数学-速算-心算法

最近看了个印度人讲的速算法则，这才知道这心算原来也是有技巧的啊。  当然印度人的英语发音实在不能恭维，适应了这么多年我还是听了感觉想死的心都有。

anyway，整理了一下，跟大家分享. 

1. Multiplicand & Multiplier
eg: 43 x 23 : 43 is Multiplicand, and 23 is Multiplier.


2. Multiplication by 11 
<1>: The first number of the multiplicand is put down as the left-hand number of the answer.
<2>: each successive number of the multiplicand is added to its neighbor at the left. 
if the addtion results in two figures, carry over the 1.
<3>: The last number of the multiplicand becomes the right hand number of the answer.
乘11相比较来说还是比较好算的: 第一位数字和最后一位数字不变，中间的数字就是两个数字的相加。 

eg:
22 x 11 = 242 (2 | (2+2) | 2 )
423 x 11 = 4653
726 x 11 = 7986
7986 x 11 = 87846
4653 x 11 = 51183


2. Multiplication by 12
<1>: The first number of the multiplicand is put down as the left-hand number of the answer.
<2>: Each number of the multiplicand is multiplied by 2 and added to its neighbor at the right.
if the addition results in two figures, carry out the 1.
<3>: The last number of the multiplicand is multiplied by 2 and it becomes the right hand number
of the answer.

乘12,  第1位数字保持不变，被乘数的数字每个 乘 2 然后加上右边的数字成为1位，最后一位则乘2后成为新数字的最后一位。

323 x 12 = 3876   ( 3 | (3*2+3) | (2*2+3) | 3*2 ) 
3876 x 12 = 46512   ( 3 | (3*2+8)=14 | (8*2+7)=23 | (7*2+6)=20| 6*2=12 )  
152 x 12 = 1824    (1 | (1*2+5)=7 | (5*2+2)=12 | 2*2=4)
254 x 12 = 3048     (2 | (2*2+5)=9 | (5*2+4)=14 | 4*2=8)


3. Simplified Halves , used by Multiplication by 5, 6 & 8 , odd numbers drop fractions.
这就是Python 的地板除.
5 / 2 = 2 
4 / 2 = 2


4. Multiplication by 6
<1>: The first number of the multiplicand is divided by 2(halved),and put down as the left-hand
number of the answer.
<2>: Each following number of the multiplicand is divided by 2(halved) and added to its neighbor at the left.
if the neighbor is odd, 5 is added.
<3>: The last number of the multiplicand becomes the right hand number of the answer. if odd, 5 is added.
乘6:
乘6: 第一位数字除2(地板除)成为第一位，每一位数字除2(地板除)加上数字左边的数字，如果这一位数字是奇数，则加上5，成为一位，大于10则进位。
最后一位如果是奇数则加上5成为最后一位，否则不变。

246 x 6 = 1476  (2/2 | 4/2+2=4 | 6/2+4=7 | 6  )
231 x 6 = 1386  (2/2 | 3/2+2=3 | 1/2+3+5=8 | 1+5 )
451 x 6 = 2706  (4/2 | 5/2+4=6 | 1/2+5+5=10 | 1+5 )
1476 x 6 = 8856 (1/2 | 4/2+1+5=8 | 7/2+4=7 |6/2+7+5=15 |6)


5. Multiplication by 7
<1>: The first number of the multiplicand is divided by 2(halved),and put down as the left-hand
number of the answer.
<2>: Each following number of the multiplicand is multiplied by 2 and added to the half of its neighbor at the right.
if the number is odd, 5 is added.
<3>: The last number of the multiplicand is doubled and then becomes the right hand number of the answer. if odd, 5 is added.
乘7:
第一位除2(地板除)作为第一位.
每一位都乘2加上右边数字除2(地板除)，如果这一位数字是奇数，则加5. 
最后一位乘2成为最后一位，如果是奇数，则加5.

231 x 7 = 1617  (2/2=1 | 2*2+3/2=5 |3*2+1/2+5=11 | 1*2+5=7 )
563 x 7 = 3941  (5/2=2 | 5*2+6/2+5=18 | 6*2+3/2=13 |3*2+5=11 )
365 x 7 = 2555  (3/2=1 | 3*2+6/2+5=14 | 6*2+5/2=14 |5*2+5=15 )
3941 x 7 = 27587    (3/2=1 | 3*2+9/2+5=15 | 9*2+4/2+5=25 | 4*2+1/2=8 | 1*2+5=7 )


6. Multiplication by 5
<1>: The first number of the multiplicand is divided by 2(halved),and put down as the left-hand
number of the answer.
<2>: The right neighbor of each number of the multiplicand is divided by 2(halved),if the number is odd, 5 added.
<3>: If the last number of multiplicand is even, 0 is added as the right hand number of the answer, 
if odd, 5 is added instead of 0.
乘5:
第一位除2(地板除)，成为新数字的第一位.
每个数字的右边一位数字 除2(地板除), 如果这位数字是奇数，加上5。
最后一位如果是偶数，末位是0，如果奇数，末位是5.

