identify the number which is irrational. a -1...

# Explanation: ## Step1: Recall rational - number definition A rational number can be written as $\frac{p}{q}$ where $p,q\in\mathbb{Z}$ and $q\neq0$, or as a terminating or repeating decimal. ## Step2: Analyze option A $- 1.\overline{23}$ is a repeating decimal. Repeating decimals are rational. Let $x = - 1.2323\cdots$. Then $100x=-123.2323\cdots$ and $100x - x=-123.2323\cdots-(-1.2323\cdots)$, $99x=-122$, $x =-\frac{122}{99}$, so it is rational. ## Step3: Analyze option B $\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}$. Since $\sqrt{2}$ is an irrational number (it cannot be written as a fraction $\frac{p}{q}$ with $p,q\in\mathbb{Z}$ and $q\neq0$) and $2\sqrt{2}$ is a non - zero multiple of $\sqrt{2}$, $\sqrt{8}$ is irrational. ## Step4: Analyze option C $\frac{2}{3}$ is in the form of $\frac{p}{q}$ where $p = 2$ and $q = 3$ ($p,q\in\mathbb{Z}$ and $q\neq0$), so it is rational. ## Step5: Analyze option D $0$ can be written as $\frac{0}{1}$, so it is rational. # Answer: B. $\sqrt{8}$