The numbers between two numbers, and an infinitely small positive number

Asked Sep 26 '18 at 04:52

Active Sep 26 '18 at 15:43

Viewed 46 times

This question sounded a bit foolish to me, but at the same time, a bit intriguing if nothing else.

Given any arbitrary number $a$ and $c$ where $a≠c$, the set containing all numbers on the interval $[a, a+c]$ contains infinitely many numbers.

Can a number Φ exist such that $0<Φ<c$ and the set containing all numbers on the interval $[a,a+Φ]$ contains only $a$ and $a+Φ$, or would that by definition only include $a$? I remember reading this somewhere but did not understand it at the time. I only recently remembered it.

edited Sep 26 '18 at 15:43

asked Sep 26 '18 at 04:52

Descartes Before the Horse

7,038

3

It's not possible in the real numbers. If an interval contains $a$ and $a+\Phi$, then it contains $a + \Phi/2$ as well, and the latter is distinct from the other two. – Sep 26 '18 at 04:55
@Bungo do you specify the reals because it does not apply in another context? – Descartes Before the Horse Sep 26 '18 at 04:56
3

It is possible in the integers, for example. There $\Phi=1$ will work. – hmakholm left over Monica Sep 26 '18 at 04:59
@HenningMakholm Ah, that's neat. I am studying graph theory and the behavior of multiple variables in n-dimensional graphs that have different types of numbers; I am trying to make some relations between the ideas I stated here and how it applies to, say, the complex plane. Thanks! – Descartes Before the Horse Sep 26 '18 at 05:03
1

That's not what "graph theory" means, Matt; that's analytic geometry. – Gerry Myerson Sep 26 '18 at 07:33

The numbers between two numbers, and an infinitely small positive number

0 Answers0