Prove by reducing to absurdity.

Let a be irrational, b be nonzero rational, and c=ab.

Suppose c is a rational number, then a = c/b.

C and B on the right are rational numbers, so c/b is rational.

So left A can only be rational and contradictory.

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Extended data:

The discovery of irrational numbers:

In 500 BC, hippasus, a disciple of Pythagoras School, discovered an amazing fact: the diagonal of a square is incommensurable with the length of one side (if the side length of a square is 1, the length of the diagonal is not a rational number), which is quite different from Pythagoras School's philosophy of "everything is a number" (referring to a rational number).

This discovery frightened the leaders of the school, thinking that it would shake their dominant position in the academic world, so they tried their best to stop the spread of this truth, and Herbesos was forced into exile. Unfortunately, he met his disciples on a seagoing ship. Was brutally thrown into the water by Pisces disciples and killed. Thus began the history of science, but it was a tragedy.