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2023年11月21日 · Rational Number Definition. A rational number is a number that can be be expressed as a ratio of two integers, meaning in the form p q. In other words, rational numbers are fractions. The set of ...
Aug 19, 2023 at 3:58. One simple way to find x ∈ [0 ∘, 90 ∘] such that sinx is a rational number, is to make the substitution x = arcsin(m / n) where m,n are both positive integers and m ≤ n. A good question follows, does this expression exhaust all x ∈ [0 ∘, 90 ∘] such that sinx is a rational number.
2013年10月29日 · I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrational. Eg. √2 + 1 can be expressed as a continuous fraction, and through looking at the fraction, it can be assumed √2 + 1 is ...
Yes, for every rational number (with the exception of 0 itself, of course), there's an irrational number that's closer to 0. But there's no irrational number which is closer to 0 than every rational number, because for each irrational number there's a rational number which is even closer to 0. – celtschk.
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Therefore every rational number is represented by a decimal that either terminates or repeats. Formal proof attempt: Claim: if a number is rational, then it's decimal expansion either terminates or repeats. Proof: let a/b be a rational number. Then by the definition of
This happens because the decimal representation is obtained via division of p by q and hence the only possible choices for remainder are 0, 1, 2, …, q − 1. If at some point we get a 0 remainder then the decimal representation is finite; otherwise the remainder has to repeat and lead to a repeating non-terminating decimal representation.
Any set that can be put in one-to-one correspondence in this way with the natural numbers is called countable. In some sense, this means there is a way to label each element of the set with a distinct natural number, and all natural numbers label some element of the set.$\endgroup$. – Jared.
2011年1月26日 · Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
2012年6月13日 · Corollary. $$\mathbb{Q} +\mathbb{Q}^c \subseteq \mathbb{Q}^c$$ That is: "A rational number plus an irrational number will always itself be irrational." The opening observation of user17762's answer is trickier.
