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I am just curious if 0 is a real number. The definition of a real number is all rational and irrational numbers. And the def definition for rational number is that "$\mathbb{Q}={a\div b|a,b\in\mathbb{Z}}$". But in done say that $0$ is a whole number and some day that it is not. In some websites some say that $0$ is a real number, and in some not.

Please help me. Thanks in advance

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2 Answers

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$\mathbb{Q}$, the set of rational numbers, is the set $\{\frac{a}{b}:a,b\in\mathbb{Z}, b\neq 0\}$

$\Rightarrow 0=\frac0b$ is a rational number, where $0,b\in\mathbb{Z}, b\neq 0$.

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By $\mathbb{Z}$'s definition, $0\in\mathbb{Z}$. So then for all $b\in\mathbb{Z}\backslash\{0\}$, we have that $0=\frac{0}{b}\in\mathbb{Q}$. Because of this, then $\frac{0}{b}=0\in\mathbb{R}$. To be said, $0$ is a real number.

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