Calculabo

An elegant yet powerful calculator app.

# Infinite numbers

Calculabo extends the real number line and the complex plane with various infinite numbers. Each of them has an absolute value of infinity ($∞$).

## Directed infinity

Directed infinity is an infinite number with a specific argument, i.e. a specific direction in the complex plane in which the limit to infinity was evaluated.

Calculabo can store four directed infinities, namely:

• positive infinity ($+∞$),
• negative infinity ($−∞$),
• infinity in the positive imaginary direction ($+∞i$), and
• infinity in the negative imaginary direction ($−∞i$).

See below for infinities in other directions. To type $∞$ in Calculabo, press Ctrl+M and type 8.

## Complex infinity

Complex infinity, denoted $$ is a number whose absolute value is infinite, but its argument is indeterminate. It is hence a special case of an indeterminate number. It may either be positive infinity, negative infinity, or any other infinity in the complex plane. To type $$ in Calculabo, press Ctrl+M, type 8 and press Space.

Complex infinity for example results from division by zero. In this case, the (complex) sign of the limit $1∕x$ would depend on whether $x$ approaches $0$ from above, from below or from any other complex direction. The argument of the limit is therefore indeterminate, whereas its absolute value unambiguously equals $∞$. Consequently, if the inversion is repeated, you get back $0$, since the inverse of a infinite number is always zero regardless of its argument.

### Example

$1∕0$
$$
$1∕ans$
$0$

In Calculabo, any infinity whose argument is not a multiple of $π∕2$, is also implicitly converted into complex infinity, because internally Calculabo can only store directed infinities in one of the four basic directions listed above. The information about the argument of the infinity is then lost.

### Example

$(1+i)∕√2⋅∞$
$$