वास्तविक संख्याएँ और उनके दशमलव प्रसार


${\frac {1}{7}}$								${\frac {10}{3}}$							${\frac {7}{8}}$
	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 (for ${\frac {10}{3}}$ , $3$ repeats in the quotient and for ${\frac {1}{7}}$ , repeating block $142857$ in the quotient)

The above pattern using the examples above 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.

The decimal expansion of real numbers can be classified into three types. They are: