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可忽视

在数学中,你有时会处理一些无限小的项。在计算过程中,这些项被省略了,因为它们是可以忽略的。

 


小幅过渡

当使用 Δx 进行区分时,会发生以下的转换

     以及     

f g 可得

增加的部分包括三个条款

对于乘f g 的导数,我们得到的是

后一个词是可以忽略的,因为

而这无论如何都会变成 0。因此,积规则

f g )' = f′ g + f g′

如果经过计算后还剩下无限小的项,你可以忽略这些项。

 


Small increments

When calculating the number e we use the formula

and see how you can get the correct result. We start with usual numbers







and see that the outcome increases during each step, although the value of the fraction decreases steadily. Eventually you work with infinitely small terms, but you must not neglect those here.

 


Differentials

The differential of the logarithm gives

Subtraction of logarithms produces

Substitution of this in the power series for the logarithm gives

Because all the differentials of the second order and higher are neglectable, you may write

After substitution you get

 


The number 1

You can write the number 1 with an infinite number of decimals as

The three points indicate that there are infinitely many decimal places. You can calculate that with a fraction

It is said that 0.999999… approaches 1 in infinity. That sounds impressive, but no one can imagine what infinity is. We feel that there must be a neglectable difference between 0.999999… and 1. That's however wrong. It are just two different ways in which the same number can be written.

 


Limits

In principle an infinitely small value in a calculation may be neglected. If it occurs infinitely often, however, it should not. That is a rule of thumb. We know that

And then you may certainly not neglect

∞ × ∆x = ∞

In all calculations you should strictly apply the mathematical rules. Therefore you must use limits in these cases, because then you know what you are doing. Where necessary to avoid confusion you write

     or     

In itself it is obvious when you can neglect an infinitesimal value.

 


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