynomialRemainder", ["Add", ["Power", "x", 3], ["Multiply", 2, "x"], 1], ["Add", "x", 1], "x"]
// ➔ -2

PolynomialGCD(a, b, var)

Return the greatest common divisor of two polynomials a and b with respect to the variable var.

["PolynomialGCD", ["Subtract", ["Power", "x", 2], 1], ["Subtract", "x", 1], "x"]
// ➔ ["Subtract", "x", 1]

This represents \gcd(x^2 - 1, x - 1) = x - 1.

Cancel(expr, var)

Cancel common polynomial factors in the numerator and denominator of the rational expression expr with respect to the variable var.

["Cancel", ["Divide", ["Subtract", ["Power", "x", 2], 1], ["Subtract", "x", 1]], "x"]
// ➔ ["Add", "x", 1]

This represents \frac{x^2 - 1}{x - 1} = x + 1 after canceling the common factor (x - 1).