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## Main Question or Discussion Point

I'm an undergrad math major who has recently taken an interest in normal numbers. I've just decided to begin to seriously read about them, and I was wondering if anyone here could point me in the right direction (in terms of which papers to read, for example). I'm wondering what other characterizations there are of normal numbers and normal numbers in base b. For example, Wolfram MathWorld says a normal number is:

Additionally, since it seems that normal numbers are related somehow to transcendental numbers, could anyone recommend a good book on that subject?

Specifically, I suspect that a number is normal in base 2 if and only if its base-2 expansion is a concatenation of every integer. I have sketched a proof though I have not filled in the details, so there remains a significant possibility that my reasoning is wrong. Is anyone here knowledgeable on the subject? Is this blatantly false? Obvious? Simply not a useful characterization?...an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base...

Additionally, since it seems that normal numbers are related somehow to transcendental numbers, could anyone recommend a good book on that subject?