calculus - were Irrational numbers discovered at Archimedes's age?

Archimedes axiom states a property of real numbers, while the real numbers include all the rational numbers and all the irrational numbers.

I wonder were Irrational numbers discovered at Archimedes's age?

I think the question is equivalent to ask : Does Hippasus( he is sometimes credited with the discovery of the existence of irrational numbers)
live earlier than Archimedes?

Hippasus of Metapontum (/ˈhɪpəsəs/; Greek: Ἵππασος, Híppasos; fl. 5th
century BC)

Archimedes of Syracuse (/ˌɑːkɪˈmiːdiːz/;2 Greek: Ἀρχιμήδης; c. 287

BC – c. 212 BC)

P.S. I am Chinese , I don't understand these BCs

Answer

The OP wrote: "Archimedes axiom states a property of real numbers, while the real numbers include all the rational numbers and all the irrational numbers." It should be clear that the property in question is referred to as the Archimedes axiom only since about 1880 when the term was introduced by Otto Stolz.

The OP further asks: "I wonder were Irrational numbers discovered at Archimedes's age?"

The answer is most likely negative. The Greeks thought mostly in terms of proportions among whole numbers. The idea of systematizing the number system to include roots and other irrationals is mainly due to Simon Stevin who lived almost two thousand years later.