What is the diference between zero (0) and infinitive(infinitesimal) ?

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I believe that you meant an infinitesimal.

Throughout history, especially in the Western tradition beginning with the ancient greeks, there has been a problem concerning infinities. Realizing an infinity creates paradoxes (see Zeno's paradoxes for example), but there are so many uses for problems that involve infinities.

When the calculus was invented (discovered?), the solution to problems relied on an infinite process. This is a difficult thing for most to imagine, so a group of mathematicians reworked the calculus so that it did not rely on infinite processes by using the infinitesimal.

**The infinitesimal is a number that is smaller than any real number yet greater than zero.** (In effect they replaced infinity with its reciprocal.) This of course creates new problems (you cannot divide an infintesimal by 2, etc... so normal arithmetic does not work with infinitesimals).

**Sources:**

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