Wednesday, August 29, 2012

math - multiplication for kids

2010-1-8 07:28
0   1,2  3,4    5  6,7   8,9    10

* TIP: fingers
* TIP: fingers, b
* TIP: x 10
* TIP: x 5
* TIP: square

=========================


Big 2010-1-8 07:29
* TIP: fingers

9 x _ = hold up ten fingers, put down _th finger (v)
ex. 9 x 2 = * v * * *   * * * * * = 1 v 8 = 18

8 x _ = hold up ten fingers, put down _th finger (v) and following _ fingers (x)
ex. 8 x 2 = * v x x  *   * * * * * = 1 v 6 = 16


* hold up ten fingers, thumb to thumb (10); ^ up, v down;
(6,7,8,9,0) x (0,9,8,7,6)
ex. 8 x 6 = (v,v,v,^,^) x (^,^,^,^,v) = (3v + 1v) x 10 + 2^ x 4^
     = 40 + 8 = 48

(5,6,7,8,9) x (9,8,7,6,5)
ex. 8 x 7 = (^,^,^,v,v) x (v,v,v,^,^) = (3^ + 2^) x 10 + 2v x 3v
     = 50 + 6 = 56
ex. 6 x 6 = (^,v,v,v,v) x (v,v,v,v,^) = (1^ + 1^) x 10 + 4v x 4v
     = 20 + 16 = 36

ex. 8 x 7 = [(8 - 5)+(7 - 5)]x10 + (10 - 8)x(10 - 7) = [3+2]x10 +2x3
     = 50 + 6 = 56
flicker 彩虹炫 | 编辑 删除Big 2010-1-8 07:31
* TIP: x 10

9 x _ = _ x 10 - _ x (10 - 9) = _ x 10 - _ x 1
ex. 9 x 2 = 2 x 10 - 2 = 20 - 2 = 18

8 x _ = _ x 10 - _ x (10 - 8) = _ x 10 - _ x 2
ex. 8 x 2 = 2 x 10 - 2 x 2 = 20 - 4 = 16

7 x _ = _ x 10 - _ x (10 - 7) = _ x 10 - _ x 3
ex. 7 x 2 = 2 x 10 - 2 x 3 = 20 - 6 = 14

6 x _ = _ x 10 - _ x (10 - 6) = _ x 10 - _ x 4
ex. 6 x 2 = 2 x 10 - 2 x 4 = 20 - 8 = 12

4 x _ = _ x 10 - _ x (10 - 4)
ex. 4 x 2 = 2 x 10 - 2 x 6 = 20 - 12 = 8
flicker 彩虹炫 | 编辑 删除Big 2010-1-8 07:35
* TIP: x 5

6 x _ = _ x 5 + _ x (6 - 5) = _ x 5 + _ x 1
ex. 6 x 4 = 4 x 5 + 4 = 20 + 4 = 24

ex. 6 x 4 = 4//2 x 10 + 4 = 20 + 4 = 24
ex. 6 x 7 = 7//2 x 10 + 7 + 5 == 30 + 7 + 5 = 42, if x-odd, + 5

7 x _ = _ x 5 + _ x (7 - 5) = _ x 5 + _ x 2
ex. 7 x 4 = 4 x 5 + 4 x 2 = 20 + 8 = 28

8 x _ = _ x 5 + _ x (8 - 5) = _ x 5 + _ x 3
ex. 8 x 4 = 4 x 5 + 4 x 3 = 20 + 12 = 32

9 x _ = _ x 5 + _ x (9 - 5) = _ x 5 + _ x 4
ex. 9 x 4 = 4 x 5 + 4 x 4 = 20 + 16 = 36

* * * * * * * * * * * *

4 x _ = _ x 5 - _ x (5 - 4)
ex. 4 x 4 = 4 x 5 - 4 = 20 - 4 = 16

3 x _ = _ x 5 - _ x (5 - 3)
ex. 3 x 4 = 4 x 5 - 4 x 2 = 20 - 8 = 12
flicker 彩虹炫 | 编辑 删除Big 2010-1-8 07:45
* TIP: fingers, b

9 x _ = (_ - 1) x 10 + (10 - _)
ex. 9 x 3 = (3 - 1) x 10 + (10 - 3) = 20 + 7 = 27

8 x _ = (_ - 2) x 10 + (10 - _) x 2
ex. 8 x 3 = (3 - 2) x 10 + (10 - 3) x 2 = 10 + 14 = 24

7 x _ = (_ - 3) x 10 + (10 - _) x 3
ex. 7 x 3 = (3 - 3) x 10 + (10 - 3) x 3 = 0 + 21 = 21
ex. 7 x 7 = (7 - 3) x 10 + (10 - 7) x 3 = 40 + 9 = 49

(5,6,7,8,9) x (9,8,7,6,5)
ex. 8 x 7 = [(8 - 5)+(7 - 5)]x10 + (10 - 8)x(10 - 7) = [3+2]x10 +2x3
     = 50 + 6 = 56
flicker 彩虹炫 | 编辑 删除Big 2010-1-8 07:54
* TIP: square

0
   1 2 3
   4 5 6
   7 8 9

a. start with _x1
b. next to _x2, then column to column, or row to row
c. odd-x goes Z shape
d. even-x goes N shape

g. go left, then +10; not down-left
h. go up, then +10

* * * * * * * * * * *
odd-x

   1 2 3
   4 5 6
   7 8 9

a. start with _x1
b. next to _x2, then column to column, or row to row
c. odd-x goes Z shape

g. go left, then +10; not down-left
h. go up, then +10



* * * * * * * * * * *
even-x

0 2 4
   6 8

a. start with _x1
b. next to _x2, then column to column, or row to row
d. even-x goes N shape

g. go left, then +10; not down-left
h. go up, then +10
flicker 彩虹炫 | 编辑 删除Big 2010-1-21 11:06
4 x _ = _ + _ + _ + _ , double double
ex, 4 x 6 = 6 + 6 + 6 + 6 = 12 + 12 = 24

6 x _ = _ %2 x10 + _ [+ 5, if odd]
ex, 6 x 8 = 4 x 10 + 8 = 48

4 x _ = 5 x _ - _
ex, 4 x 8 = 5 x 8 - 8 = 32

5 x _ =
four fingers up, 5    /10\  15   /20\   25  /30\   35   /40\   45
                           x1   x2    x3    x4    x5    x6    x7    x8    x9
ex, 5 x 5 = . /\ . /\ . = 2/\ x 10 + 5 = 25

6 x _ = 5 x _ + _
ex, 6 x 7 = 5 x 7 + 7 = 42

* * * * * * * * * * * * * * * * * * * * *

8 x _ =
ten fingers up,
if _ < 6, _th finger down, more _ fingers down;
if _ > 5, _th finger down, back one finger down, double the number of right fingers;

