Documentation

## Units of Measurement Tutorial

Use units of measurement with Symbolic Math Toolbox™. This page shows how to define units, use units in equations (including differential equations), and verify the dimensions of expressions.

### Define and Convert Units

u = symunit;

Specify a unit by using u.unit. For example, specify a distance of 5 meters, a weight of 50 kilograms, and a speed of 10 kilometers per hour. In displayed output, units are placed in square brackets [].

d = 5*u.m
w = 50*u.kg
s = 10*u.km/u.hr
d =
5*[m]
w =
50*[kg]
s =
10*([km]/[h])

### Tip

Use tab expansion to find names of units. Type u., press Tab, and continue typing.

Units are treated like other symbolic expressions and can be used in any standard operation or function. Units are not automatically simplified, which provides flexibility. Common alternate names for units are supported. Plurals are not supported.

Add 500 meters and 2 kilometers. The resulting distance is not automatically simplified.

d = 500*u.m + 2*u.km
d =
2*[km] + 500*[m]

Simplify d by using simplify. The simplify function automatically chooses the unit to simplify to.

d = simplify(d)
d =
(5/2)*[km]

Instead of automatically choosing a unit, convert d to a specific unit by using unitConvert. Convert d to meters.

d = unitConvert(d,u.m)
d =
2500*[m]

There are more unit conversion and unit system options. See Unit Conversions and Unit Systems.

Find the speed if the distance d is crossed in 50 seconds. The result has the correct units.

t = 50*u.s;
s = d/t
s =
50*([m]/[s])

### Use Temperature Units in Absolute or Difference Forms

By default, temperatures are assumed to represent differences and not absolute measurements. For example, 5*u.Celsius is assumed to represent a temperature difference of 5 degrees Celsius. This assumption allows arithmetical operations on temperature values.

To represent absolute temperatures, use kelvin, so that you do not have to distinguish an absolute temperature from a temperature difference.

Convert 23 degrees Celsius to kelvin, treating it first as a temperature difference and then as an absolute temperature.

u = symunit;
T = 23*u.Celsius;
diffK = unitConvert(T,u.K)
diffK =
23*[K]
absK = unitConvert(T,u.K,'Temperature','absolute')
absK =
(5923/20)*[K]

### Verify Dimensions

In longer expressions, visually checking for units is difficult. You can check the dimensions of expressions automatically by verifying the dimensions of an equation.

First, define the kinematic equation ${v}^{2}={v}_{0}{}^{2}+2as$, where v represents velocity, a represents acceleration, and s represents distance. Assume s is in kilometers and all other units are in SI base units. To demonstrate dimension checking, the units of a are intentionally incorrect.

syms v v0 a s
u = symunit;
eqn = (v*u.m/u.s)^2 == (v0*u.m/u.s)^2 + 2*a*u.m/u.s*s*u.km
eqn =
v^2*([m]^2/[s]^2) == v0^2*([m]^2/[s]^2) + (2*a*s)*(([km]*[m])/[s])

Observe the units that appear in eqn by using findUnits. The returned units show that both kilometers and meters are used to represent distance.

findUnits(eqn)
ans =
[ [km], [m], [s]]

Check if the units have the same dimensions (such as length or time) by using checkUnits with the 'Compatible' input. MATLAB® assumes symbolic variables are dimensionless. checkUnits returns logical 0 (false), meaning the units are incompatible and not of the same physical dimensions.

checkUnits(eqn,'Compatible')
ans =
logical
0

Looking at eqn, the acceleration a has incorrect units. Correct the units and recheck for compatibility again. eqn now has compatible units.

eqn = (v*u.m/u.s)^2 == (v0*u.m/u.s)^2 + 2*a*u.m/u.s^2*s*u.km;
checkUnits(eqn,'Compatible')
ans =
logical
1

Now, to check that each dimension is consistently represented by the same unit, use checkUnits with the 'Consistent' input. checkUnits returns logical 0 (false) because meters and kilometers are both used to represent distance in eqn.

checkUnits(eqn,'Consistent')
ans =
logical
0

Convert eqn to SI base units to make the units consistent. Run checkUnits again. eqn has both compatible and consistent units.

eqn = unitConvert(eqn,'SI')
eqn =
v^2*([m]^2/[s]^2) == v0^2*([m]^2/[s]^2) + (2000*a*s)*([m]^2/[s]^2)
checkUnits(eqn)
ans =
struct with fields:

Consistent: 1
Compatible: 1

After you finish working with units and only need the dimensionless equation or expression, separate the units and the equation by using separateUnits.

