数学小记之常用数值

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本文链接：https://blog.csdn.net/tkokof1/article/details/100732835

本文简单列举了一些常用数值,熟记这些数值可以方便我们进行数学运算

• π\piπ 相关的数值想必大部分朋友都比较熟悉了:

π≈3.14π2≈1.572π≈6.28 \begin{aligned} & \pi \approx 3.14 \\ & \frac{\pi}{2} \approx 1.57 \\ & 2\pi \approx 6.28 \end{aligned} ​π≈3.142π​≈1.572π≈6.28​

• 自然数 eee 的数值:

e≈2.718 e \approx 2.718 e≈2.718

• 一些常用的根号值:

1=12≈1.4143≈1.7324=25≈2.2366≈2.449497≈2.645758=22≈2.8289=310≈3.162 \begin{aligned} & \sqrt{1} = 1 \\ & \sqrt{2} \approx 1.414 \\ & \sqrt{3} \approx 1.732 \\ & \sqrt{4} = 2 \\ & \sqrt{5} \approx 2.236 \\ & \sqrt{6} \approx 2.44949 \\ & \sqrt{7} \approx 2.64575 \\ & \sqrt{8} = 2\sqrt{2} \approx 2.828 \\ & \sqrt{9} = 3 \\ & \sqrt{10} \approx 3.162 \end{aligned} ​1​=12​≈1.4143​≈1.7324​=25​≈2.2366​≈2.449497​≈2.645758​=22​≈2.8289​=310​≈3.162​

• 一些常用的三角函数值:

sin(0)=sin(0°)=0sin(π6)=sin(30°)=0.5sin(π4)=sin(45°)=22≈0.707sin(π3)=sin(60°)=32≈0.866sin(π2)=sin(90°)=1sin(2π3)=sin(120°)=32≈0.866sin(5π6)=sin(150°)=0.5sin(π)=sin(180°)=0cos(0)=cos(0°)=1cos(π6)=cos(30°)=32≈0.866cos(π4)=cos(45°)=22≈0.707cos(π3)=cos(60°)=0.5cos(π2)=cos(90°)=0cos(2π3)=cos(120°)=−0.5cos(5π6)=cos(150°)=−32≈−0.866cos(π)=cos(180°)=−1 \begin{aligned} & sin(0) = sin(0\degree) = 0 \\ & sin(\frac{\pi}{6}) = sin(30\degree) = 0.5 \\ & sin(\frac{\pi}{4}) = sin(45\degree) = \frac{\sqrt{2}}{2} \approx 0.707 \\ & sin(\frac{\pi}{3}) = sin(60\degree) = \frac{\sqrt{3}}{2} \approx 0.866 \\ & sin(\frac{\pi}{2}) = sin(90\degree) = 1 \\ & sin(\frac{2\pi}{3}) = sin(120\degree) = \frac{\sqrt{3}}{2} \approx 0.866 \\ & sin(\frac{5\pi}{6}) = sin(150\degree) = 0.5 \\ & sin(\pi) = sin(180\degree) = 0 \\ \\\hline \\ & cos(0) = cos(0\degree) = 1 \\ & cos(\frac{\pi}{6}) = cos(30\degree) = \frac{\sqrt{3}}{2} \approx 0.866 \\ & cos(\frac{\pi}{4}) = cos(45\degree) = \frac{\sqrt{2}}{2} \approx 0.707 \\ & cos(\frac{\pi}{3}) = cos(60\degree) = 0.5 \\ & cos(\frac{\pi}{2}) = cos(90\degree) = 0 \\ & cos(\frac{2\pi}{3}) = cos(120\degree) = -0.5 \\ & cos(\frac{5\pi}{6}) = cos(150\degree) = -\frac{\sqrt{3}}{2} \approx -0.866 \\ & cos(\pi) = cos(180\degree) = -1 \end{aligned} ​sin(0)=sin(0°)=0sin(6π​)=sin(30°)=0.5sin(4π​)=sin(45°)=22​​≈0.707sin(3π​)=sin(60°)=23​​≈0.866sin(2π​)=sin(90°)=1sin(32π​)=sin(120°)=23​​≈0.866sin(65π​)=sin(150°)=0.5sin(π)=sin(180°)=0cos(0)=cos(0°)=1cos(6π​)=cos(30°)=23​​≈0.866cos(4π​)=cos(45°)=22​​≈0.707cos(3π​)=cos(60°)=0.5cos(2π​)=cos(90°)=0cos(32π​)=cos(120°)=−0.5cos(65π​)=cos(150°)=−23​​≈−0.866cos(π)=cos(180°)=−1​​