9 x _ =
ten fingers up, _th finger down

a x b = b x 10 - b x  (10 - a)
ex, 8 x 3 = 30 - 3 x 2
flicker 彩虹炫 | 编辑 删除Big 2010-1-23 00:08
3 x 6 = 30 - 3 x (10 - 6) = 30 - 12 = 18
or, 3 x 6 = (3 x 5) x 2 - 3 x (5 + (5 - 6)) = 30 - 12 = 18
9 x 8 = 90 - 9 x 2 = 72

* * * * * * * * * * * *

a x b = b x 10 - b x (10 - a)

9 x 8 = 80 - 8 = 72
8 x 9 = 90 - 18
9 x 2 = 20 - 2 = 18
flicker 彩虹炫 | 编辑 删除Big 2011-9-7 05:01
小数
1。有限小数  1/2=0.5
2。无限小数  3.14159.....
2.1。循环小数  1/3=0.'333.....
2.1.1。有前导部  1/6=0.1'666.....
2.1.2。无前导部 1/7=0.'142857.....
            142,857 : 1 +3 4 -2 2, 8 -3 5 +2 7 : +3 -2, -3 +2
            142,857 : 1 -7 4 +8 2, 8 -3 5 +2 7 : -7 +8, -3 +2
2.1.3。循环部为偶数长
2.1.3.1。循环部叠对位和为9   1/7=0.'142,857..... 1+8=4+5=2+7=9
2.1.3.2。循环部叠对位和为6   1/21=0.'047,619.....  0+6=6, 47+19=66,
2.1.3.3。循环部叠对位和为3   2/21=0.'095,238"..... 095+238=333
2.1.3.4。循环部叠对位和大于10   1/21=0.'047,619.....  7+9=16
2.1.4。循环部为奇数长  1/27=0.'037"...
2.2。非循环小数  e=2.71828.....
flicker 彩虹炫 | 编辑 删除Big 2012-2-9 12:55
0.2_2... => 2 / (10 - 1)
0.27_27... => 27 / (100 - 1)
0.11_345_345... => (11_345 - 11) / (100 - 1) * 100

---------------------------------------------

R0 = non-recurrsive
R2 = recurrsive, even number long of recurrsive segment
R3 = recurrsive, odd number long of recurrsive segment
len = length of recurrsive segment

_decimal_21.htm

1/7=0.14285714285714285
1/7=142857142857142857142857142857142857142857142857142857142857142857


_decimal_22.htm

1 / 2 = 0.5000000000
1 / 3 = 0.33333333333333333333333333333333333333333333333333333333333333333333
1 / 4 = 0.25000000000
1 / 7 = 0.1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428


_decimal_23.htm

1 / 2 = 0.5, nr
1 / 3 = 0. +3, r 3
1 / 7 = 0.1428 +571428, r 571428
1 / 97 = 0 +.0103092783505154639175257731958762886597938144329896907216494845360824742268041237113402061855670103, r .0103092783505154639175257731958762886597938144329896907216494845360824742268041237113402061855670103
1 / 98 = 0.0102040816326530 +612244897959183673469387755102040816326530, r 612244897959183673469387755102040816326530


_decimal_24.htm

1 / 2 = 0.5, nr
1 / 3 = 0. +3, r 3
1 / 7 = 0. +142857, r 142857
1 / 97 = 0. +010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567, r 010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567
1 / 98 = 0.0 +102040816326530612244897959183673469387755, r 102040816326530612244897959183673469387755


