Arithmetic

Constants

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

Relational Operators

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

See below for additonal relational operators: Congruent, etc…

Functions

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

Transcendental Functions

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

Rounding

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

Other Relational Operators

["Congruent", a, b, modulus]

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

26 \equiv 11 \pmod{5}

\[ 26 \equiv 11 \pmod{5} \]

["Congruent", 26, 11, 5]
// ➔ True

Other Functions

["BaseForm", value:integer]

["BaseForm", value:integer, base]

Format an integer in a specific base, such as hexadecimal or binary.

If no base is specified, use base-10.

The sign of integer is ignored.

value should be an integer.
base should be an integer from 2 to 36.

["Latex", ["BaseForm", 42, 16]]
 
// ➔ (\text(2a))_{16}

Latex(BaseForm(42, 16))
// ➔ (\text(2a))_{16}
String(BaseForm(42, 16))
// ➔ "'0x2a'"

["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