數與數線(4)
Author: 藍邦偉　老師

(A)0.2\overline{15}=\frac{215-2}{990}=\frac{213}{990}=\frac{71}{330}

(B)\frac{203}{140}=\frac{29}{20}=1.45

(C)設a=α+\sqrt{β},b=α-\sqrt{β}>0,α∈\mathbb{Q}^+,\sqrt{β}∈\mathbb{Q}'

⇒a+b=2α∈\mathbb{Q},a-b=2\sqrt{β}∈\mathbb{Q}'

⇒a²-b²=4α\sqrt{β}∈\mathbb{Q}'

b-\frac{\sqrt{2}a+\sqrt{3}b}{\sqrt{6}}=\frac{(\sqrt{6}-\sqrt{3})b-\sqrt{2}a}{\sqrt{6}}

又(\sqrt{6}-\sqrt{3})²=9-6\sqrt{2}<\sqrt{2}²=2

∴b-\frac{\sqrt{2}a+\sqrt{3}b}{\sqrt{6}}<0

$ |a-2b|≤|a| +2|b|<|a|+3|b|$

n = 0 (slider from 0 to 10, step 1)