_decimal_25.htm

1 / 2 = 0.5, nr
1 / 3 = 0. + 3, r3,
1 / 7 = 0. + 142_857 // 9, r2,

1 / 27 = 0. + 0_3_7, R3, len 3
1 / 31 = 0. + 0322580_6_4516129, R3, len 15
1 / 37 = 0. + 0_2_7, R3, len 3
1 / 41 = 0. + 02_4_39, R3, len 5
1 / 43 = 0. + 02325 58139_5_34883 72093, R3, len 21
1 / 53 = 0. + 018867_9_245283, R3, len 13
1 / 54 = 0.0 + 1_8_5, R3, len 3
1 / 62 = 0.0 + 1612903_2_2580645, R3, len 15
flicker 彩虹炫 | 编辑 删除Big 2012-2-9 12:56
1 / 2 = 0.5, R0
1 / 3 = 0. + 3, R3, len 1
1 / 4 = 0.25, R0
1 / 5 = 0.2, R0
1 / 6 = 0.1 + 6, R3, len 1
1 / 7 = 0. + 142_857 // 9, R2, len 6
1 / 8 = 0.125, R0
1 / 9 = 0. + 1, R3, len 1
1 / 10 = 0.1, R0
1 / 11 = 0. + 0_9 // 9, R2, len 2
1 / 12 = 0.08 + 3, R3, len 1
1 / 13 = 0. + 076_923 // 9, R2, len 6
1 / 14 = 0.0 + 714_285 // 9, R2, len 6
1 / 15 = 0.0 + 6, R3, len 1
1 / 16 = 0.0625, R0
1 / 17 = 0. + 05882352_94117647 // 9, R2, len 16
1 / 18 = 0.0 + 5, R3, len 1
1 / 19 = 0. + 052631578_947368421 // 9, R2, len 18
1 / 20 = 0.05, R0
1 / 21 = 0. + 047_619 // 6, R2, len 6
1 / 22 = 0.0 + 4_5 // 9, R2, len 2
1 / 23 = 0. + 04347826086_95652173913 // 9, R2, len 22
1 / 24 = 0.041 + 6, R3, len 1
1 / 25 = 0.04, R0
1 / 26 = 0.0 + 384_615 // 9, R2, len 6
1 / 27 = 0. + 037, R3, len 3
1 / 28 = 0.03 + 571_428 // 9, R2, len 6
1 / 29 = 0. + 03448275862068_96551724137931 // 9, R2, len 28
1 / 30 = 0.0 + 3, R3, len 1
1 / 31 = 0. + 032258064516129, R3, len 15
1 / 32 = 0.03125, R0
1 / 33 = 0. + 0_3 // 3, R2, len 2
1 / 34 = 0.0 + 29411764_70588235 // 9, R2, len 16
1 / 35 = 0.0 + 285_714 // 9, R2, len 6
1 / 36 = 0.02 + 7, R3, len 1
1 / 37 = 0. + 027, R3, len 3
1 / 38 = 0.0 + 263157894_736842105 // 9, R2, len 18
1 / 39 = 0. + 025_641 // 6, R2, len 6
1 / 40 = 0.025, R0
1 / 41 = 0. + 02439, R3, len 5
1 / 42 = 0.0 + 238_095 // 2, R2, len 6
1 / 43 = 0. + 023255813953488372093, R3, len 21
1 / 44 = 0.02 + 2_7 // 9, R2, len 2
1 / 45 = 0.0 + 2, R3, len 1
1 / 46 = 0.0 + 21739130434_78260869565 // 9, R2, len 22
1 / 47 = 0. + 02127659574468085106382_97872340425531914893617 // 9, R2, len 46
1 / 48 = 0.0208 + 3, R3, len 1
1 / 49 = 0. + 020408163265306122448_979591836734693877551 // 9, R2, len 42
1 / 50 = 0.02, R0
1 / 51 = 0. + 01960784_31372549 // 3, R2, len 16
1 / 52 = 0.01 + 923_076 // 9, R2, len 6
1 / 53 = 0. + 0188679245283, R3, len 13
1 / 54 = 0.0 + 185, R3, len 3
1 / 55 = 0.0 + 1_8 // 9, R2, len 2
1 / 56 = 0.017 + 857_142 // 9, R2, len 6
1 / 57 = 0. + 017543859_649122807 // 6, R2, len 18
1 / 58 = 0.0 + 17241379310344_82758620689655 // 9, R2, len 28
1 / 59 = 0. + 01694915254237288135593220338_98305084745762711864406779661 // 9, R2, len 58
1 / 60 = 0.01 + 6, R3, len 1
1 / 61 = 0. + 016393442622950819672131147540_983606557377049180327868852459 // 9, R2, len 60
1 / 62 = 0.0 + 161290322580645, R3, len 15
1 / 63 = 0. + 015_873 // 8, R2, len 6
1 / 64 = 0.015625, R0
1 / 65 = 0.0 + 153_846 // 9, R2, len 6
1 / 66 = 0.0 + 1_5 // 6, R2, len 2
1 / 67 = 0. + 014925373134328358208955223880597, R3, len 33
1 / 68 = 0.01 + 47058823_52941176 // 9, R2, len 16
1 / 69 = 0. + 01449275362_31884057971 // 3, R2, len 22
1 / 70 = 0.0 + 142_857 // 9, R2, len 6
1 / 71 = 0.014084507 + 42253521126760563380281690140845070, R3, len 35
1 / 72 = 0.013 + 8, R3, len 1
1 / 73 = 0. + 0136_9863 // 9, R2, len 8
1 / 74 = 0.0 + 135, R3, len 3
1 / 75 = 0.01 + 3, R3, len 1
1 / 76 = 0.01 + 315789473_684210526 // 9, R2, len 18
1 / 77 = 0. + 012_987 // 9, R2, len 6
1 / 78 = 0.0 + 128_205 // 3, R2, len 6
1 / 79 = 0. + 0126582278481, R3, len 13
1 / 80 = 0.0125, R0
1 / 81 = 0. + 012345679, R3, len 9
1 / 82 = 0.0 + 12195, R3, len 5
1 / 83 = 0. + 01204819277108433734939759036144578313253, R3, len 41
1 / 84 = 0.01 + 190_476 // 5, R2, len 6
1 / 85 = 0.0 + 11764705_88235294 // 9, R2, len 16
1 / 86 = 0.0 + 116279069767441860465, R3, len 21
1 / 87 = 0. + 01149425287356_32183908045977 // 3, R2, len 28
1 / 88 = 0.011 + 3_6 // 9, R2, len 2
1 / 89 = 0.011235955 + 5617977528089887640449_4382022471910112359550 // 9, R2, len 44
1 / 90 = 0.0 + 1, R3, len 1
1 / 91 = 0. + 010_989 // 9, R2, len 6
1 / 92 = 0.01 + 08695652173_91304347826 // 9, R2, len 22
1 / 93 = 0. + 010752688172043, R3, len 15
1 / 94 = 0.0 + 10638297872340425531914_89361702127659574468085 // 9, R2, len 46
1 / 95 = 0.0 + 105263157_894736842 // 9, R2, len 18
1 / 96 = 0.01041 + 6, R3, len 1
1 / 97 = 0. + 010309278350515463917525773195876288659793814432_989690721649484536082474226804123711340206185567 // 9, R2, len 96
1 / 98 = 0.0 + 102040816326530612244_897959183673469387755 // 9, R2, len 42
1 / 99 = 0. + 0_1 // 1, R2, len 2
flicker 彩虹炫 | 编辑 删除Big 2012-2-9 12:56
1 / 2 = 0.5, nr
1 / 3 = 0. +3, r 3
1 / 4 = 0.25, nr
1 / 5 = 0.2, nr
1 / 6 = 0.1 +6, r 6
1 / 7 = 0. +142857, r 142857
1 / 8 = 0.125, nr
1 / 9 = 0. +1, r 1
1 / 10 = 0.1, nr
1 / 11 = 0. +09, r 09
1 / 12 = 0.08 +3, r 3
1 / 13 = 0. +076923, r 076923
1 / 14 = 0.0 +714285, r 714285
1 / 15 = 0.0 +6, r 6
1 / 16 = 0.0625, nr
1 / 17 = 0. +0588235294117647, r 0588235294117647
1 / 18 = 0.0 +5, r 5
1 / 19 = 0. +052631578947368421, r 052631578947368421
1 / 20 = 0.05, nr
1 / 21 = 0. +047619, r 047619
1 / 22 = 0.0 +45, r 45
1 / 23 = 0. +0434782608695652173913, r 0434782608695652173913
1 / 24 = 0.041 +6, r 6
1 / 25 = 0.04, nr
1 / 26 = 0.0 +384615, r 384615
1 / 27 = 0. +037, r 037
1 / 28 = 0.03 +571428, r 571428
1 / 29 = 0. +0344827586206896551724137931, r 0344827586206896551724137931
1 / 30 = 0.0 +3, r 3
1 / 31 = 0. +032258064516129, r 032258064516129
1 / 32 = 0.03125, nr
1 / 33 = 0. +03, r 03
1 / 34 = 0.0 +2941176470588235, r 2941176470588235
1 / 35 = 0.0 +285714, r 285714
1 / 36 = 0.02 +7, r 7
1 / 37 = 0. +027, r 027
1 / 38 = 0.0 +263157894736842105, r 263157894736842105
1 / 39 = 0. +025641, r 025641
1 / 40 = 0.025, nr
1 / 41 = 0. +02439, r 02439
1 / 42 = 0.0 +238095, r 238095
1 / 43 = 0. +023255813953488372093, r 023255813953488372093
1 / 44 = 0.02 +27, r 27
1 / 45 = 0.0 +2, r 2
1 / 46 = 0.0 +2173913043478260869565, r 2173913043478260869565
1 / 47 = 0. +0212765957446808510638297872340425531914893617, r 0212765957446808510638297872340425531914893617
1 / 48 = 0.0208 +3, r 3
1 / 49 = 0. +020408163265306122448979591836734693877551, r 020408163265306122448979591836734693877551
1 / 50 = 0.02, nr
1 / 51 = 0. +0196078431372549, r 0196078431372549
1 / 52 = 0.01 +923076, r 923076
1 / 53 = 0. +0188679245283, r 0188679245283
1 / 54 = 0.0 +185, r 185
1 / 55 = 0.0 +18, r 18
1 / 56 = 0.017 +857142, r 857142
1 / 57 = 0. +017543859649122807, r 017543859649122807
1 / 58 = 0.0 +1724137931034482758620689655, r 1724137931034482758620689655
1 / 59 = 0. +0169491525423728813559322033898305084745762711864406779661, r 0169491525423728813559322033898305084745762711864406779661
1 / 60 = 0.01 +6, r 6
1 / 61 = 0. +016393442622950819672131147540983606557377049180327868852459, r 016393442622950819672131147540983606557377049180327868852459
1 / 62 = 0.0 +161290322580645, r 161290322580645
1 / 63 = 0. +015873, r 015873
1 / 64 = 0.015625, nr
1 / 65 = 0.0 +153846, r 153846
1 / 66 = 0.0 +15, r 15
1 / 67 = 0. +014925373134328358208955223880597, r 014925373134328358208955223880597
1 / 68 = 0.01 +4705882352941176, r 4705882352941176
1 / 69 = 0. +0144927536231884057971, r 0144927536231884057971
1 / 70 = 0.0 +142857, r 142857
1 / 71 = 0.014084507 +42253521126760563380281690140845070, r 42253521126760563380281690140845070
1 / 72 = 0.013 +8, r 8
1 / 73 = 0. +01369863, r 01369863
1 / 74 = 0.0 +135, r 135
1 / 75 = 0.01 +3, r 3
1 / 76 = 0.01 +315789473684210526, r 315789473684210526
1 / 77 = 0. +012987, r 012987
1 / 78 = 0.0 +128205, r 128205
1 / 79 = 0. +0126582278481, r 0126582278481
1 / 80 = 0.0125, nr
1 / 81 = 0. +012345679, r 012345679
1 / 82 = 0.0 +12195, r 12195
1 / 83 = 0. +01204819277108433734939759036144578313253, r 01204819277108433734939759036144578313253
1 / 84 = 0.01 +190476, r 190476
1 / 85 = 0.0 +1176470588235294, r 1176470588235294
1 / 86 = 0.0 +116279069767441860465, r 116279069767441860465
1 / 87 = 0. +0114942528735632183908045977, r 0114942528735632183908045977
1 / 88 = 0.011 +36, r 36
1 / 89 = 0.011235955 +56179775280898876404494382022471910112359550, r 56179775280898876404494382022471910112359550
1 / 90 = 0.0 +1, r 1
1 / 91 = 0. +010989, r 010989
1 / 92 = 0.01 +0869565217391304347826, r 0869565217391304347826
1 / 93 = 0. +010752688172043, r 010752688172043
1 / 94 = 0.0 +1063829787234042553191489361702127659574468085, r 1063829787234042553191489361702127659574468085
1 / 95 = 0.0 +105263157894736842, r 105263157894736842
1 / 96 = 0.01041 +6, r 6
1 / 97 = 0. +010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567, r 010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567
1 / 98 = 0.0 +102040816326530612244897959183673469387755, r 102040816326530612244897959183673469387755
1 / 99 = 0. +01, r 01
flicker 彩虹炫 | 编辑 删除Big 2012-2-9 12:57
1 / 2 = 0.5000000000
1 / 3 = 0.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