[eqn,units] = separateUnits(eqn)
eqn =
v^2 == v0^2 + 2000*a*s
units =
1*([m]^2/[s]^2)

You can return the original equation with units by multiplying eqn with units and expanding the result.

expand(eqn*units)
ans =
v^2*([m]^2/[s]^2) == v0^2*([m]^2/[s]^2) + (2000*a*s)*([m]^2/[s]^2)

To calculate numeric values from your expression, substitute for symbolic variables using subs, and convert to numeric values using double or vpa.

Solve eqn for v. Then find the value of v where v0 = 5, a = 2.5, and s = 10. Convert the result to double.

v = solve(eqn,v);
v = v(2);		% choose the positive solution
vSol = subs(v,[v0 a s],[5 2.5 10]);
vSol = double(vSol)
vSol =
223.6627

### Use Units in Differential Equations

Use units in differential equations just as in standard equations. This section shows how to use units in differential equations by deriving the velocity relations v = v0 + at and ${v}^{2}={v}_{0}{}^{2}+2as$ starting from the definition of acceleration $a=\frac{dv}{dt}$.

Represent the definition of acceleration symbolically using SI units. Given that the velocity V has units, V must be differentiated with respect to the correct units as T = t*u.s and not just t.

syms V(t) a
u = symunit;
T = t*u.s;		% time in seconds
A = a*u.m/u.s^2;	% acceleration in meters per second
eqn1 = A == diff(V,T)
eqn1(t) =
a*([m]/[s]^2) == diff(V(t), t)*(1/[s])

Because the velocity V is unknown and does not have units, eqn1 has incompatible and inconsistent units.

checkUnits(eqn1)
ans =
struct with fields:

Consistent: 0
Compatible: 0

Solve eqn1 for V with the condition that the initial velocity is v0. The result is the equation v(t) = v0 + at.

syms v0
cond = V(0) == v0*u.m/u.s;
eqn2 = V == dsolve(eqn1,cond)
eqn2(t) =
V(t) == v0*([m]/[s]) + a*t*([m]/[s])

Check that the result has the correct dimensions by substituting rhs(eqn2) into eqn1 and using checkUnits.

checkUnits(subs(eqn1,V,rhs(eqn2)))
ans =
struct with fields:

Consistent: 1
Compatible: 1

Now, derive ${v}^{2}={v}_{0}{}^{2}+2as$. Because velocity is the rate of change of distance, substitute V with the derivative of distance S. Again, given that S has units, S must be differentiated with respect to the correct units as T = t*u.s and not just t.

syms S(t)
eqn2 = subs(eqn2,V,diff(S,T))
eqn2(t) =
diff(S(t), t)*(1/[s]) == v0*([m]/[s]) + a*t*([m]/[s])

Solve eqn2 with the condition that the initial distance covered is 0. Get the expected form of S by using expand.

cond2 = S(0) == 0;
eqn3 = S == dsolve(eqn2,cond2);
eqn3 = expand(eqn3)
eqn3(t) =
S(t) == t*v0*[m] + ((a*t^2)/2)*[m]

You can use this equation with the units in symbolic workflows. Alternatively, you can remove the units by returning the right side using rhs, separating units by using separateUnits, and using the resulting unitless expression.

[S units] = separateUnits(rhs(eqn3))
S(t) =
(a*t^2)/2 + v0*t

units(t) =
[m]

When you need to calculate numeric values from your expression, substitute for symbolic variables using subs, and convert to numeric values using double or vpa.

Find the distance traveled in 8 seconds where v0 = 20 and a = 1.3. Convert the result to double.

S = subs(S,[v0 a],[20 1.3]);
dist = S(8);
dist = double(dist)
dist =
201.6000