Calculabo

An elegant yet powerful calculator app.

# Indeterminate numbers

Calculabo is able to represent and store indeterminate numbers, of which the actual value is partially or completely ambiguous.

Calculabo denotes indeterminate numbers as $¿$. To get the $¿$ symbol in Calculabo, simply type ?. When used in Calculabo input, this represents a fully indeterminate scalar.

## Occurrence

Indeterminate numbers can typically be generated by feeding certain mathematical operations with zeros or infinities in a particular way.

### Example

$⟨0∕0,0⋅∞,∞∕∞,∞−∞,0^0,∞^0,1^∞⟩$
$⟨¿,¿,¿,¿,¿,¿,¿⟩$

In theory, to find the actual value of such an expression, you would need to analytically evaluate the limit of the formula that resulted in the particular mathematical operation. Since Calculabo does not operate at this analytical level, it simply reports the form as indeterminate.

Additionally, in most cases, if one of the inputs of an operation is indeterminate, the output will also be indeterminate.

### Example

$¿+1$
$¿$

Finally, Calculabo may return indeterminate numbers if you try to invert a singular matrix, or if you enter a mathematical expression Calculabo cannot evaluate for practical reasons. In both cases, a warning will be shown.

### Example

$[1,2;2,4]^−1$
Mathematical warning: Matrix is not invertible
$[¿,¿;¿,¿]$
$0.5!$
Calculator warning: Factorial is not supported for non-integer arguments
$¿$

## Partially indeterminate numbers

Calculabo can store partially indeterminate numbers for which either the real component or the imaginary component is actually known. In Calculabo output, these are also denoted $¿$, but further analysis can extract this information. For example, when extracting Cartesian coordinates from an indeterminate complex number, it is obvious that such a coordinate cannot itself have an imaginary component, as further analysis will reveal.

### Example

$Re(¿)$
$¿$
$Im(ans)$
$0$

Complex infinity, denoted $$, is a special indeterminate number in Calculabo with an absolute value of infinity ($∞$), while its argument is unknown, and hence its real and imaginary components are unknown as well.