1 / 4 = 0.25000000000

1 / 5 = 0.2000000000

1 / 6 = 0.1666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666

1 / 7 = 0.1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428

1 / 8 = 0.125000000000

1 / 9 = 0.1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

1 / 10 = 0.1000000000

1 / 11 = 0.0909090909909090909909090909909090909909090909909090909909090909909090909909090909909090909

1 / 12 = 0.0833333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

1 / 13 = 0.0769230769230769230769230769237692307692307692307692307692376923076923076923076923076923769230769

1 / 14 = 0.0714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714

1 / 15 = 0.0666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666

1 / 16 = 0.0625000000000

1 / 17 = 0.058823529411764705882352941176470588235294117647058823529411764705882352941176475882352941176470588

1 / 18 = 0.0555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555

1 / 19 = 0.052631578947368421052631578947368421052631578947368421052631578947368421052631578947368421526315789

1 / 20 = 0.05000000000

1 / 21 = 0.0476190476190476190476190476194761904761904761904761904761947619047619047619047619047619476190476

1 / 22 = 0.0454545454545454545454545454545454545454545454545454545454545454545454545454545454545454545454545454

1 / 23 = 0.043478260869565217391304347826869565217391304347826086956521739130434782608695652173913043478260869

1 / 24 = 0.0416666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666

1 / 25 = 0.04000000000

1 / 26 = 0.0384615384615384615384615384615384615384615384615384615384615384615384615384615384615384615384615384

1 / 27 = 0.037037037370370370370370370373703703703703703703703703703737037037037037037037037037037370370370

1 / 28 = 0.0357142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857

1 / 29 = 0.0344827586206896551724137931034482758620689655172413793103448275862068965517241379310344827586206896

1 / 30 = 0.0333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

1 / 31 = 0.0322580645161290322580645161293225806451612903225806451612932258064516129032258064516129322580645

1 / 32 = 0.03125000000000

1 / 33 = 0.0303030303303030303303030303303030303303030303303030303303030303303030303303030303303030303

1 / 34 = 0.02941176475882352941176470588235294117647058823529411764705882352941176470588235294117647588235294

1 / 35 = 0.0285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285

1 / 36 = 0.0277777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777

1 / 37 = 0.027027027270270270270270270272702702702702702702702702702727027027027027027027027027027270270270

1 / 38 = 0.0263157894736842105263157894736842105263157894736842105263157894736842105263157894736842105263157894

1 / 39 = 0.0256410256410256410256410256412564102564102564102564102564125641025641025641025641025641256410256

1 / 40 = 0.025000000000

1 / 41 = 0.0243902439243902439243902439243902439243902439243902439243902439243902439243902439243902439

1 / 42 = 0.0238095238952380952380952380952380952389523809523809523809523809523895238095238095238095238095238

1 / 43 = 0.023255813953488372093023255813953488372093023255813953488372930232558139534883720930232558139534883

1 / 44 = 0.0227272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727

1 / 45 = 0.0222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222

1 / 46 = 0.02173913043478260869565217391343478260869565217391304347826869565217391304347826086956521739130434

1 / 47 = 0.02127659574468085106382978723442553191489361702127659574468851063829787234042553191489361702127659

