Hello maths(🐞) and hivers(🐝)
I hope you are strong and stout , doing good in life.
Today' topic is recurring decimals.This is a very interesting topic.let me show you how it looks like. Check it in the following picture:
There are two kinds of decimal. One which repeats and the other which doesn't repeat.The one which repeats , we call it recurring decimals and it also comes in the set of Rational numbers. The non repeating decimal comes under the set of irrational number.
↪️Rational numbers:
A kind of numbers which comes under real number and can be written as p/q where both p and q are integers but q≠0, are called a rational numbers.
Examples:
(1) all natural numbers are rational.Eg: 1, 2, 3 etc.
(2) all whole numbers are rational. Eg: 0,1,2 etc.
(3) all integers are rational.Eg: -2,-1,0,1,2 etc.
(4) Fraction also comes here like 2/3, -5/4 , 22/7 , 24/7 etc.
All set of numbers mentioned above can be written as p/q, q≠0. So they are rational numbers.
All those numbers perfectly divisible or just divisible taking decimal after natural numbers are rational numbers.All four kinds of number mentioned above satisfy this condition.Hence they are rational numbers and also comes under recurring decimals.
↪️Irrational Number:
Numbers which do not come under recurring decimal that means they are non repeating or non recurring and hence they are called Irrational numbers.
Examples: √5, √7, π etc
Note: π≈22/7 but π≠22/7 means approximate value of π is 22/7 but not exact value. We can't determine the exact value of π because after decimal value never repeat and so we can't terminate it after decimals. But value of 22/7 repeats after decimals and hence we can terminate it.That is why 22/7 comes under Rational Number but π in Irrational Number.
So when we talk about recurring decimals,it means we gonna talk about rational numbers actually but today I just want to show you how a fraction can be converted into decimal fraction when it is a recurring and vice-versa.
Converting a fraction to recurring decimal is very easy ; it can be done with simple long division method.So, now I wanna show you how to convert recurring decimal to decimal (p/q).
⚛️Recurring decimal:
It can be divided into two categories. Proper recurring decimals and mixed recurring decimals. We call it proper when recurring point or the bar comes after decimal point. When the bar doesn't comes right after the decimal point , we call it mixed recurring decimals.
(1️⃣) Proper Recurring Decimal:
To write it as fraction,we take all the digits in NEUMERATOR removing the point and the bar.In DENOMINATOR we just take as many 9s as number of digits.Check it below:
(2️⃣) Mixed/Improper Recurring Decimal:
To write it as fraction, the NEUMERATOR will be subtraction of all the digit without point & bar and digits on the left side of the bar. In the DENOMINATOR there will as many 9s as number of digits having bar over it and the after 9(s), as many zero(0) as digits between the point and the bar. Things may seem getting complicated but explanation comes later.Check it below:
Proof:
Let's prove how do we come to above two conclusions.I gonna use liner equation to prove it.Check it below:
You may get confused how to understand by which multiple of tens (10), we should multiply the recurring decimal before subtracting( In case of mixed recurring decimal). Well it's simple, first we should multiply the decimal by multiple of tens that can bring the decimal point after right-end-digit and then again need to multiply it by such multiple of ten that brings Decimal point just before bar. Then we gota subtract both of them. Like (10x - x )= (3.333... -0.333...) for the first proof.And for the second proof (100y-10y)= (78.888... -7.888...).Here x and y represent recurring decimals used in above two proofs.
I am going to finish it here. If you have any problem to understand or find any mistake ,please feel free to ask for.
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