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In Ancient Greek, most people like Pythagoras thought 1 (monad, unity) is no number, but it is ruler and beginning of all other numbers. And Pythagoras thought everything is number. But they found irrational numbers which can not be measured by 1, and their theory broke up because of it. But even though ancient Greek found irrational numbers, for example, Euclid who was born after Pythagoras and also knew about irrational number, said every number is made by 1 (monad). Not only Ancient Greek, but also Medival arthimeticians say 1 is source of all other numbers.

Q1. Why did they say 1 is source of numbers even though they found existence of irrational numbers? Is it because they only treated natural numbers as numbers?

Q2. If so, (only natural numbers can be numbers) what was irrational number for them?

Q3. According to ancient Greeks, If 'one' can not be divided, how can fractions exist? (1/2 or 1/3 etc..)

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    $\begingroup$ They did not find "irrational numbers", that is just sloppy talk in some books concerning the incommensurable ratios of magnitudes. The only numbers recognized by Pythagoreans were positive integers, even ratios of integers were not numbers. Ratios were a separate class of objects handled differently than numbers, they could not be added, for example. $\endgroup$ – Conifold Aug 20 '19 at 19:35
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    $\begingroup$ Possible duplicate of How were irrational numbers that are not constructible accepted by mathematicians? $\endgroup$ – Conifold Aug 20 '19 at 19:36
  • $\begingroup$ See also hsm.stackexchange.com/questions/6721/… $\endgroup$ – sand1 Aug 21 '19 at 15:26
  • $\begingroup$ @Conifold "Ratios were a separate class of objects handled differently than numbers, they could not be added, for example." Do you mean that they didn't know how to, or that they didn't have the use/wanted to add ratios/fractions? $\endgroup$ – EigenDavid Sep 3 '19 at 12:39
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    $\begingroup$ @PeterShor They were good at not needing it. The modern way of doing things is not the only one, and would not have fit well with their background and interests. $\endgroup$ – Conifold Sep 11 '19 at 17:44
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In the time of Pythagoras, it was widely believed among Greek mathematicians and philosophers that any line (or object) is made up of basic elements of same length (also known as atom). Thus if the length of natural number 1 consists of n atoms, then each atom would have the length of $1/n$, and a line of any length consisting of $m$ atoms would have the length of a rational number $m/n$.

The discovery of the fact that $\sqrt{2}$ is not a rational number completely shattered this belief because it means that the line can not be made up of atoms, for otherwise there would be many holes left on the line that can not be measured by rational numbers. Since no one could explain the nature of $\sqrt{2}$, it was known as an irrational number, which means number of no sense. The name of irrational number has become widely in use until today.

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    $\begingroup$ At the time of Pythagoras atomism was not yet formulated. Nor did it have anything to do with Pythagorean beliefs, Democritus was not a Pythagorean. At no time in antiquity was atomism "widely believed". There was no shattering either, atomists simply rejected Euclidean axioms. The name "irrational number" dates to late middle ages, a millenium later. $\endgroup$ – Conifold Aug 29 '19 at 23:13
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    $\begingroup$ @MathWizard ... You need to add references. Some of your historical assertions are now known to be false. $\endgroup$ – Gerald Edgar Aug 30 '19 at 13:21
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    $\begingroup$ There was no "similar idea", you may have heard of Tannery's thesis from 1877, which has been discredited since then. The name "rational number" came from "ratio", and long after Pythagoreans. The root is a Latin translation of Greek logos (reason) The statement "$\sqrt{2}$ is not a rational number" could not even have been made in antiquity, mathematics was formulated in terms of magnitudes and ratios. The Hippasus story is a late anecdote that even Wikipedia identifies as a fake. $\endgroup$ – Conifold Sep 1 '19 at 9:53
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    $\begingroup$ There is no source for this "well-known and long accepted" story that links Hippasus to the incommensurability rather than the construction of the dodecahedron. Or any source linking him to mathematics that is less than six centuries removed from when he supposedly lived. Currency of anecdotes in popular literature does not reflect their historical accuracy, or the view of historians. $\endgroup$ – Conifold Sep 1 '19 at 23:48
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    $\begingroup$ @PeterShor Greek alogon was not an irrational number, it was a term for two magnitudes not having a (number) ratio, see e.g. Fowler, Ratio in early Greek mathematics. $\endgroup$ – Conifold Sep 11 '19 at 17:41

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