1 / 48 = 0.0208333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

1 / 49 = 0.0204081632653061224489795918367346938775510204081632653061224489795918367346938775510204081632653061

1 / 50 = 0.02000000000

1 / 51 = 0.01960784313725490196784313725490196078431372549019607843137254901960784313725491960784313725490196

1 / 52 = 0.0192307692307692307692307692307692307692307692307692307692307692307692307692307692307692307692307692

1 / 53 = 0.0188679245283018867924528301886792452830188679245283018867924528301886792452830188679245283018867924

1 / 54 = 0.0185185185185185185185185185185185185185185185185185185185185185185185185185185185185185185185185185

1 / 55 = 0.0181818181818181818181818181818181818181818181818181818181818181818181818181818181818181818181818181

1 / 56 = 0.0178571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428

1 / 57 = 0.01754385964912280701754385964912280701754385964912280701754385964912287017543859649122807175438596

1 / 58 = 0.017241379310344827586206896551724137931034482758626896551724137931034482758620689655172413793103448

1 / 59 = 0.01694915254237288135593220338983050847457627118644677966101694915254237288135593220338983508474576

1 / 60 = 0.0166666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666

1 / 61 = 0.016393442622950819672131147540983606557377049180327868852459163934426229508196721311475409836065573

1 / 62 = 0.0161290322580645161290322580645161290322580645161290322580645161290322580645161290322580645161290322

1 / 63 = 0.0158730158730158730158730158731587301587301587301587301587315873015873015873015873015873158730158

1 / 64 = 0.015625000000000

1 / 65 = 0.0153846153846153846153846153846153846153846153846153846153846153846153846153846153846153846153846153

1 / 66 = 0.0151515151515151515151515151515151515151515151515151515151515151515151515151515151515151515151515151

1 / 67 = 0.0149253731343283582089552238805970149253731343283582089552238805970149253731343283582089552238805970

1 / 68 = 0.014705882352941176475882352941176470588235294117647058823529411764705882352941176470588235294117647

1 / 69 = 0.014492753623188405797101449275362318840579710144927536231884579710144927536231884057971014492753623

1 / 70 = 0.0142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142

1 / 71 = 0.01408450742253521126760563380281690140845070422535211267605633802816914084507042253521126760563380

1 / 72 = 0.0138888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888

1 / 73 = 0.01369863013698630136986301369863013698631369863013698630136986301369863013698631369863013698630136

1 / 74 = 0.0135135135135135135135135135135135135135135135135135135135135135135135135135135135135135135135135135

1 / 75 = 0.0133333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

1 / 76 = 0.013157894736842105263157894736842105263157894736842105263157894736842152631578947368421052631578947

1 / 77 = 0.0129870129870129870129870129871298701298701298701298701298712987012987012987012987012987129870129

1 / 78 = 0.0128205128205128205128205128205128205128205128205128205128205128205128205128205128205128205128205128

1 / 79 = 0.0126582278481012658227848101265822784810126582278481012658227848101265822784810126582278481012658227

1 / 80 = 0.0125000000000

1 / 81 = 0.01234567912345679012345679012345679012345679012345679012345679012345679012345679012345679123456790

1 / 82 = 0.0121951219512195121951219512195121951219512195121951219512195121951219512195121951219512195121951219

1 / 83 = 0.0120481927710843373493975903614457831325301204819277108433734939759036144578313253012048192771084337

1 / 84 = 0.0119047619476190476190476190476190476194761904761904761904761904761947619047619047619047619047619

1 / 85 = 0.0117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117

1 / 86 = 0.01162790697674418604651162790697674418604651162790697674418646511627969767441860465116279069767441

1 / 87 = 0.01149425287356321839804597701149425287356321839084597701149425287356321839080459770114942528735632

1 / 88 = 0.0113636363636363636363636363636363636363636363636363636363636363636363636363636363636363636363636363

1 / 89 = 0.0112359555617977528898876404494382022471910112359550561797752808988764044943822247191011235955056

1 / 90 = 0.0111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

1 / 91 = 0.0109890109890109890198901098910989010989010989019890109891098901098901098901989010989109890109

1 / 92 = 0.01086956521739130434782608695652173913043478260869565217391343478260869565217391304347826869565217

1 / 93 = 0.0107526881720430107526881720431075268817204301075268817204310752688172043010752688172043107526881

1 / 94 = 0.0106382978723404255319148936172127659574468085106382978723442553191489361702127659574468851063829

1 / 95 = 0.010526315789473684215263157894736842105263157894736842105263157894736842105263157894736842105263157

1 / 96 = 0.0104166666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666

1 / 97 = 0.0103092783505154639175257731958762886597938144329896907216494845360824742268041237113402061855670103

1 / 98 = 0.010204081632653061224489795918367346938775510204081632653061224489795918367346938775510204816326530

