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Neglectable

In mathematics you sometimes work with terms that are infinitesimal. In the course of a calculation these are omitted because they are neglectable.

 


Small transitions

When differentiating using Δx, transitions occur of

     and of     

The product f g yields

The increase consists of the three terms

For the derivative of the product f g we get

The latter term is neglectable, because

and that goes to 0 anyway. The product rule is therefore

f g )' = f′ g + f g′

If after a calculation infinitely small terms remain, you may neglect those.

 


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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