数学小记之常用数值

  • 2019 年 10 月 4 日
  • 筆記

版权声明:本文为博主原创文章,遵循 CC 4.0 BY-SA 版权协议,转载请附上原文出处链接和本声明。

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

  • πpiπ 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 \ & 2pi 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} = 2sqrt{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(0degree) = 0 \ & sin(frac{pi}{6}) = sin(30degree) = 0.5 \ & sin(frac{pi}{4}) = sin(45degree) = frac{sqrt{2}}{2} approx 0.707 \ & sin(frac{pi}{3}) = sin(60degree) = frac{sqrt{3}}{2} approx 0.866 \ & sin(frac{pi}{2}) = sin(90degree) = 1 \ & sin(frac{2pi}{3}) = sin(120degree) = frac{sqrt{3}}{2} approx 0.866 \ & sin(frac{5pi}{6}) = sin(150degree) = 0.5 \ & sin(pi) = sin(180degree) = 0 \ \hline \ & cos(0) = cos(0degree) = 1 \ & cos(frac{pi}{6}) = cos(30degree) = frac{sqrt{3}}{2} approx 0.866 \ & cos(frac{pi}{4}) = cos(45degree) = frac{sqrt{2}}{2} approx 0.707 \ & cos(frac{pi}{3}) = cos(60degree) = 0.5 \ & cos(frac{pi}{2}) = cos(90degree) = 0 \ & cos(frac{2pi}{3}) = cos(120degree) = -0.5 \ & cos(frac{5pi}{6}) = cos(150degree) = -frac{sqrt{3}}{2} approx -0.866 \ & cos(pi) = cos(180degree) = -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~