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Mathematical Operations

Numeric operators applied to mixed-type operands (such as a Number and a mathematical value) are not defined and should be considered an editorial error in this specification.

This specification denotes most numeric values in base 10; it also uses numeric values of the form 0x followed by digits 0-9 or A-F as base-16 values.

When the term integer is used in this specification, it refers to a mathematical value which is in the set of integers, unless otherwise stated. When the term integral Number is used in this specification, it refers to a Number value whose mathematical value is in the set of integers.

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Conversions between mathematical values and Numbers or BigInts are always explicit in this document. A conversion from a mathematical value or extended mathematical value _x_ to a Number is denoted as "the Number value for _x_" or 𝔽(_x_), and is defined in . A conversion from an integer _x_ to a BigInt is denoted as "the BigInt value for _x_" or ℤ(_x_). A conversion from a Number or BigInt _x_ to a mathematical value is denoted as "the mathematical value of _x_", or ℝ(_x_). The mathematical value of *+0*𝔽 and *-0*𝔽 is the mathematical value 0. The mathematical value of non-finite values is not defined. The extended mathematical value of _x_ is the mathematical value of _x_ for finite values, and is +∞ and -∞ for *+∞*𝔽 and *-∞*𝔽 respectively; it is not defined for *NaN*.

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Conversions between mathematical values and Numbers or BigInts are always explicit in this document. A conversion from a mathematical value or extended mathematical value _x_ to a Number is denoted as "the Number value for _x_" or 𝔽(_x_), and is defined in . A conversion from an integer _x_ to a BigInt is denoted as "the BigInt value for _x_" or ℤ(_x_). A conversion from a Number or BigInt _x_ to a mathematical value is denoted as "the mathematical value of _x_", or ℝ(_x_). The mathematical value of *+0*𝔽 and *-0*𝔽 is the mathematical value 0. The mathematical value of non-finite values is not defined. The extended mathematical value of _x_ is the mathematical value of _x_ for finite values, and is +∞ and -∞ for *+∞*𝔽 and *-∞*𝔽 respectively; it is not defined for *NaN*.

The mathematical function abs(_x_) produces the absolute value of _x_, which is -_x_ if _x_ < 0 and otherwise is _x_ itself.

The mathematical function min(_x1_, _x2_, … , _xN_) produces the mathematically smallest of _x1_ through _xN_. The mathematical function max(_x1_, _x2_, ..., _xN_) produces the mathematically largest of _x1_ through _xN_. The domain and range of these mathematical functions are the extended mathematical values.

The notation “_x_ modulo _y_” (_y_ must be finite and non-zero) computes a value _k_ of the same sign as _y_ (or zero) such that abs(_k_) < abs(_y_) and _x_ - _k_ = _q_ × _y_ for some integer _q_.