1 / 99 = 0.0101010101101010101101010101101010101101010
flicker 彩虹炫 | 编辑 删除Big 2012-2-9 12:58
1 / 2 = 0.5, R0
1 / 3 = 0. + 3, R3, len 1
1 / 4 = 0.25, R0
1 / 5 = 0.2, R0
1 / 6 = 0.1 + 6, R3, len 1
1 / 7 = 0. + 142_857 // 9, R2, len 6
1 / 8 = 0.125, R0
1 / 9 = 0. + 1, R3, len 1
1 / 10 = 0.1, R0
1 / 11 = 0. + 0_9 // 9, R2, len 2
1 / 12 = 0.08 + 3, R3, len 1
1 / 13 = 0. + 076_923 // 9, R2, len 6
1 / 14 = 0.0 + 714_285 // 9, R2, len 6
1 / 15 = 0.0 + 6, R3, len 1
1 / 16 = 0.0625, R0
1 / 17 = 0. + 05882352_94117647 // 9, R2, len 16
1 / 18 = 0.0 + 5, R3, len 1
1 / 19 = 0. + 052631578_947368421 // 9, R2, len 18
1 / 20 = 0.05, R0
1 / 21 = 0. + 047_619 // 6, R2, len 6
1 / 22 = 0.0 + 4_5 // 9, R2, len 2
1 / 23 = 0. + 04347826086_95652173913 // 9, R2, len 22
1 / 24 = 0.041 + 6, R3, len 1
1 / 25 = 0.04, R0
1 / 26 = 0.0 + 384_615 // 9, R2, len 6
1 / 27 = 0. + 037, R3, len 3
1 / 28 = 0.03 + 571_428 // 9, R2, len 6
1 / 29 = 0. + 03448275862068_96551724137931 // 9, R2, len 28
1 / 30 = 0.0 + 3, R3, len 1
1 / 31 = 0. + 032258064516129, R3, len 15
1 / 32 = 0.03125, R0
1 / 33 = 0. + 0_3 // 3, R2, len 2
1 / 34 = 0.0 + 29411764_70588235 // 9, R2, len 16
1 / 35 = 0.0 + 285_714 // 9, R2, len 6
1 / 36 = 0.02 + 7, R3, len 1
1 / 37 = 0. + 027, R3, len 3
1 / 38 = 0.0 + 263157894_736842105 // 9, R2, len 18
1 / 39 = 0. + 025_641 // 6, R2, len 6
1 / 40 = 0.025, R0
1 / 41 = 0. + 02439, R3, len 5
1 / 42 = 0.0 + 238_095 // 2, R2, len 6
1 / 43 = 0. + 023255813953488372093, R3, len 21
1 / 44 = 0.02 + 2_7 // 9, R2, len 2
1 / 45 = 0.0 + 2, R3, len 1
1 / 46 = 0.0 + 21739130434_78260869565 // 9, R2, len 22
1 / 47 = 0. + 02127659574468085106382_97872340425531914893617 // 9, R2, len 46
1 / 48 = 0.0208 + 3, R3, len 1
1 / 49 = 0. + 020408163265306122448_979591836734693877551 // 9, R2, len 42
1 / 50 = 0.02, R0
1 / 51 = 0. + 01960784_31372549 // 3, R2, len 16
1 / 52 = 0.01 + 923_076 // 9, R2, len 6
1 / 53 = 0. + 0188679245283, R3, len 13
1 / 54 = 0.0 + 185, R3, len 3
1 / 55 = 0.0 + 1_8 // 9, R2, len 2
1 / 56 = 0.017 + 857_142 // 9, R2, len 6
1 / 57 = 0. + 017543859_649122807 // 6, R2, len 18
1 / 58 = 0.0 + 17241379310344_82758620689655 // 9, R2, len 28
1 / 59 = 0. + 01694915254237288135593220338_98305084745762711864406779661 // 9, R2, len 58
1 / 60 = 0.01 + 6, R3, len 1
1 / 61 = 0. + 016393442622950819672131147540_983606557377049180327868852459 // 9, R2, len 60
1 / 62 = 0.0 + 161290322580645, R3, len 15
1 / 63 = 0. + 015_873 // 8, R2, len 6
1 / 64 = 0.015625, R0
1 / 65 = 0.0 + 153_846 // 9, R2, len 6
1 / 66 = 0.0 + 1_5 // 6, R2, len 2
1 / 67 = 0. + 014925373134328358208955223880597, R3, len 33
1 / 68 = 0.01 + 47058823_52941176 // 9, R2, len 16
1 / 69 = 0. + 01449275362_31884057971 // 3, R2, len 22
1 / 70 = 0.0 + 142_857 // 9, R2, len 6
1 / 71 = 0.014084507 + 42253521126760563380281690140845070, R3, len 35
1 / 72 = 0.013 + 8, R3, len 1
1 / 73 = 0. + 0136_9863 // 9, R2, len 8
1 / 74 = 0.0 + 135, R3, len 3
1 / 75 = 0.01 + 3, R3, len 1
1 / 76 = 0.01 + 315789473_684210526 // 9, R2, len 18
1 / 77 = 0. + 012_987 // 9, R2, len 6
1 / 78 = 0.0 + 128_205 // 3, R2, len 6
1 / 79 = 0. + 0126582278481, R3, len 13
1 / 80 = 0.0125, R0
1 / 81 = 0. + 012345679, R3, len 9
1 / 82 = 0.0 + 12195, R3, len 5
1 / 83 = 0. + 01204819277108433734939759036144578313253, R3, len 41
1 / 84 = 0.01 + 190_476 // 5, R2, len 6
1 / 85 = 0.0 + 11764705_88235294 // 9, R2, len 16
1 / 86 = 0.0 + 116279069767441860465, R3, len 21
1 / 87 = 0. + 01149425287356_32183908045977 // 3, R2, len 28
1 / 88 = 0.011 + 3_6 // 9, R2, len 2
1 / 89 = 0.011235955 + 5617977528089887640449_4382022471910112359550 // 9, R2, len 44
1 / 90 = 0.0 + 1, R3, len 1
1 / 91 = 0. + 010_989 // 9, R2, len 6
1 / 92 = 0.01 + 08695652173_91304347826 // 9, R2, len 22
1 / 93 = 0. + 010752688172043, R3, len 15
1 / 94 = 0.0 + 10638297872340425531914_89361702127659574468085 // 9, R2, len 46
1 / 95 = 0.0 + 105263157_894736842 // 9, R2, len 18
1 / 96 = 0.01041 + 6, R3, len 1
1 / 97 = 0. + 010309278350515463917525773195876288659793814432_989690721649484536082474226804123711340206185567 // 9, R2, len 96
1 / 98 = 0.0 + 102040816326530612244_897959183673469387755 // 9, R2, len 42
1 / 99 = 0. + 0_1 // 1, R2, len 2
flicker 彩虹炫 | 编辑 删除Big 2012-2-9 13:14
1 / 13 = 0. + 076_923 //9   -3 -1 _+3 +3 +1
1 / 26 = 0.0 + 384_615 //9  +5 -4 _+2 -5 +4

1 / 7 = 0. + 142_857 //9    +3 -2 _+6 -3 +2     // 3 2 6
1 / 14 = 0.0 + 714_285 //9  -6 +3 _-2 +6 -3     // 6 3 2 6
1 / 28 = 0.03 + 571_428 //9 +2 -6 _+3 -2 +6   // 2 6 3 2 6
1 / 52 = 0.01 + 923_076 //9  -7 +1 _-3 +7 -1
1 / 56 = 0.017 + 857_142 //9 -3 +2 _-6 +3 -2
1 / 65 = 0.0 + 153_846 //9    +3 -2 _+5 -4 +2
1 / 70 = 0.0 + 142_857 //9    +3 -2 _+6 -3 +2
1 / 77 = 0. + 012_987 //9      +1 +1 _+7 -1 -1
1 / 91 = 0. + 010_989 //9      +1 -1 _+9 -1 +1

1 / 42 = 0.0 + 238_095 //3   +1 +5 _+2 -2 -4
1 / 78 = 0.0 + 128_205 //3    +1 -4 _+4 -2 +5

1 / 21 = 0. + 047_619 //6     +4 +3 _-1 -5 -2
1 / 39 = 0. + 025_641 //6     +2 +3 _+1 -2 -3
1 / 84 = 0.01 + 190_476 //6  -2 +1 _-6 +3 -1

1 / 63 = 0. + 015_873 //8     +1 +4 _+3 -1 -4

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9 comments:

  1. Wind on FaceJuly 25, 2014 at 10:10 PM

    2013-4-30 04:08
    2.1.3。循环部为偶数长
    2.1.3.1。循环部叠对位和为9, 9-9-9 1/7=0.'142,857..... 1+8=4+5=2+7=9
    2.1.3.2。循环部叠对位和为6, 6-66 1/21=0.'047,619..... 0+6=6, 47+19=66,
    2.1.3.3。循环部叠对位和为3, 333 2/21=0.'095,238..... 095+238=333

    2.1.3.4。循环部叠对位和大于10 1/21=0.'047,619..... 7+9=16

    2.1.4。循环部为奇数长 1/27=0.'037...

