Recurrent and Non -Recurrent Decimal Numbers
Recurrent and Non-Recurrent Decimal Numbers: Before knowing what are Recurrent and Non -Recurrent Decimal Numbers it is important for us to understand the types of decimal numbers, decimal numbers are of two kinds, finite decimal and infinite decimal, as example finite decimal is the outcome of 1/2=0.5, it is finite because after dividing 1 by 2, the remainder comes 0, taking another example 1/3=0.33333…., in this case, the remainder never comes 0 and as a result after the decimal 3 is repeating continually so such decimal numbers are called Infinite recurring decimal numbers, taking third example √2=1.41421356237……., this is also an infinite decimal number but after decimal the numbers are not repeating, such decimal numbers are known as non-recurrent infinite decimal numbers.
Questions based on Recurrent and Non-recurrent decimal numbers:
Question1:Are all infinite decimal numbers are Rational numbers?
Answers: No,The numbers which are written in the form of P/Q are known as rational numbers.The numbers 1/2=0.5 and 1/3=0.333—-,so both decimal numbers finite and infinite recurrent decimal numbers can be written in the form of P/Q, so these decimal numbers are k are rational numbers.
Question 2:What are non-recurrent infinite decimal numbers called?
Non-recurrent infinite decimal numbers are called irrational numbers as example √2,√3,π.
The decimal expansion of √2,√3,π
√3=1.73205080757……
π=3.14159…….
The decimal expansion of all unresolved roots(√2,√3 etc) is always a non-recurring decimal and the numbers like π,e is also non-recurring decimal, so all non-recurrent infinite decimals are known as irrational numbers.
Question 3:π is written 22/7, then why is it called an irrational number?
Answers:π is the ratio between circumference and diameter, practically its value is evaluated by many mathematicians which comes to π=3.14159265358979…….,it can be assumed that either circumference or diameter or both can be irrational. The value of π=3.14 or 22/7 is actually an approximate value that we use in mathematics, and of course, this value 3.14 or 22/7 is a rational number.So the extraction is π is an irrational number but 3.14 or 22/7 is rational number.
Q4:How can you write 3.121212…in the form of P/Q?
Answer:
step 1:Write the given number equivalent to a variable
x = 3.121212…..(i)
step 2:Since after decimal two digits are repeating,so place two zeroes after 1 that makes 100,multiply equation (1) by 100
100x =312.121212…..(ii)
Subtract equation (i) from equation (ii),we get
99x =308
x =308/99
Therefore
3.121212…=308/99
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Chapter 1-Sets | Chapter 9-Sequences and Series |
Chapter 2- Relations and functions | Chapter 10- Straight Lines |
Chapter 3- Trigonometry | Chapter 11-Conic Sections |
Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |
Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |
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Chapter 1-Relations and Functions | Chapter 9-Differential Equations |
Chapter 2-Inverse Trigonometric Functions | Chapter 10-Vector Algebra |
Chapter 3-Matrices | Chapter 11 – Three Dimensional Geometry |
Chapter 4-Determinants | Chapter 12-Linear Programming |
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