1 Operations, Types, and Structs

6.3.90.900

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1 Operations, Types, and Structs

1.1 Operations

1.1.1 Operations on Measures

syntax
(m: expr-or-op ...)
(m: expr-or-op ... : type)

expr-or-op = expr
| op

op = +
| -
| *
| /
| ^

This macro lets you write measures and operations on measures using infix notation with the symbols +, -, *, /, and ^.

It is syntactic sugar for using m+, m-, m*, m1/, and mexpt in prefix notation.

Examples:

> (require typed/racket)
> (require typed/measures-with-dimensions)
> (m: 1 meter)
- : Length-Real-Measure
(m: 1 meter)
> (m: 50 centimeter)
- : Length-Real-Measure
(m: 50 centimeter)
> (m: 1 meter + 50 centimeter)
- : Length-Real-Measure
(m: 3/2 meter)
> (m: 1 meter - 50 centimeter)
- : Length-Real-Measure
(m: 1/2 meter)
> (m: 1 meter ^ 2)
- : Measure
(m: 1 (usqr meter))
> (m: 1 meter ^ 2 + 100 centimeter ^ 2)
- : Measure
(m: 101/100 (usqr meter))
> (m: 1 foot + 3 inch)
- : Length-Real-Measure
(m: 5/4 foot)
> (m: 1 meter - 50 centimeter : Length-Real-Measure)
- : Length-Real-Measure
(m: 1/2 meter)

procedure
(m+ m ...) → Measure
m : Measureish

adds the measures together.

Note for typed racket users: when the typechecker typechecks m+, it doesn’t actually check that its arguments have the same dimensions (it does check at runtime). Instead it typechecks as returning a measure with a union of the dimensions of its arguments. For example, (m+ (make-measure 1 meter) (make-measure 1 second)) typechecks as something that could have either the length dimension or the time dimension. You can fix this by either requiring the type of the result to be what you want, or using inst to instantiate it for a certain dimension.

Examples:

> (require typed/racket)
> (require typed/measures-with-dimensions)
> (ann (m+) Zero-Measure)
- : Zero-Measure
(m: 0 1-unit)
> (m+ (m: 1 meter))
- : Length-Real-Measure
(m: 1 meter)
> (m+ (m: 1 meter) (m: 50 centimeter))
- : Length-Real-Measure
(m: 3/2 meter)
> (m: 1 foot + 3 inch)
- : Length-Real-Measure
(m: 5/4 foot)
> ((inst m+ Length-Dimension) (m: 1 meter) (m: 50 centimeter))
- : Length-Real-Measure
(m: 3/2 meter)

procedure
(m- m) → Measure
m : Measureish

Negates a measure. To do subtraction, either use the m: macro or use a pattern like (m+ a (m- b)).

Examples:

> (require typed/racket)
> (require typed/measures-with-dimensions)
> (m- (m: 1 meter))
- : Length-Real-Measure
(m: -1 meter)
> (m: 1 meter - 50 centimeter)
- : Length-Real-Measure
(m: 1/2 meter)

procedure
(m* m ...) → Measure
m : Measureish

Multiplies the measures.

procedure
(m1/ m) → Number-Measure
m : Number-Measureish

Takes the multiplicative inverse of the measure. To do division, either use the m: macro or use a pattern like (m* a (m1/ b)).

procedure
(mexpt m1 m2) → Number-Measure
m1 : Number-Measureish
m2 : Number-Measureish

Takes m1 to the exponent of m2.

procedure
(mabs m) → Real-Measure
m : Real-Measure

Takes the absolute value of m.

procedure
(convert m u) → Measure
  m : Measure
  u : Unitish
(convert n u1 u2) → Measure
  n : (U Number (Vectorof Real))
  u1 : Unitish
  u2 : Unitish

The first form converts the measure m to the unit u. The second form converts the number or vector n from u1 to u2.

Examples:

> (require typed/racket)
> (require typed/measures-with-dimensions)
> (convert (m: 1 meter) centimeter)
- : Length-Real-Measure
(m: 100 centimeter)
> (convert 1 meter centimeter)
- : Length-Measure
(m: 100 centimeter)

procedure
(measure=? m1 m ...) → Boolean
m1 : Measureish
m : Measureish

Returns true if the measures represent the same quantities.

