## Moving one digit from one end to the other of a number

$1/7 = 0.142857142857142857142857142857142857142857142857142857142857...$

$0. \overline{142857}$   (period 6)

Move 7 :
$714285 = 5 \; \times \: 142857$

$2/7 = 0.285714285714285714285714285714285714285714285714285714285714...$

$0. \overline{285714}$   (period 6)

Move 4 :
$428571 = 3/2 \; \times \: 285714$

$3/7 = 0.428571428571428571428571428571428571428571428571428571428571...$

$0. \overline{428571}$   (period 6)

Move 1 :
$428571 = 3 \; \times \: 142857$

$4/7 = 0.571428571428571428571428571428571428571428571428571428571428...$

$0. \overline{571428}$   (period 6)

Move 8 :
$857142 \; = \; 3/2 \; \times \: 571428$

$5/7 = 0.714285714285714285714285714285714285714285714285714285714285...$

$0. \overline{714285}$   (period 6)

Move 5 :
$571428 \; = \; 4/5 \; \times \: 714285$

$6/7 = 0.857142857142857142857142857142857142857142857142857142857142...$

$0. \overline{857142}$   (period 6)

Move 2 :
$857142 \; = \; 3 \; \times \: 285714$

$2/13 = 0.153846153846153846153846153846153846153846153846153846153846...$

$0. \overline{153846}$   (period 6)

Move 6 :
$615384 \; = \; 4 \; \times \: 153846$

$4/39 = 0.102564102564102564102564102564102564102564102564102564102564...$

$0. \overline{102564}$   (period 6)

Move 4 :
$410256 \; = \; 4 \; \times \: 102564$

$5/39 = 0.128205128205128205128205128205128205128205128205128205128205...$

$0. \overline{128205}$   (period 6)

Move 5 :
$512820 \; = \; 4 \; \times \: 128205$

$7/39 = 0.179487179487179487179487179487179487179487179487179487179487...$

$0. \overline{179487}$   (period 6)

Move 7 :
$717948 \; = \; 4 \; \times \: 179487$

$8/79 = 0.101265822784810126582278481012658227848101265822784810126582...$

$0. \overline{1012658227848}$   (period 13)

move 8 :
$8101265822784 \; = \; 8 \; \times \: 1012658227848$

$9/79 = 0.113924050632911392405063291139240506329113924050632911392405...$

$0. \overline{1139240506329}$   (period 13)

move 9 :
$9113924050632 \; = \; 8 \; \times \: 1139240506329$

$7/19 = 0.368421052631578947368421052631578947368421052631578947368421...$

$0. \overline{368421052631578947}$   (period 18)

Move 7 :
$736842105263157894 \; = \; 2 \; \times \: 368421052631578947$

$9/19 = 0.473684210526315789473684210526315789473684210526315789473684...$

$0.\overline{473684210526315789}$   (period 18)

move 9 :
$947368421052631578 \; = \; 2 \; \times \: 473684210526315789$

$3/23 = 0.130434782608695652173913043478260869565217391304347826086956...$

$0. \overline{1304347826086956521739}$   (period 22)

Move 9 :
$9130434782608695652173 \; = \; 7 \; \times \: 1304347826086956521739$

$9/29 = 0.310344827586206896551724137931034482758620689655172413793103...$

$0. \overline{3103448275862068965517241379}$   (period 28)

Move 9 :
$9310344827586206896551724137 \; = \; 3 \; \times \: 3103448275862068965517241379$

$9/49 = 0.183673469387755102040816326530612244897959183673469387755102...$

$0. \overline{183673469387755102040816326530612244897959}$   (period 42)

Move 9 :
$918367346938775510204081632653061224489795 = 5 \times \: 183673469387755102040816326530612244897959$

$9/89 = 0.101123595505617977528089887640449438202247191011235955056179...$

$0. \overline{10112359550561797752808988764044943820224719}$   (period 44)

Move 9 :
$10112359550561797752808988764044943820224719$
$\; \times \: 9$
$= \; 91011235955056179775280898876404494382022471$

$9/59 = 0.152542372881355932203389830508474576271186440677966101694915...$

$0. \overline{1525423728813559322033898305084745762711864406779661016949}$   (period 58)

Move 9 :

$1525423728813559322033898305084745762711864406779661016949$
$\; \times \: 6$
$= \; 9152542372881355932203389830508474576271186440677966101694$

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