Quantity Arithmetic

[ Addition and Subtraction ] [ Multiplication and Division ] [ Exponentials ]

Physical quantities may be manipulated with basic multiply, division and power operations.

ADDITION AND SUBTRACTION

Let Variable1 and Variable2 be physical quantities of the form

    Variable1 = Number1 [Length]^a . [Mass]^b . [Time]^c . [Temp]^d

    Variable2 = Number2 [Length]^e . [Mass]^f . [Time]^g . [Temp]^h

where a-g are exponent constants. The quentity sum/difference arithmetic operation

    Quantity Sum          ==>      Variable1 + Variable2 
    Quantity Difference   ==>      Variable1 - Variable2 

will be defined only if a = e, b = f, c = g, and d = h.

ALADDIN computes, by default, a physical quantity result with units matching the second physical quantity in the arithmetic operation.

Example 1 : The script of code:

    x = 1.0  m;
    y = 100 cm;

    print "x + y = ", x + y, "\n";
    print "y + x = ", y + x, "\n";

generates the output:

    x + y =        200 cm 
    y + x =          2 m 

Example 2 : An important feature of quantity operations is the check for consistent units. Suppose that we try to add a quantity with time units to a second quantity having units of length, as in the script:

    x = 1 in; y = 1 sec;
    z = x + y;

ALADDIN provides an appropriate fatal error message

    FATAL ERROR >> In Add() : Inconsistent Dimensions.
    FATAL ERROR >> Compilation Aborted.

followed by the termination of program execution.

MULTIPLICATION AND DIVISION

Let Variable1 and Variable2 be physical quantities of the form

    Variable1 = Number1 [Length]^a . [Mass]^b . [Time]^c . [Temp]^d

    Variable2 = Number2 [Length]^e . [Mass]^f . [Time]^g . [Temp]^h

where a-g are exponent constants. The quentity product and quentity division arithmetic operations are given by

    Quantity Product      ==>      Variable1 * Variable2 
    Quantity Division     ==>      Variable1 / Variable2 

The units in Variable1 * Variable2 will be

    [Length]^(a+e) . [Mass]^(b+f) . [Time]^(c+g) . [Temp]^(d+h)

and the in Variable1 / Variable2

    [Length]^(a-e) . [Mass]^(b-f) . [Time]^(c-g) . [Temp]^(d-h)

Example 3 : Suppose that the velocity and distance traveled by a free-falling object is given by

    velocity = gravity * time
    distance = 1/2 . gravity * time * time

where ``gravity'' is the acceleration due to gravity, and ``time'' is the time of free-fall.

The script of code

    gravity = 9.81 m/sec^2;
    t = 3 sec;

    print "acceleration = ", gravity , "\n";
    print "velocity     = ", gravity*t , "\n";
    print "distance     = ", 1/2*gravity*t*t, "\n";

generates the output:

    acceleration =      9.81 m/sec^2
    velocity     =      29.43 m/sec 
    distance     =      44.14 m 

Example 4 : The script of code

    force = 20    N;
    mass  =  5   kg;
    area  = 20 cm^2;

    print "force      = ", force       , "\n";
    print "mass       = ", mass        , "\n";
    print "area       = ", area (cm^2) , "\n";
    print "force/mass = ", force/mass , "\n";
    print "force/area = ", force/area , "\n";

generates the output:

    force      =         20 N 
    mass       =          5 kg 
    area       =         20 cm^2 
    force/mass =          4 m/sec^2 
    force/area =      1e+04 Pa 

EXPONENTIALS

Power operations on quantities do not work in quite the same way as the basic add, subtract, multiply and divide operations on quantities. If quantity1 and quantity2 are physical quantities, then the operation

 
      quantity1 ^ quantity2;

is defined for dimensionless quantity1 and quantity2 , and cases where one of the two operands, but not both operands, has units.

Order of Evaluation

Unlike the basic arithmetic operations, which are evaluated left-to-right, expressions involving power operations are evaluated from right-to-left. Power operations also have higher precedence than the add/subtract operators, and the muldiply/divide operators.

Example 5 : The statement

    print "2^2^3 = ", 2^2^3 , "\n"; 

    2^2^3 =        256 

   2^2^3 

   2^(2^3) 

Example 6 : The following statements show how power operations apply to quantities having units -- the script:

    print "(2 m)^2    = ", (2 m)^2      , "\n"; 
    print "2^2 m      = ", 2^2 m        , "\n"; 
    print "10^2^2 N/m = ", 10^2^2  N/m  , "\n"; 

    (2 m)^2    =          4 m^2 
    2^2 m      =          4 m 
    10^2^2 N/m =      1e+04 N/m 

You should should carefully note how parentheses are used to bind a unit to a constant, and how the right-to-left evaluation of numerical quantities takes precedence over the unit itself.

Example 7 : Now let's look at a family of expressions where power operations are mixed in with multiply and divide operations. The block of code

    print "0.25*2^2   m  = ", 0.25*2^2   m, "\n"; 
    print "1/2*2      m  = ",  1/2*2      m, "\n";
    print "(1/2*2     m) = ", (1/2*2     m), "\n";
    print "(1/2^2     m) = ", (1/2^2     m), "\n";
    print "(1/2^2)*(1 m) = ", (1/2^2)*(1 m), "\n";
    print "1/2^2*(1 m)   = ",  1/2^2 *(1 m), "\n";

    0.25*2^2 m    =     1 m 
    1/2*2    m    =     1 m 
    (1/2*2  m)    =     1 m 
    (1/2^2  m)    =  0.25 1/m 
    (1/2^2)*(1 m) =  0.25 m 
    1/2^2*(1 m)   =  0.25 m 

Notice how parentheses can be used to adjust the precedence of evaluation, and how the multiply and divide operations are evaluated left-to-right.

Example 8 : In this example, we demonstrate the exponential operator on quantity variables:

    x = 100 kg; z = 10 m;

    print "z^-3          = ", z^-3, "\n";     
    print "(x*z)^2       = ", (x*z)^2, "\n";  

    z^-3          =  0.001 1/m^3 
    (x*z)^2       =  1e+06 m^2.kg^2 

Example 9 : Power operations such as

      x = (1 m)^(2 sec);

are illegal, and will cause ALADDIN to terminate its execution.