Examples:

> (require typed/racket)
> (require typed/measures-with-dimensions)
> (measure=? (m: 1 meter)
             (m: 100 centimeter))
- : Boolean
#t
> (measure=? (m: 1 meter)
             (m: 1 kilogram))
- : Boolean
#f
> (measure=? (m: 1 radian)
             (m: 1 degree))
- : Boolean
#f

procedure
(->measure m) → Measure
m : Measureish

Converts from something "measureish" to a Measure.

procedure
(measure->num/vec m) → (U Number (Vectorof Real))
  m : Measureish
procedure
(measure->number m) → Number
  m : Number-Measureish
procedure
(measure->real m) → Real
  m : Real-Measureish
procedure
(measure->vector m) → (Vectorof Real)
  m : Vector-Measureish

Functions for converting dimensionless measures to numbers or vectors. For measures that have a unit with a scalar, it converts to 1-unit first. So mols will be multiplied by approx. 6.02*10^23, percents will be divided by 100, degrees will be multiplied by 2π/360, etc.

1.1.2 Operations on Units and Dimensions

procedure
(u* u ...) → Unit
  u : Unitish
procedure
(u1/ u) → Unit
  u : Unitish
procedure
(u/ u1 u ...+) → Unit
  u1 : Unitish
  u : Unitish
procedure
(uexpt u n) → Unit
  u : Unitish
  n : Exact-Rational
procedure
(u+ u ...+) → Unit
  u : Unitish
procedure
(->unit u) → Unit
  u : Unitish
procedure
(unit-rename u n) → Unit
  u : Unit
  n : Any
procedure
(unit=? u1 u ...) → Boolean
  u1 : Unitish
  u : Unitish

Operations on units

procedure
(d* u ...) → Dimension
  u : Dimension
procedure
(d1/ d) → Dimension
  d : Dimension
procedure
(d/ d1 d ...+) → Dimension
  d1 : Dimension
  d : Dimension
procedure
(dexpt d n) → Dimension
  d : Dimension
  n : Exact-Rational
procedure
(dimension=? d1 d ...) → Boolean
  d1 : Dimension
  d : Dimension

Operations on dimensions

1.2 Types and Structs

type
Measure

equivalent to (Measureof (U Number (Vectorof Real)) Unit).

type
(Measureof n u)

equivalent to (measure n u Sig-Figs).

type
Number-Measure

equivalent to (Number-Measureof Unit).

type
Vector-Measure

equivalent to (Vector-Measureof Unit).

type
(Number-Measureof u)

equivalent to (Measureof Number u).

type
(Vector-Measureof u)

equivalent to (Measureof (Vectorof Real) u).

type
Measureish
type
Number-Measureish
type
Vector-Measureish
type
Real-Measureish

type
Unit

equivalent to (Unitof Dimension).

type
(Unitof d)

the type for a unit with the dimension d.

type
Unitish

type
Dimension

equivalent to the type (dimension Integer Integer Integer Integer Integer).

type
Sig-Figs

equivalent to (U Positive-Integer +inf.0).

struct
(struct dimension (M-expt L-expt T-expt Q-expt Θ-expt))
  M-expt : Integer
  L-expt : Integer
  T-expt : Integer
  Q-expt : Integer
  Θ-expt : Integer
procedure
(make-dimension #:M^ M-expt
#:L^ L-expt
#:T^ T-expt
#:Q^ Q-expt
#:Θ^ Θ-expt) → Dimension
  M-expt : Integer
  L-expt : Integer
  T-expt : Integer
  Q-expt : Integer
  Θ-expt : Integer

a struct representing a dimension.

struct
(struct unit (name scalar dimension))
  name : Any
  scalar : Positive-Real
  dimension : Dimension

a struct representing a unit.

struct
(struct measure (number unit sig-figs))
  number : (U Number (Vectorof Real))
  unit : Unit
  sig-figs : Sig-Figs
procedure
(make-measure number unit [sig-figs]) → Measure
  number : (U Number (Vectorof Real))
  unit : Unit
  sig-figs : Sig-Figs = +inf.0
procedure
(measure-dimension m) → Dimension
  m : Measure

a struct representing a measure.

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1	Operations, Types, and Structs
2	Units and Dimensions
3	Physical Constants
4	Temperatures
5	Short Names
6	Chemistry