Are all numbers properly classified in the real number system?

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Have I correctly classified these sets of numbers? I am trying to diagram algebraic, transcendental, irrational...etc, numbers. Please see the attached picture.
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First correction: put the natural numbers ellipse completely inside the integers. The attachment has been updated.
 
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...comments?

Surely this isn't a difficult question.
 
naturals, integers, and rationals are contained in the reals by every definition I've seen (though there is the notion of the gaussian integers, it's field of fractions, etc). Irrationals can go either way, but the most common definition has them contained in the reals as well.
 

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