• 两个常用的对数值:

log102≈0.3010log103≈0.4771 \begin{aligned} & log_{10}2 \approx 0.3010 \\ & log_{10}3 \approx 0.4771 \end{aligned} ​log10​2≈0.3010log10​3≈0.4771​

• 222 的幂次在计算机领域应该是最常见的了~

20=121=222=423=824=1625=3226=6427=12828=25629=512210=1024211=2048212=4096213=8192214=16384215=32768216=65536 \begin{aligned} & 2^0 = 1 \\ & 2^1 = 2 \\ & 2^2 = 4 \\ & 2^3 = 8 \\ & 2^4 = 16 \\ & 2^5 = 32 \\ & 2^6 = 64 \\ & 2^7 = 128 \\ & 2^8 = 256 \\ & 2^9 = 512 \\ & 2^{10} = 1024 \\ & 2^{11} = 2048 \\ & 2^{12} = 4096 \\ & 2^{13} = 8192 \\ & 2^{14} = 16384 \\ & 2^{15} = 32768 \\ & 2^{16} = 65536 \\ \end{aligned} ​20=121=222=423=824=1625=3226=6427=12828=25629=512210=1024211=2048212=4096213=8192214=16384215=32768216=65536​

有时候出于方便,遇到 2102^{10}210 时,我们可以近似的将其当作 100010001000 来进行处理,譬如估算内存占用时我们得到了 1000KB1000KB1000KB 大小的数值,则可以近似认为是 1MB1MB1MB(实际而言, 1MB1MB1MB 应该等于 1024KB(210KB)1024KB(2^{10}KB)1024KB(210KB))

• 000 到 202020 的平方数也很常用~

02=012=222=432=942=1652=2562=3672=4982=6492=81102=100112=121122=144132=169142=196152=225162=256172=289182=324192=361202=400 \begin{aligned} & 0^2 = 0 \\ & 1^2 = 2 \\ & 2^2 = 4 \\ & 3^2 = 9 \\ & 4^2 = 16 \\ & 5^2 = 25 \\ & 6^2 = 36 \\ & 7^2 = 49 \\ & 8^2 = 64 \\ & 9^2 = 81 \\ & 10^2 = 100 \\ & 11^2 = 121 \\ & 12^2 = 144 \\ & 13^2 = 169 \\ & 14^2 = 196 \\ & 15^2 = 225 \\ & 16^2 = 256 \\ & 17^2 = 289 \\ & 18^2 = 324 \\ & 19^2 = 361 \\ & 20^2 = 400 \\ \end{aligned} ​02=012=222=432=942=1652=2562=3672=4982=6492=81102=100112=121122=144132=169142=196152=225162=256172=289182=324192=361202=400​

番外

求解个位数为 555 的数值(譬如 656565)的平方有个小技巧,可以加速我们的运算:

以 656565 为例,这个数字的十位数字为 666,我们首先计算 666 与 (6+1)(6 + 1)(6+1) 的乘积

6∗7=42 6 * 7 = 42 6∗7=42

再将计算得到的 424242 与 252525 组合(42∣2542|2542∣25),即可得 656565 的平方

652=42∣25 65^2 = 42|25 652=42∣25

总结一下上面的规则就是: 对于形如 a5a5a5(即 10∗a+510 * a + 510∗a+5) 这种形式的数字,我们有:

(10∗a+5)2=a∗(a+1)∗100+25 (10 * a + 5)^2 = a * (a + 1) * 100 + 25 (10∗a+5)2=a∗(a+1)∗100+25

有兴趣的朋友可以评论补充更多的常用数值~