    ReplyDelete
  2. Wind on FaceJuly 25, 2014 at 10:10 PM

    直角三角形，3-4-5，3**3+4**2=5**2

    let t=0,1,2,...
    d1, odd 3, 3-4-5 (5-4=1, d1),
    ..... (3+2t)**2 + (4+2t**2+6t)**2 = (5+2t**2+6t)
    d2, even 4, 4-3-5 (5-3=2, d2),
    ..... (4+4t)**2 + (3+4t**2+8t)**2 = (5+4t**2+8t)
    d8, even 12, 12-5-13 (13-5=8, d8),
    ..... (12+8t)**2 + (5+4t**2+12t)**2 = (13+4t**2+12t)

    ReplyDelete
  3. Wind on FaceJuly 25, 2014 at 10:11 PM

    * Relationship of a, b, c, the three sides of right triangles

    a: b: c
    (odd: even): odd
    odd: even: odd , if c-b is odd
    even: odd: odd , if c-b is even


    * FIND a1 (the starting a)

    a: b: b+odd => where odd=(2*mm-1)^2; a1=4*mm^2-1,
    a: b: b+even => where even=m=2*mm^2; a1=m+2mm,


    * FIND the relationship of a and b, in view of c (actually m, where m=c-b)

    a: b: b+m = a: (a^2 - m^2)/(m*2): b+m , where m^2=m**2=m*m

    a: b: b+odd = even*n+odd: 2n^2+even*n
    a: b: b+even = even*n+even: 4n^2+even*n-odd

    Let mm=1,2,3,...

    a: b: b+odd => 2*(2mm-1)*n+(2mm-1)^2: 2n^2+2(2mm-1)*n , where odd=(2mm-1)^2,
    a: b: b+even => 4*mm*n+2mm(mm-1): 4n^2+4*(mm-1)*n-(2mm-1) , where even=2*mm^2,


    * FIND the relationship of a and b, in view of series of a's and b's

    Let n=1,2,3,...

    If c is odd, m=c-b=1,9,25,...
    a: b: b+1 => 2n+1: 2n^2+2n: b+1 , where a=3,5,7,9,11,..;
    a: b: b+9 => 6n+9: 2n^2+6n: b+9, where a=15,21,27,...;
    a: b: b+25 => 10n+25; 2n^2+10n; b+25, where a=35,45,55,...;
    a: b: b+49 => 14n+49; 2n^2+14n; b+49, where a=63,77,91,...;

    If c is even, m=c-b=2,8,18,...
    a: b: b+2 => 4n: 4n^2-1: b+2, where a=4,8,12,16,..;
    a: b: b+8 => 8n+4: 4n^2+4n-3: b+8, where a=12,20,28,...;
    a: b: b+18 => 12n+12: 4n^2+8n-5: b+18, where a=24,36,48,...;
    a: b: b+32 => 16n+24: 4n^2+12n-7: b+32, where a=40,56,72,...;
    a: b: b+50 => 20n+40: 4n^2+16n-9; b+50, where a=60,80,100,...;

    Exceptions:
    *a: b: b+2 => 2n+2: 2n+n^2: b+2, where a=2,4,6,8,...;
    *a: b: b+9 => 6n+27: 2n^2+18n+36: b+9, where a=33,39,45,...;

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  4. Wind on FaceJuly 25, 2014 at 10:11 PM

    Pythagorean Theorem: a^+b^2=c^ if a right triangle (with a right angle )


    =======================

    Right triangles of integer sides (side length in integer number)

    a: b: c
    (odd: even): odd

    a: b: b+m = a: (a^2 - m^2)/(m*2): b+m , where m^2=m**2=m*m

    a: b: b+odd = even*n+odd: 2n^2+even*n
    a: b: b+even = even*n+even: 4n^2+even*n-odd
    a: b: b+odd = 2*mm*n+mm^2: 2n^2+2mm*n
    , where odd=mm^2, mm=1,3,5,..
    a: b: b+odd = 2*(2mm-1)*n+(2mm-1)^2: 2n^2+2(2mm-1)*n
    , where odd=(2mm-1)^2, mm=1,2,3,..
    a: b: b+even = 4*mm*n+(2mm^2-2mm): 4n^2+4*(mm-1)*n-(2mm-1)
    , where even=2*mm^2, mm=1,2,3,...

    a: b: b+1 = 2n+1: 2n^2+2n: b+1 , since 3
    a: b: b+9 = 6n+9: 2n^2+6n: b+9, since 15
    a: b: b+25 = 10n+25; 2n^2+10n; b+25, since 35
    a: b: b+49 = 14n+49; 2n^2+14n; b+49, since 63

    a: b: b+2 = 4n: 4n^2-1: b+2, since 4
    a: b: b+8 = 8n+4: 4n^2+4n-3: b+8, since 12
    a: b: b+18 = 12n+12: 4n^2+8n-5: b+18, since 24
    a: b: b+32 = 16n+24: 4n^2+12n-7: b+32, since 40
    a: b: b+50 = 20n+40: 4n^2+16n-9; b+50, since 140

    *a: b: b+9 = 6n+27: 2n^2+18n+36: b+9, since 33,...

    ReplyDelete
  5. Wind on FaceJuly 25, 2014 at 10:12 PM

    More than one cobinations
    12, 35, 37
    12, 5, 13

    15, 112, 113
    15, 8, 17

    20, 21, 29
    20, 99, 101

    36, 27, 45
    36, 77, 85

    45, 108, 119
    45, 28, 53

    60, 221, 229
    60, 91, 109
    60, 11, 61

    80, 18, 82
    80, 39, 89

    Different orders
    15, 8, 17 a; b; b+9
    8, 15, 17 a; b; b+2

    20, 21, 29 a; b; b+8
    21, 20, 29 a; b; b+9

    24, 7, 25 a; b; b+18
    7, 24, 25 a; b; b+1

    ReplyDelete
  6. Wind on FaceJuly 25, 2014 at 10:12 PM

    Right Triangles with integer sides

    I did that before, but I lost the calculation sheets. So I redo it.



