3.5 Magnitudes and Numbers

3.5.1 Basic Types

Magnitude

the abstract protocol for linearly-comparable objects, following <, >, <=, >=, and =.

Number

the abstract type of dimensionless quantities.

Integer

integral quantities, generally.

SmallInteger

machine-word-limited literals. Their normal protocol will not produce errors inconsistent with mathematic behavior of Integers, however.

BigInteger

larger Integers, implemented as WordArrays.

Fraction

An exact representation of a quotient, or rational number.

Float

A low-level floating-point numeric representation, being inexact.

Complex

A complex number, similar to a pair of real numbers.

3.5.2 Basic Operations

All of the normal arithmetic operations (i.e. +, -, *, /) are supported primitively between elements of the same type. Type coercion has to be done entirely in code; no implicit coercions are performed by the virtual machine. However, the standard library includes methods which perform this coercion. The interpreter also transparently provides unlimited-size integers, although the bootstrapped system may not do so implicitly.

The following are the rest of the primitive operations, given with an indication of their "signatures":

Float raisedTo: Float

is simple floating-point exponentiation.

Integer as: Float

extends an integer into a float.

Float as: Integer

truncates a float.

Integer bitOr: Integer

performs bitwise logical OR.

Integer bitXor: Integer

performs bitwise logical XOR.

Integer bitAnd: Integer

performs bitwise logical AND.

Integer bitShift: Integer

performs bitwise logical right-shift (left-shift if negative).

Integer bitNot

performs bitwise logical NOT.

Integer >> Integer

performs logical right-shift.

Integer << Integer

performs logical left-shift.

Integer quo: Integer

returns a quotient (integer division).

Many more useful methods are defined, such as mod:, reciprocal, min:, max:, between:and:, lcm:, and gcd:. Slate also works with Fractions when dividing Integers, keeping them lazily reduced.

3.5.3 Non-core Operations

zero

The zero element for the type of number.

isZero

Whether the number is the zero element for its type.

isPositive/isNegative

Whether its positive or negative.

abs

The absolute value of the number.

sign

The sign of the number.

negated

Returns -x for x.

gcd:

Greatest common divisor.

lcm:

Least common multiple.

factorial

Factorial.

mod:/rem:/quo:

Modulo division, remainder, and quotient.

reciprocal

Constructs a new fraction reciprocal.

min:

The lesser of the arguments. The least in cases of min:min:.

max:

The greater of the arguments. The greatest in cases of max:max:.

a between: b and: c

Whether a is between b and c.

truncated/fractionPart

Returns the greatest integer less than the number, and the corresponding difference as a fraction (or a float for Float).

reduced

Only defined on Fraction, this is the lazily-applied reducer; it will be invoked automatically for arithmetic operations as necessary, but is useful when only the reduced form is needed.

3.5.4 Limit Numerics

PositiveInfinity

is greater than any other Magnitude.

NegativeInfinity

is lesser than any other Magnitude.

LargeUnbounded

A Magnitude designed to represent non-infinite, but non-bounded (``as large as you like'') quantities.

PositiveEpsilon

is as small as you like, but positive and greater than zero.

NegativeEpsilon

is as small as you like, but negative and lesser than zero.

3.5.5 Dimensioned Units

There is an entire system for handling dimensioned units and their various combinations and mathematical operations. See the 'src/dimensioned.slate' file for an overview.