[isaac21259]

To clarify where I live limits are introduced in high school, irrational numbers just much earlier.

[afiori]

Multiplication is repeated additions is the informal way of stating the distributive property of multiplication and addition.

Probably you were taught how to multiply irrational by the property of powers (a^b * c^b = (a*c)^b etc.).

You were not taught a grand unifying theory of multiplication, you were taught how to manipulate operations to turn them into more useful operations.

Teaching these laws also prepares you for when a and b are just symbolic reals with no structure and those laws are the only thing you can use to manipulate them.

[naniwaduni]

You don't need a rigorous notion of limits to informally notice that irrationals have arbitrarily close rational approximations, e.g. by adding successive digits.