Positive infinity
In mathematics there is the concept of infinity. You use the symbol ∞ for it. But be careful: Infinity is not a number. You can perform calculations with it, and sometimes you will obtain astonishing results.
Explanation
Let us first make an addition
∞ + 1 = ∞
It works the same for any number, and even for very big numbers. It also works with
∞ + ∞ = ∞
You may also subtract
∞ − 1 = ∞
That is the same for any number. However
That is quite logical, as ∞ has no fixed value. Multiplication is also possible. Here we use the positive constant c and get
∞ × c = ∞
Now let us look at a division. You should always take care, and certainly here, then
and now you will notice that as a general rule
Here you can still imagine something to explain it, but it remains strange. In addition, you will see that it has weird effects. And beware, because
Infinity has no fixed value. If this division emerges in another way, you get a solution, because
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Here you work with n→∞ and it means: it approaches infinity, but it is not. In the numerator and the denominator is concerns the same (infinite) value, and you may therefore divide. And now for the icing on the cake:
0 × ∞ = ?
It can be anything – even zero. However, that is not as weird as you may think. The number 0 (mathematicians have decided that it is a number) can cause problems, and ∞ is not even a number. You saw this already slowly looming at the explanation of the division. For the sake of completeness also a division by infinitely small. Sometimes you write that as
Here you work with Δx→0+ and it means: it is approaching from the positive to 0, but is not 0, so you may therefore perform a division. You can always create even crazier examples, and then you write something like
,
,
,
and so on.
That's kind of fun, but we will stop here.