231 x 5 = 1155  (2/2=1 | 3/2=1 &(2 is Even) | 1/2+5=5 (1 is odd) |5 )
844 x 5 = 4220  (8/2=4 | 4/2=2 &(8 is Even) | 4/2=2 &(4 is Even) | 0 )
657 x 5 = 3285  (6/2=3 | 5/2=2 &(6 is Even) | 7/2+5=8 &(5 is odd) | 5 )
3285 x 5 = 16425    (3/2=1 | 2/2+5=6 &(3 is odd) | 8/2=4 | 5/2=2 &(8 is Even) |5 )


7. Multiplication by 9
<1>: The first number of the multiplicand is subtracted by 1,and put down as the left-hand
number of the answer.
<2>: Each following number of the multiplicand is subtracted from 9 and added to its neighbor on the right.
<3>: The last number of the multiplicand is subtracted from 10 becomes the right hand number of the answer. 
No Step 2 for one digit number 
乘9:
第一位减去1成为第一位.
每一位被9减，然后加上右边的数字。
最后一位被10减. 

771 x 9 = 6939 (7-1=6 | 9-7+7=9 |9-7+1=3 | 10-1=9 )
523 x 9 = 4707  (5-1=4 | 9-5+2=6 |9-2+3=10 | 10-3=7 )
7 x 9 = 63      (7-1=6 | 10-7=3 )


8. Multiplication by 8
<1>: The first number of the multiplicand is subtracted by 2,and put down as the left-hand
number of the answer.
<2>: Each following number of the multiplicand is subtracted from 9 and doubled before adding it to its neighbor on the right.
<3>: The last number of the multiplicand is subtracted by 10 and doubled becomes the right hand number of the answer. 
乘8 
第一位减去2，成为第一位。
每一位都被9减去然后乘2加上右边的数字.
最后一位被10来减后乘2成为最后一位.
eg:
986 x 8 = 7888  (9-2=7 | (9-9)*2+8=8 | (9-8)*2+6=8  | 2*(10-6)=8 )
753 x 8 = 6024  (7-2=5 | (9-7)*2+5=9 | (9-5)*2+3=11 | 2*(10-3)=14 )
878 x 8 = 7024  (8-2=6 | (9-8)*2+7=9 | (9-7)*2+8=12 | 2*(10-8)=4 )

9. Multiplication by 4
<1>: The first number of the multiplicand is divided by 2(halved) and subtracted by 1,and put down as the left-hand
number of the answer.
<2>: Each following number of the multiplicand is subtracted from 9 and added to the half of its neighbor on the right.
if the number if odd, 5 is added.
<3>: The last number of the multiplicand is subtracted by 10 becomes the right hand number of the answer. 
if the number is odd, 5 is added.

eg:
772 x 4 = 3088   ( 7/2-1=2 | 9-7+7/2+5=10 | 9-7+2/2+5=8 | 10-2=8 )
456 x 4 = 1824   ( 4/2-1=1 | 9-4+5/2=7 | 9-5+6/2+5=12 | 10-6=4 )
872 x 4 = 3488   ( 8/2-1=3 | 9-8+7/2=4 | 9-7+2/2+5=8 | 10-2=8 )


Squaring: 
1. Numbers  from 1 to 99 ,Ending with 5 
<1>. Multiply the first digit with the next higher digit to get the first numbers of the answer.
<2>. Attach 25 to the first numbers to get the full answer.
eg:
65 x 65 = 4225
85 x 85 = 7225

2. Other number from 1 to 99.


64 x 64 =
<1>: round up to the nearest multiple of 10 , and change another number accordingly, multiply them.
60 x 68 =  4080
<2>: square the number rounded up/down and add it the number from previous step.
64 x 64 = 4080 + 4 x 4 = 4096

3. 1000以内的数相乘
eg:
32 x 13 = 416   (3x1=3 | 2x1+3*3=11 | 2x3=6)
416 x 13 = 5408 (3x1=4 | 1x1+3x4=13 |1x3+6x1=9 | 3x6=18 )
323 x 132 = 42436    (3x1=3 | 3x3+1x2=11 |3x2+2x3+3x1=13 |2x2+3x3=13 |6)


<1>. Multiply the first number of the multiplicand with the first number of the multiplier and put 
the result as the left hand number of the answer.
<2>. Multiply the outside pair, and multiply the inside pair.
<3>. Add the products of the outside and inside pairs together to get the second figure of the answer.
<4>. repeat the second set of pairs.(Repetition of step 2)
<5>. Add the products of the second outside and inside pairs together to get the third figure of the answer.
<6>. Multiply the last set of pairs(Repetion of step 2)
<7>. Multiply the last number of the multiplicand with the second number of the multiplier and put 
the result as the right hand number of the answer.