what is the smallest non zero positive real number and how can it be expressed?

Answer 1

The number is not defined.

Lots of people are saying things like...
"0.000...00001"
"1/x as x approaches infinity"

These numbers, technically, are the same as 0.

"n^0" --- ok that's just not thinking. If n <> 0 or n <> inf, then any n^0 = 1.

Proof:
Let 0* be a number infinitely close to 0.
0* = 1 - .999....
.999... = 3 * (.333...) // Because 3*3 = 9 no matter how many time you do it.
.999... = 3 * (1/3 ) // Exact fractional representation
.999... = 1
Therefore, 0* = 1 - 1, and 0* = 0.

As for the answer, the "smallest", non-zero real number does not exist. Well, we know that it "should" exist. It just doesn't.

Except on computers. That's another problem entirely. Because computers have a data limit, they do actually have a smallest real number >0

Answer 2

There is no smallest non-zero positive real number. If x>0, then x>x/2>0, so x/2 is even smaller than x. This is true for rational numbers, so it really isn't something strange about real numbers.

However, if x>0, there is a natural number n so that 0<1/n

Answer 3

As was pointed out in a couplre of prior posts.....

There is NO 'smallest' positive real number..

That the reals ..intuitively speaking..have no gaps (holes) is
basic...this is the usual intuited imagery of a 'smooth'
continuum....BUT you should contemplate :

1) Archimedean property of the reals: given positive reals
A and B.....there exists integer N such that NA>B

2) Any Cauchy sequence of reals converges to a real...
i.e. a real exists to which the sequence actually converges
(look up the definition of 'Cauchy sequence' AND review what
it means for a sequence to converge)

3) A set of reals with upper bound has a Least Upper Bound
i.e. the LUB is less than or equal to any other upper bound.

Two of the above are logically equivqlent to the remaining one!!!
Can you figure out which two imply the other??

They are three DISTINCT concepts which happen to coincide
for the familiar reals BUT they are NOT one and the same.

The reals are an example of an 'ordered' field...but there are
many others.

For example the rational numbers satisfy precisely one of the
three properties but not the other two.

Answer 4

It is 1^n

Answer 5

There is no "smallest positive real number".

If you select a small, positive real, you can always take half of it and get something smaller.

Answer 6

There is no smallest positive real number.

Answer 7

the region is that it is not precisely precise. Pi is an irrational huge type precisely because, like sq. root of two, etc., that's endless. also, 22/7 is purely an unfinished branch situation. end the dept situation and one receives 3.142857, it really is totally close yet no longer excellent. The irrational is the better properly proper.

Answer 8

n^0

Answer 9

The answer is:
0.0000000000000001
there can be infinte number of zeroes and so there is no definite way to express it

Answer 10

10^-n