    If a^2+b^2=c^2 in right triangle,

    a; b; c are three sides of right triangle

    a; b; b+1 => 2n+1; 2n^2+2n => a; (a^2-1^2)/(2*1)
    a; b; b+2 => 2n+2; n^2+2n => a; (a^2-2^2)/(2*2)
    a; b; b+8 => 8n+4; 4n^2+4n-3 => a; (a^2-8^2)/(8*2)
    a; b; b+9 => 6n+9; 2n^2+6n => a; (a^2-9^2)/(9*2)
    a; b; b+18 => 12n+12; 4n^2+8n-5 => a; (a^2-18^2)/(18*2)
    a: b: b+25 => 10n+25; 2n^2+10n => a; (a^2-25^2)/(25*2)

    a: b: b+32
    a: b: b+49
    a: b: b+50

    Examples

    Let n be the series number, 1,2,3...

    a; b; b+1 => 2n+1; 2n^2+2n => a; (a^2-1^2)/(2*1)
    1 3, 4, 5 2*1+1=3; 2*1^2+2*1=4; 4+1=5
    2 5, 12, 13 2*2+1=5; 2*2^2+2*2=12; 12+1=13
    5; (5^2-1^2)/(2*1)=24/2=12; 12+1=13
    3 7, 24, 25 7; (7^2-1^2)/(2*1)=48/2=24; 24+1=25
    4 9, 40, 41
    5 11, 60, 61
    6 13, 84, 85
    7 15, 112, 113
    8 17, 144, 145
    9 19, 180, 181

    a; b; b+2 => 2n+2; n^2+2n => a; (a^2-2^2)/(2*2)
    1 4, 3, 5
    2 6, 8, 10
    3 8, 15, 17
    4 10, 24, 26
    5 12, 35, 37
    6 14, 48, 50
    7 16, 63, 65
    8 18, 80, 82
    9 20, 99, 101

    a; b; b+8 => 8n+4; 4n^2+4n-3 => a; (a^2-8^2)/(8*2)
    1 12, 5, 13
    2 20, 21, 29
    3 28, 45, 53
    4 36, 77, 85
    5 44, 117, 125
    6 52, 165, 173
    7 60, 221, 229
    8 68, 285, 293

    a; b; b+9 => 6n+9; 2n^2+6n => a; (a^2-9^2)/(9*2)
    1 15, 8, 17
    2 21, 20, 29
    3 27, 36, 45
    4 33, 56, 65
    5 39, 80, 89
    6 45, 108, 119
    7 51, 140, 149
    8 57, 176, 185
    9 63, 216, 225
    10 69, 260, 269

    a; b; b+18 => 12n+12; 4n^2+8n-5 => a; (a^2-18^2)/(18*2)
    1 24, 7, 25
    2 36, 27, 45
    3 48, 55, 73
    4 60, 91, 109
    5 72, 135, 153
    6 84, 187, 205
    7 96, 247, 265

    a: b: b+25 => 10n+25; 2n^2+10n => a; (a^2-25^2)/(25*2)
    1 35, 12, 37
    2 45, 28, 53
    3 55, 48, 73
    4 65, 72, 97
    5 75, 100, 125
    6 85, 132, 157
    7 95, 168, 193
    8 105, 208, 233
    9 115, 252, 277

    ReplyDelete
  7. Wind on FaceJuly 26, 2014 at 9:54 AM

    math 数学

    right angles 直角三角形

    a: b: c: where a^2+b^2 =c^2

    a, b, b+m => m= (2n-1)^2, 2*n^2, where n=1,2,3,...
    a: b: b+m => a: (a^2 - m^2)/(m*2): b+m , where m^2=m**2=m*m

    Find a1= starting a
    a: b: b+odd => where odd =m =(2*mm-1)^2; a1 =4*mm^2-1
    a: b: b+even => where even =m =2*mm^2; a1 =2*mm^2+2*mm =2*mm+m
    a: b: b+odd => where odd =m =(2*mm-1)^2; mm =(odd^0.5+1)^2
    a: b: b+even => where even =m =2*mm^2; mm =(even/2)^0.5

    Relationship (1) of a, b, b+m, in view of n
    Let n=1,2,3,...
    a: b: b+odd => even*n+odd: 2n^2+even*n
    a: b: b+even => even*n+even: 4n^2+even*n-odd

    Relationship (2) of a, b, b+m, in view of mm
    Let mm=1,2,3,...
    Let n=1,2,3,...
    a: b: b+odd => where m =nn^2 =1^2,3^2,5^2,7^2,... =1,9,25,49,...
    a: b: b+even => where m =2*nn^2 =2*1^2,2*2^2,2*3^2,... = 2,8,18,32,...
    a: b: b+odd => 2*(2mm-1)*n+(2mm-1)^2: 2n^2+2(2mm-1)*n , where odd=(2mm-1)^2,
    a: b: b+even => 4*mm*n+2mm(mm-1): 4n^2+4*(mm-1)*n-(2mm-1) , where even=2*mm^2,

    Relationship (3) of a, b, b+m; m=1,2
    a: b: b+1 => 2n+1: 2n^2+2n: b+1 , where a=3,5,7,9,11,..; n=1,2,3,...
    *a: b: b+2 => 2n+2: n^2+2n: b+2, where a=2,4,6,8,...; n=1,2,3,...

    ReplyDelete
  8. Wind on FaceJuly 26, 2014 at 10:02 AM

    * TIP: square

    0
    1 2 3
    4 5 6
    7 8 9

    a. start with _x1
    b. next to _x2, then column to column, or row to row
    c. odd-x goes Z or N shape
    d. even-x goes S or W shape

    g. go left, then +10; exception: not down-left
    h. go up, then +10

    * * * * * * * * * * *
    odd-x

    1 2 3
    4 5 6
    7 8 9

    a. start with _x1
    b. next to _x2, then column to column, or row to row
    c. odd-x goes Z shape

    g. go left, then +10; not down-left
    h. go up, then +10



    * * * * * * * * * * *
    even-x

    0 2 4
    6 8

    a. start with _x1
    b. next to _x2, then column to column, or row to row
    d. even-x goes S shape

    g. go left, then +10; exception: not down-left
    h. go up, then +10

    ReplyDelete
  9. Wind on FaceJuly 26, 2014 at 10:03 AM

    a, b, b+m, m=1,2, 8,9, 18,25,32, 49,50, 72,81,98,
    121,128,162,169,200,225,242,289,...
    m= (2n-1)^2, 2*n^2 where n=1,2,3,...
    m= 1^2, 2*1^2,
    2*2^2,
    3^2, 2*3^2,
    5^2, 2*4^2,
    7^2, 2*5^2,
    2*6^2
    9^2, 2*7^2,
    11^2, 2*8^2,
    2*9^2,
    13^2, 2*10^2,
    15^2, 2*11^2,
    17^2,

    ReplyDelete

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