The Tolerances package provides an environment to perform computations with quantities involving tolerances.

After issuing the with(Tolerances) command, quantities involving tolerances can now be entered using a natural notation and any computations involving such quantities will return a value with the associated tolerance result.

The Tolerances package uses interval arithmetic to perform its computations.

Tolerances are entered by specifying the value, followed by the (absolute) tolerance, separated by a plus/minus ($\pm$) sign. To enter a plus/minus sign:

Use the palettes. See View>Palettes>Common Symbols. (Standard Worksheet interface only)

Use symbol completion.  Enter plusmn, and then press Esc. (Standard Worksheet interface 2-D input only)  For more information, see symbol completion.

Enter &+-.

with(Tolerances):

a := 2 &+- 0.1;

$a≔2.000\pm 0.100$

r := (Pi/2) &+- 0.2;

$r≔1.571\pm 0.200$

Note: The plus/minus operator takes precedence over most operators, like ^ and *. As a result, parentheses are required around any non-atomic expression for a nominal value or a tolerance value. For more information, see operators/precedence.

In addition to simple arithmetic, the following functions can be used with Tolerances:

with(Tolerances):

a := 2 &+- 0.1;

$a≔2.000\pm 0.100$

b := 3 &+- 0.05;

$b≔3.000\pm 0.050$

a+b;

$5.000\pm 0.150$

a*b;

$6.005\pm 0.400$

a^b;

$8.127\pm 1.484$

sin(a);

$0.905\pm 0.042$

exp(1/b);

$1.396\pm 0.008$

By default, the Tolerances package displays values with 3 digits even though computations are always performed at the precision indicated by the Digits variable.  Use interface(displayprecision) to control the number of displayed digits.

interface(displayprecision=10);

$\mathrm{-1}$

a := (1/3) &+- 0.1;

$a≔0.3333333332\pm 0.1000000000$

interface(displayprecision=3);

$10$

a := (1/3) &+- 0.1;

$a≔0.333\pm 0.100$

The NominalValue command computes the nominal value of a quantity with a tolerance. The nominal value is the center of the confidence interval.

The ToleranceValue command computes the tolerance value of a quantity with a tolerance.  The tolerance is the width of the confidence interval divided by two.

a := 2 &+- 0.1;

$a≔2.000\pm 0.100$

b := 3 &+- 0.05;

$b≔3.000\pm 0.050$

a+b;

$5.000\pm 0.150$

NominalValue(a+b);

$5.000$

ToleranceValue(a+b);

$0.150$

Tolerances and Units can be used together in the same computation

with(Units):

a := 150 &+- 10 * Unit(m);

$a≔150.000\pm 10.000⟦m⟧$

b := 0.2 &+- 0.001 * Unit(km);

$b≔0.200\pm 0.001⟦\mathrm{km}⟧$

C := 2*a+2*b;

$C≔700.000\pm 22.000⟦m⟧$

A := a*b;

$A≔3.001\times {10}^4\pm 2.150\times {10}^3⟦m^2⟧$