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No! Consider this number:

0.101001000100001..... (continue the obvious pattern)

This number is irrational but there is an obvious pattern to the digits. The pattern is not random. It's an open question on whether or not the digits of pi are randomly distributed. If so then what you write is correct.

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Doesn't the difference between just being irrational (e.g., sqrt(2)) and being transcendental (e.g., \pi, e) matter here?

The claim I responded to had to do with the randomness of the digits of a number. The person appeared to have the belief that irrational numbers have to have random digits. This isn't true. This isn't true for transcendental numbers either. The number I gave is transcendental but doesn't have random digits.

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