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Preface

Mathematics are a very flexible language. There are many ways to express the same computation. As an example you can take the simple division

This can also be written as

or as

There is not much difference in that yet. Only a crook will see how steep the fraction bar is drawn. A bookkeeper prefers the notation

and that looks quite strange. In primary school you learned how a division is computed

That shows, how it is made. You see, there are many possibilities to express the same operation. And that is almost always the case in mathematics. The theme is: Many roads lead to Rome. In high school you learned still another method

Calculations are nowadays done with 10 digits, the so-called arabic numerals. By the way, they were invented in India. What is in a name? If you add numbers, you work from right to left, although we always write from left to right. Because of this, digits must be right-aligned, that comes from the Arabs, but probably you never really noticed. The notation is

0 1 2 3 4 5 6 7 8 9

Here you start with 0. You were told that it is a somewhat special number, and care should be taken. For example, a division by zero is not permitted. In former times, roman numerals were used. You sometimes find them as a decoration on buildings. The notation is

The number 0 does not exist here. Zero is just nothing. Hundred of years it has been disputed whether 0 is a real number at all. This has now been settled. However, you will still be confronted with some peculiar details. Then, why actually reads

100 = 1

The answer is

That applies for every number, but what about

Then a division by zero is not allowed. A multiplication with 0 is possible, because

02 = 0
01 = 0
00 = ?

Oh, that is amusing. For this kind of problem, mathematicians have found a rather elegant solution. It is determined by definition. In this way 00 = 1. It has been shown that most computations give a correct result if you take 00 = 1, but you must pay extra attention, and carefully check the result, as in your special case it could well be different. The exception confirms the rule.
Numbers were invented by men. In the real world, things work out different. There you will find things that are infinite. In mathematics you use the symbol . But be careful: infinity is not a number. You can perform calculations with it, and sometimes you will obtain astonishing results

∞ + ∞ = ∞

and even

0 × ∞ = ?

So, a multiplication with zero does not always yield zero. Maybe you are by now not surprised that much anymore. But it can get even worse. There exist constants, that you can not state as an exact number. The most well-known constant is pi, that is written with the Greek letter π. You know it from the formulas for circles

circumference = 2πr
area = πr2

The ancient Greeks had already noticed that a constant describes the ratio between the circumference and the diameter D of a circle

    π =   circumference 
         D 

and that it also describes the ratio between the area and radius r of the circle

    π =   area 
   r2 

This is a real world constant. The value amounts to

π = 3.1415…

One quintillion figures behind decimal point were already computed, and there never develops a pattern that is repeated. Mathematicians proved that this will never happen, and therefore call it a transcendental number. Another famous transcendental constant is e, the base of the Naperian logarithm. You need this to calculate how bacteria multiply, or to compute how radioactive contamination decreases in time. Its value is

e = 2.7182…

On this website it is explained how the number was discovered and where to apply it in calculations. Moreover, mathematicians have proven that there are an infinite number of transcendental constants, but nobody knows them, and we have no idea what they should be used for. That is higher mathematics. Now back to something more simple. If you add the infinite series

you can continue forever. But there is a faster way. Please watch this. You can double a term, and immediately subtract it again. Then you obtain the original value, because

2 × 3 − 3 = 3

or

2 apples – 1 apple = 1 apple

Apply this scheme to the series, so

then after calculation it gives

and this is again

It is exactly 1, and not something mysterious like "In the infinite it approaches 1". There is a fine difference between theoretically infinite and physically infinite. Let us now switch to the infinitely small. In mathematics this is often written as Δx→0 and means: It approaches 0, but is not equal zero, and therefore division by Δx is permitted. You must sometimes even distinguish between

Δx→0+  and  Δx→0

If you think this is confusing, then please look at the following

That seems clear. But the following can also easily be explained

Omitting the brackets in both calculations leads to

1 − 1 + 1 − 1 + 1 − 1 + ··· = ?

and now we don't know the answer anymore. What is going on here? Mathematician dislike computations where this phenomenon arises. Do you like these matters? Would you like to know more about nothing, the infinite or more than infinite? Then you must proceed further in this document.

Sometimes you must apply some hocus pocus when computing. And by the way: What do you actually need mathematics for? Well, that is up to you. In most professions you can perfectly work without it. But perhaps it is interesting to know what is possible – or just impossible.

One cannot predict the future with mathematics.

 


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