The mathematician Leibniz based his calculus on the assumption that there were i

Question

The mathematician Leibniz based his calculus on the assumption that there were infinitesimals, positive real numbers that are extremely small: smaller than all positive rational numbers certainly. Some calculus students also believe, apparently, in the existence of such numbers since they can imagine a number that is just next to zero. Is there a positive real number smaller than all positive rational numbers? Show how you might prove your assertion using terms a calculus student would understand.

Explanation / Answer

Solution :- We have a question that " Is there a positive real number smaller than all positive rational numbers".

Suppose there is a smallest positive real number x.

x is such that , x > 0 and x |R.

Let y = x/10

Then we get contradiction.

This implies that y < x.

==> We can always construct a rational number that is less that the smallest positive real number.

This is process has no end.

So there is no positive real number smaller than all positive rational numbers.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---