数学小记之常用数值

怼烎@ 2024-04-18 19:53 67阅读 0赞

> 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 \\pi π 相关的数值想必大部分朋友都比较熟悉了:

π ≈ 3.14 π 2 ≈ 1.57 2 π ≈ 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  e e e 的数值:

e ≈ 2.718 e \\approx 2.718 e≈2.718

*  一些常用的根号值:

1 = 1 2 ≈ 1.414 3 ≈ 1.732 4 = 2 5 ≈ 2.236 6 ≈ 2.44949 7 ≈ 2.64575 8 = 2 2 ≈ 2.828 9 = 3 10 ≈ 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:

s i n ( 0 ) = s i n ( 0 ° ) = 0 s i n ( π 6 ) = s i n ( 30 ° ) = 0.5 s i n ( π 4 ) = s i n ( 45 ° ) = 2 2 ≈ 0.707 s i n ( π 3 ) = s i n ( 60 ° ) = 3 2 ≈ 0.866 s i n ( π 2 ) = s i n ( 90 ° ) = 1 s i n ( 2 π 3 ) = s i n ( 120 ° ) = 3 2 ≈ 0.866 s i n ( 5 π 6 ) = s i n ( 150 ° ) = 0.5 s i n ( π ) = s i n ( 180 ° ) = 0 c o s ( 0 ) = c o s ( 0 ° ) = 1 c o s ( π 6 ) = c o s ( 30 ° ) = 3 2 ≈ 0.866 c o s ( π 4 ) = c o s ( 45 ° ) = 2 2 ≈ 0.707 c o s ( π 3 ) = c o s ( 60 ° ) = 0.5 c o s ( π 2 ) = c o s ( 90 ° ) = 0 c o s ( 2 π 3 ) = c o s ( 120 ° ) = − 0.5 c o s ( 5 π 6 ) = c o s ( 150 ° ) = − 3 2 ≈ − 0.866 c o s ( π ) = c o s ( 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:

l o g 10 2 ≈ 0.3010 l o g 10 3 ≈ 0.4771 \\begin\{aligned\} & log\_\{10\}2 \\approx 0.3010 \\\\ & log\_\{10\}3 \\approx 0.4771 \\end\{aligned\} log102≈0.3010log103≈0.4771

*   2 2 2 的幂次在计算机领域应该是最常见的了~

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 \\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  2 10 2^\{10\} 210 时,我们可以近似的将其当作  1000 1000 1000 来进行处理,譬如估算内存占用时我们得到了  1000 K B 1000KB 1000KB 大小的数值,则可以近似认为是  1 M B 1MB 1MB(实际而言,  1 M B 1MB 1MB 应该等于  1024 K B ( 2 10 K B ) 1024KB(2^\{10\}KB) 1024KB(210KB))

*   0 0 0 到  20 20 20 的平方数也很常用~

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 1 0 2 = 100 1 1 2 = 121 1 2 2 = 144 1 3 2 = 169 1 4 2 = 196 1 5 2 = 225 1 6 2 = 256 1 7 2 = 289 1 8 2 = 324 1 9 2 = 361 2 0 2 = 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  5 5 5 的数值(譬如  65 65 65)的平方有个小技巧,可以加速我们的运算:

以  65 65 65 为例,这个数字的十位数字为  6 6 6,我们首先计算  6 6 6 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  42 42 42 与  25 25 25 组合( 42 ∣ 25 42|25 42∣25),即可得  65 65 65 的平方

6 5 2 = 42 ∣ 25 65^2 = 42|25 652=42∣25

总结一下上面的规则就是: 对于形如  a 5 a5 a5(即  10 ∗ a + 5 10 \* a + 5 10∗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~