Integer which is Multiplied by 9 when moving Last Digit to First/Decimal Expansion

Lemma for Integer which is Multiplied by 9 when moving Last Digit to First

$\dfrac 9 {89} = 0 \cdotp \dot 10112 \, 35955 \, 05617 \, 97752 \, 80898 \, 87640 \, 44943 \, 82022 \, 471 \dot 9$

Proof

From Reciprocal of 89 we have:

$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$

Hence:

 0.01123 59550 56179 77528 08988 76404 49438 20224 7191
x                                                     9
 ------------------------------------------------------
 0.10112 35955 05617 97752 80898 87640 44943 82022 4719
   11235 844 5 51786 6427  88776 53 44 83371  2246 18

The lowest line is the carry digit.

As there is no overall carry digit on the left of the product, it follows that:

$\dfrac 9 {89} = 0 \cdotp \dot 10112 \, 35955 \, 05617 \, 97752 \, 80898 \, 87640 \, 44943 \, 82022 \, 471 \dot 9$

$\blacksquare$