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Mathematics
General Math
Why are "irrational" and "transcendental" so commonly used to describe numbers
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[QUOTE="fresh_42, post: 5780166, member: 572553"] It is no hierarchy, it is a property. Numbers which fulfill a polynomial equation are called algebraic, and numbers which do not fulfill such an equation as e.g. ##\pi## over ##\mathbb{Q}## are called transcendental. It is not obvious whether a number is transcendental or not. It could simply be the case that we haven't found an appropriate equation yet and for many numbers, it is still not clear whether they are algebraic or transcendental. The proof that ##\pi## is transcendental dates back to 1882 (F. v. Lindemann). This is not that long ago considered since when we deal with ##\pi##. Irrational simply refers to numbers which are not elements of ##\mathbb{Q}## - not rational. So ##\pi## as well as ##\sqrt{7}## are irrational. But while ##\pi## is transcendental with respect to ##\mathbb{Q}##, ##\sqrt{7}## is algebraic with respect to ##\mathbb{Q}##. The fact that indeed ##\pi \notin \mathbb{Q}## is known since 1767. So even this "obvious" result took its time before it could have been considered certain. [/QUOTE]
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Mathematics
General Math
Why are "irrational" and "transcendental" so commonly used to describe numbers
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