Real Numbers and their Decimal Expansions

Decimal expansion ${\frac {1}{7}},{\frac {10}{3}},{\frac {7}{8}}$

${\frac {1}{7}}$								${\frac {10}{3}}$								${\frac {7}{8}}$
	0.142857.....									3.3333...							0.875
7	1	0							3	1	0					8	7	0
		7									9						6	4
		3	0								1	0						6	0
		2	8									9						5	6
			2	0								1	0						4	0
			1	4									9						4	0
				6	0								1	0						0
				5	6									9		Reminders: 6,4,0. Divisor: 8
					4	0								1
					3	5			Reminders: 1,1,1,1... Divisor: 3
						5	0
						4	9
							1
Reminders: 3,2,6,4,5,1,3,2,6,4,5,1... Divisor: 7

In the above division operation

• The remainders either become $0$ after some stage, or start repeating themselves.
• The number of entries in the repeating string of remainders is less than the divisor (in ${\frac {10}{3}}$ , one number repeats itself and the divisor is $3$ , in ${\frac {1}{7}}$ there are six entries $326451$ in the repeating string of remainders and the divisor is $7$ )
• If the remainders repeat, then we get a repeating block of digits in the quotient (in ${\frac {10}{3}}$ , $3$ repeats in the quotient and in ${\frac {1}{7}}$ , $142857$ repeats in the quotient)
• (ii) The number of entries in the repeating string of remainders is less than the divisor (in 10 3 one number repeats itself and the divisor is 3, in 1 7 there are six entries 326451 in the repeating string of remainders and 7 is the divisor). (iii) If the remainders repeat, then we get a repeating block of digits in the quotient (for 10 3 , 3 repeats in the quotient and for 1 7 , we get the repeating block 142857 in the quotient).

The above pattern using only the examples above, it is true for all rationals of the form ${\frac {p}{q}}$ ($q\neq 0$).

On division of $p$ by $q$, two main things happen – either the remainder becomes zero or never becomes zero and we get a repeating string of remainders.