PQTY - Physical Quantities¶
A physical quantity is a property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference
Source: International vocabulary of metrology
General Information :: Links¶
The domain PhysQty (abbrev. PQTY) implements physical quantities. The implementation is based on the following category and domains:
General Form¶
A physical quantity in PQTY comprises three components:
v, value (rational)e, uncertainty (rational)u, unit (e.g. type SI_UNIT)
All base calculations are done by interval propagation of the interval
[v-e,v+e]. Thus an element of the type PhysQty has the form
\([v-e,v+e] \times u\).
Constructors and query functions¶
pqty: (Q, Q, U) -> %
pqtyValue: % -> Q -- get the value
pqtyError: % -> Q -- get the uncertainty
pqtyUnit: % -> U -- get the unit
pqtyInterval: % -> QI -- get the interval [v-delta,v+delta]
pqtyScale: (%,Q) -> % -- scale by rational number
Basic Operations¶
"*": (%,%) -> %
"/": (%,%) -> %
"+": (%,%) -> %
"-": (%,%) -> %
"^": (%,Integer) -> %
"-": % -> %
elt: (Q, %) -> %
elt: (QI,U) -> %
elt: (Union(Q,F,R,I),U) -> %
coerce : % -> OutputForm
In order to use the libary we have to load it and all its dependencies:
)lib RIA PUNIT SI PQTY
RationalInterval is now explicitly exposed in frame initial
RationalInterval will be automatically loaded when needed from
/home/kfp/Development/physqty/Untitled Folder/RIA.NRLIB/RIA
PhysicalUnit is now explicitly exposed in frame initial
PhysicalUnit will be automatically loaded when needed from
/home/kfp/Development/physqty/Untitled Folder/PUNIT.NRLIB/PUNIT
SIunit is now explicitly exposed in frame initial
SIunit will be automatically loaded when needed from
/home/kfp/Development/physqty/Untitled Folder/SI.NRLIB/SI
PhysQty is now explicitly exposed in frame initial
PhysQty will be automatically loaded when needed from
/home/kfp/Development/physqty/Untitled Folder/PQTY.NRLIB/PQTY
Q ==> Fraction Integer
Type: Void
pqty - the main constructor¶
For instance let us create the quantity \(g=9.81\pm0.05 \frac{m}{s^2}\):
g := pqty(9.81::Q, 0.05::Q, %m(1)/%s(2))
Warning: HyperTeX macro table not found
1 - 2
9.81 ± 0.05 m s
Type: PhysQty(SIunit)
Now g has type PhysQty(SI_UNIT). If we had entered units of an
other system we would get a different representation, for instance of
type PhysQty(CGS_UNIT), however, still the same object.
Note: all numeric arguments in PQTY have to be rationals,
although this might not be seen on first sight. Either one has to
coerce the figures oneself or it will be done automatically by
certain elt functions.
How can we recover u,v,e from the object g?
pqty_interval¶
pqtyInterval g
244 493
[---,---]
25 50
Type: RationalInterval
Here we can see how the value \(9.81\pm 0.05\) has been converted to
a rational interval. Now we could use the functions lb,ub from the
domain Q_INTERVAL to get the lower and upper bound.
pqty_unit¶
This function recovers the unit:
pqtyUnit g
1 - 2
m s
Type: SIunit
pqty_scale If we want double the quantity g we can either use the
constructor pqty to create the object g2 or we may apply the
scaling function to g:
pqtyScale¶
g2 := pqtyScale(g,2)
1 - 2
19.62 ± 0.1 m s
Type: PhysQty(SIunit)
Note: although these low-level methods are the safe and recommended
one, they are not very convenient. For calculations without
uncertainties one may use the following method:
ELT method¶
area := 123.45 %m(2)
2
123.45 ± 0.0 m
Type: PhysQty(SIunit)
force := 56.777 SIderived("newton")
1 1 - 2
56.777 ± 0.0 m kg s
Type: PhysQty(SIunit)
pressure := force / area
- 1 1 - 2
0.4599189955_4475496152 ± 0.0 m kg s
Type: PhysQty(SIunit)
-- actually:
pqtyInterval pressure
56777 56777
[------,------]
123450 123450
Type: RationalInterval
-- the interval represents a value without uncertainty, indeed we have
len %
0
Type: Fraction(Integer)
At the moment this is the only method implemented to create a PQTY
object without the main constructor. Possibly there will be a method to
enter the so called concise form.
Examples of basic operations¶
-- we cannot add different units
force + g
TYPE-ERROR:
#<TYPE-ERROR expected-type: CONS datum: NIL>
-- but we can multiply or divide them
force*g
2 1 - 4
556.98237 ± 2.83885 m kg s
Type: PhysQty(SIunit)
M:=force/g
1
5.7878160018_953878895 ± 0.0294995718_7510391381 kg
Type: PhysQty(SIunit)
-- we can invert a quantity, even it makes no sense physically !?
-M
1
- 5.7878160018_953878895 ± 0.0294995718_7510391381 kg
Type: PhysQty(SIunit)
-- scaling (not 2*g !!)
2 g
1 - 2
19.62 ± 0.1 m s
Type: PhysQty(SIunit)
-- recall g2 above
g2 - 2 g
1 - 2
0.0 ± 0.2 m s
Type: PhysQty(SIunit)
-- error usually is not zero when e<>0.
pqtyError %
1
-
5
Type: Fraction(Integer)
-- exponentiation
pressure^2
- 2 2 - 4
0.2115254824_6289633415 ± 0.0 m kg s
Type: PhysQty(SIunit)
force^(3)
3 3 - 6
183027.911569433 ± 0.0 m kg s
Type: PhysQty(SIunit)
-- error is zero, because e=0.
force - force
1 1 - 2
0.0 ± 0.0 m kg s
Type: PhysQty(SIunit)
(6.344::Q) force
1 1 - 2
360.193288 ± 0.0 m kg s
Type: PhysQty(SIunit)
(%pi::Float::Q) g
1 - 2
30.8190239317_15871669 ± 0.1570796326_7948966192 m s
Type: PhysQty(SIunit)
Write your own functions¶
This low level library covers almost everything such that one can write
extensions. As an exmaple let us create a function add_error which
adds an uncertainty to a given quantity and returns the result as a new
instance.
addError(qty, err) ==
v:= pqtyValue qty
e:= (pqtyError qty)+err
u:= pqtyUnit qty
return pqty(v,e,u)
Type: Void
addError(pressure,1/3)
- 1 1 - 2
0.4599189955_4475496152 ± 0.3333333333_3333333333 m kg s
Type: PhysQty(SIunit)
-- comapre to pressure:
pressure
- 1 1 - 2
0.4599189955_4475496152 ± 0.0 m kg s
Type: PhysQty(SIunit)