pove that one is a positive number..?

Answer 1

Either 1 is positive or -1 is positive. 1 is the square of both of those, and hence is the square of a positive number.

And if you can't very easily prove that the square of a positive number -- or indeed any product of two positive numbers -- is positive, I don't know what axiom set you're using ...

Answer 2

You need the axioms:

(4) For every real number x there is a real number (-x), such that x + (-x) = 0
(7) 1 x = x for every real number x.
(11) For every real number x, exactly one of the following is true: x = 0, x is positive, or -x is positive.
(12) If x and y are both positive then xy is also positive.

Assume the opposite: (-1), which exists due to the axiom (4), is positive:

From the axiom (12) follows (-1)(-1) is positive.
From the axiom (7) follows 1 (-1) is positive.
By the axiom (11) these two statements cannot be true at the same time. Therefore, your assumption (-1 is positive) must be wrong. QED

Answer 3

1 > 0 > -1

Answer 4

p1, p2 and p3 are positive numbers (maybe they´re the same, that doesn´t matter)
n1 and n2 are negative number

basic mathematics say:
I.
p1 * p2 = p3 (when you multiply two positive numbers you get another positive number, division would also be possible)
II.
p1 * n1 = n2 (when you multiply a positive number with a negative you get a negative number, division is also possible)

so you can say:
1 * 3 = 3 (3 is positive) that shows that I. is true
1 * (-3) = -3 (-3 is negative) that shows that II. is true

=> 1 is a positive number
q.e.d.

Answer 5

2-x=-1
and in this case x=1 so 2-1 equals negative one so... 2-1=-1
1=-1
making your answere no solution proving one is not equal to a negative number making one positive