I find this tricky one. How to calculate the first 50 digits/decimals of the fractional number 1/49? Two of my calculators and MatLab gives different answers so I'm curious, how this is calculated "manual" way such that average person with simple sum and multiplication skills needs no calculator for it.
Method 1:
0,
10/49
0,0
100/49 = **2**,2
0,0**2**
20/49
0,020
200/49 = **4**,4
0,020**4**
40/49
0,02040
400/49 = **8**,8
0,02040**8**
800/49 = **16**,16
0,020408**16**
1600/49 = **32**,32
0,02040816**32**
3200/49 = **64**,64
0,0204081632**64**
6400/49 = **130**,30
0,02040816326**530**...
Multiplications and divisions getting big plus backward processing...
Method 2:
1/49 -> rule of second digit of divisor +1 = 5 multiplier. Numbers ending 9 starts with 1. Writing from right to left:
...1
1*5=5+0
...51
5*5=25+0
...551
5*5=25+2=27
...7551
7*5=35+2=37
...77551
7*5=35+3=38
...877551
8*5=40+3=43
...3877551
3*5=15+4=19
...93877551
9*5=45+1=46
...693877551
6*5=30+4=34
...4693877551
4*5=20+3=23
...34693877551
3*5=15+2=17
...734693877551
7*5=35+1=36
.
.
.
...6122448979591836734693877551
6*5=30+0=30
...306122448979591836734693877551
3*5=15
...5306122448979591836734693877551
5*5=25+1=26
...65306122448979591836734693877551
6*5=30+2=32
...265306122448979591836734693877551
2*5=10+3=13
...3265306122448979591836734693877551
3*5=15+1=16
...63265306122448979591836734693877551
6*5=30+1=31
...163265306122448979591836734693877551
1*5=5+3=8
...8163265306122448979591836734693877551
8*5=40
...408163265306122448979591836734693877551
4*5=20
...20408163265306122448979591836734693877551
2*5=10 ENDS HERE
020408163265306122448979591836734693877551
Pretty good approach while seems to be long written this way. Alternative syntax gives more right to the method:
0 2 0 4 0 8 1 6 3 2 6 5 3 0 6 1 2 2 4 4 8 9 7 9 5 9 1 8 3 6 7 3 4 6 9 3 8 7 7 5 5 1
1 3 1 1 3 2 1 1 1 2 2 4 4 3 4 2 4 4 1 3 3 1 2 3 4 1 4 3 3 2 2
Method 3:
02
0.02
..04
0.0204
....08
0.020408
......16
0.02040816
........32
0.0204081632
..........64
0.020408163265
...........128
0.02040816326530
.............256
0.020408163265306
..............512
0.02040816326530612
...............1024
0.020408163265306122
................2048
0.0204081632653061224
.................4096
0.02040816326530612244
..................8192
0.0204081632653061224489
...................16384
0.02040816326530612244897
Get complicated because numbers needs to be carried backwards, thus memorization...
HINT:
$$\frac1{49}=\frac2{100-2}=\frac2{100(1-.02)}=.02\left(1-.02\right)^{-1}$$
$$=.02\left(1+.02+(.02)^2+(.02)^3+(.02)^4+\cdots\right)$$