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前言

數學是一個非常靈活的語言。有許多方式來表達同一計算。作為一個例子,你可以簡單的劃分

這也可以寫成

或如

尚不太大的分別的。只有一個騙子將看到如何陡峭分數衝程繪製。簿記員喜歡記法

和那看起來很奇怪。在小學您學習了如何計算司

這表明,它如何製作。你看,有很多的可能性來表達相同的操作。而這幾乎總是是數學的情況。主題是: 很多條條道路通羅馬。在高中的時候,您學習了還有一種方法

計算現在完成了 10 個數字,所謂的阿拉伯數位。順便說一句,他們被發明的印度。在一個名稱是什麼?如果您添加的編號,你工作從右至左,雖然我們總是寫從左至右。正因為如此,數位必須為右對齊,這來自阿拉伯人,但可能你從來沒有真正注意到。符號是

0 1 2 3 4 5 6 7 8 9

在這裡你從 0 開始。你被告知這是一個有點特別的數位,要小心。例如,不允許被零除。在前時期,使用了羅馬數字。有時你會發現他們在建築物上的裝飾。符號是

數位 0 在這裡已經不存在了。零,只不過是。幾百年來它一直有爭議是否 0 根本是一個實數。這已經得到解決。然而,你仍然會面臨著一些特殊的細節。那麼,為何實際讀取

100 = 1

答案是

這適用于每一個數位,但什麼關於

被零除不是允許的。與 0 的乘法是可能的因為

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 of a circle and diameter d

     π =   circumference 
     diameter 

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

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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