有效数字缺失

有效数字

$$d_1.d_2\cdots d_n \times 10^p$$

\begin{aligned} &123.4567\\ -&123.4566\\ &\hline\\ =&000.0001 \end{aligned}

“减法”=“陷阱”

$$\sqrt{9.01}\approx 3.001 662=3.00\\ \sqrt{9.01}-3=0$$

解决方法

$$\sqrt{9.01}-3=\frac{(\sqrt{9.01}-3)(\sqrt{9.01}+3)}{\sqrt{9.01}+3}=\frac{|0.01}{3.00+3}=\frac{1}{6}=1.67\times 10^{-3}$$

$$E_1=\frac{1-\text{cos}x}{\text{sin}^2x}$$

$$E_2=\frac{(1-\text{cos}x)(1+\text{cos}x)}{\text{sin}^2x(1+\text{cos}x)}=\frac{1}{(1+\text{cos}x)}=E_1$$

$x$ $E_1$ $E_2$
0.1000000000000000 0.5012520862885769 0.5012520862885712
0.0100000000000000 0.5000125002084805 0.5000125002083363
0.0010000000000000 0.5000001249921894 0.5000001250000208
0.0001000000000000 0.4999999986279311 0.5000000012500000
0.0000100000000000 0.5000000413868522 0.5000000000125000
0.0000010000000000 0.5000444502913370 0.5000000000001250
0.0000001000000000 0.4996003610813219 0.5000000000000012
0.0000000100000000 0.0000000000000000 0.5000000000000000
0.0000000010000000 0.0000000000000000 0.5000000000000000
0.0000000001000000 0.0000000000000000 0.5000000000000000
0.0000000000100000 0.0000000000000000 0.5000000000000000
0.0000000000010000 0.0000000000000000 0.5000000000000000
0.0000000000001000 0.0000000000000000 0.5000000000000000
0.0000000000000100 0.0000000000000000 0.5000000000000000
0.0000000000000010 0.0000000000000000 0.5000000000000000
0.0000000000000001 0.0000000000000000 0.5000000000000000

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

$$x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}$$

$$x_1=\frac{(-b+\sqrt{b^2-4ac})\cdot(+b+\sqrt{b^2-4ac})}{2a(+b+\sqrt{b^2-4ac})}\\ =\frac{-4ac}{2a(+b+\sqrt{b^2-4ac})}=\frac{-2c}{(b+\sqrt{b^2-4ac})}$$

$$x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}\\ x_2=\frac{2c}{-b+\sqrt{b^2-4ac}}$$

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