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数学小记之常用数值

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数学小记之常用数值

tkokof1 2019-09-11 12:43:57 331

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

amp;π≈3.14amp;π2≈1.57amp;2π≈6.28amp;π≈3.14amp;π2≈1.57amp;2π≈6.28
​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

amp;1–√=1amp;2–√≈1.414amp;3–√≈1.732amp;4–√=2amp;5–√≈2.236amp;6–√≈2.44949amp;7–√≈2.64575amp;8–√=22–√≈2.828amp;9–√=3amp;10−−√≈3.162amp;1=1amp;2≈1.414amp;3≈1.732amp;4=2amp;5≈2.236amp;6≈2.44949amp;7≈2.64575amp;8=22≈2.828amp;9=3amp;10≈3.162
​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

amp;sin(0)=sin(0\degree)=0amp;sin(π6)=sin(30\degree)=0.5amp;sin(π4)=sin(45\degree)=2–√2≈0.707amp;sin(π3)=sin(60\degree)=3–√2≈0.866amp;sin(π2)=sin(90\degree)=1amp;sin(2π3)=sin(120\degree)=3–√2≈0.866amp;sin(5π6)=sin(150\degree)=0.5amp;sin(π)=sin(180\degree)=0amp;cos(0)=cos(0\degree)=1amp;cos(π6)=cos(30\degree)=3–√2≈0.866amp;cos(π4)=cos(45\degree)=2–√2≈0.707amp;cos(π3)=cos(60\degree)=0.5amp;cos(π2)=cos(90\degree)=0amp;cos(2π3)=cos(120\degree)=−0.5amp;cos(5π6)=cos(150\degree)=−3–√2≈−0.866amp;cos(π)=cos(180\degree)=−1amp;sin(0)=sin(0\degree)=0amp;sin(π6)=sin(30\degree)=0.5amp;sin(π4)=sin(45\degree)=22≈0.707amp;sin(π3)=sin(60\degree)=32≈0.866amp;sin(π2)=sin(90\degree)=1amp;sin(2π3)=sin(120\degree)=32≈0.866amp;sin(5π6)=sin(150\degree)=0.5amp;sin(π)=sin(180\degree)=0amp;cos(0)=cos(0\degree)=1amp;cos(π6)=cos(30\degree)=32≈0.866amp;cos(π4)=cos(45\degree)=22≈0.707amp;cos(π3)=cos(60\degree)=0.5amp;cos(π2)=cos(90\degree)=0amp;cos(2π3)=cos(120\degree)=−0.5amp;cos(5π6)=cos(150\degree)=−32≈−0.866amp;cos(π)=cos(180\degree)=−1
​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

amp;log102≈0.3010amp;log103≈0.4771amp;log102≈0.3010amp;log103≈0.4771
​log10​2≈0.3010log10​3≈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

amp;20=1amp;21=2amp;22=4amp;23=8amp;24=16amp;25=32amp;26=64amp;27=128amp;28=256amp;29=512amp;210=1024amp;211=2048amp;212=4096amp;213=8192amp;214=16384amp;215=32768amp;216=65536amp;20=1amp;21=2amp;22=4amp;23=8amp;24=16amp;25=32amp;26=64amp;27=128amp;28=256amp;29=512amp;210=1024amp;211=2048amp;212=4096amp;213=8192amp;214=16384amp;215=32768amp;216=65536
​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

amp;02=0amp;12=2amp;22=4amp;32=9amp;42=16amp;52=25amp;62=36amp;72=49amp;82=64amp;92=81amp;102=100amp;112=121amp;122=144amp;132=169amp;142=196amp;152=225amp;162=256amp;172=289amp;182=324amp;192=361amp;202=400amp;02=0amp;12=2amp;22=4amp;32=9amp;42=16amp;52=25amp;62=36amp;72=49amp;82=64amp;92=81amp;102=100amp;112=121amp;122=144amp;132=169amp;142=196amp;152=225amp;162=256amp;172=289amp;182=324amp;192=361amp;202=400
​02=012=222=432=942=1652=2562=3672=4982=6492=81102=100112=121122=144132=169142=196152=225162=256172=289182=324192=361202=400​

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~


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