Explain what is Terminating n Recurring decimals. what is length of period. ? And write special characteristics of Rational No. ?
N express 3×1/8 , 2/11 in decimal form by actual devision n Express as fraction to 0.9bar , 0.6bar , 0.009 bar. ?
Answers
Answer:
Every rational number is either a terminating or repeating decimal. For any given divisor, only finitely many different remainders can occur. ... If at any point in the division the remainder is 0, the expansion terminates at that point. Then the length of the repetend, also called “period”, is defined to be 0.
Answer:
Answer ⤵⤵
- What is Terminating Decimal ?
Consider a fraction S/T. We also know that each and every fraction cab be expressed as a decimal. Suppose , the decimal expression of
S/T comes to an end ( i.e Terminates) , then the decimal which is obtained is known as Terminating Decimal.
Example :-- 5/13
5 ) 13 ( 2.6
- 10
x30
30
00
In above example 2.6 comes to an end. Thus 5/13 is a Terminating Decimal.
2. What is Recurring Decimal ?
Consider a fraction M/N. We also know that each and every fraction cab be expressed as a decimal. Suppose , in the decimal expression of
M/N a digit or a set of digit repeats periodically , then the decimal form so obtained will be called as Recurring Decimal.
Example :-- 2/3 => 0.6 bar
3 ) 2 ( 0.66..
0
20
18
20
18
2
So , here we see that 0.66666..(infinite) repeats periodically. Hence , 2/3 is Recurring Decimal.
Note* We use a — sign above the repeating digit that is known as ' bar'
3. Length of Period ?
Length of period is defined as the no of times a set of particular number repeats.
Example :--
In 2.345735 bar the length of period is 6.
( count the digits after the . point )
4. Special Characteristics Of Rational No.
- Every rational no. can be expressed as a Terminating Decimal or a Recurring Decimal.
- Each and every Terminating Decimal and Recurring Decimal is a Rational number.
5. Express 3×1/8 , 2/11 in decimal form.
=> 3×1/8
=> 3×8 + 1 / 8
=> 25 / 8
=> 3.125 (answer)
=> 2 / 11
=> 0.181818...
=> 0.18 bar ( answer )
6. Express as fraction to 0.9bar , 0.6bar , 0.009 bar. ?
- 0.9 bar :--
let x = 0.9
so , x = 0.999
also, 10x = 9.999
on doing Subtraction of both we get :-
10x - x = 9.999 - 0.999
9x = 9
x = 9/9
x = 1/1
Therefore, 0.9 bar = 1/1
- 0.6 bar :--
Similarly ,
let x = 0.666
so , 10x = 6.666
on doing Subtraction of both we get :-
10x - x = 6.666 - 0.666
9 x = 6
x = 6/9
x = 2/3
Therefore , 0.6 bar = 2/3
- 0.009 bar :--
let x = 0.009009
so , 1000 x = 9.009009
on doing Subtraction of both we get :-
1000x - x = 9.009009 - 0.009009
999x = 9
x = 9/999
x = 1 / 111
Therefore , 0.009 bar = 1/111