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Express 2.927927927… as a rational number

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  • Last Updated : 01 Jun, 2022
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The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five.

Number System

A Number system or numeral system is defined as an elementary system to express numbers and figures. It is a unique way of representing numbers in arithmetic and algebraic structure.

Numbers are used in various arithmetic values applicable to carry out various arithmetic operations like addition, subtraction, multiplication, etc which are applicable in daily lives for the purpose of calculation. The value of a number is determined by the digit, its place value in the number, and the base of the number system.

Numbers generally are also known as numerals are the mathematical values used for counting, measurements, labeling, and measuring fundamental quantities. Numbers are the mathematical values or figures used for the purpose of measuring or calculating quantities. It is represented by numerals as 2, 4, 7, etc. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.

Types Of Numbers

There are different types of numbers categorized into sets by the real number system. The types are described below:

  • Natural numbers: Natural numbers are the positive numbers that count from 1 to infinity. The set of natural numbers is represented by ‘N’. It is the numbers we generally use for counting. The set of natural numbers can be represented as N = 1, 2, 3, 4, 5, 6, 7,…
  • Whole numbers: Whole numbers are positive numbers including zero, which counts from 0 to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. The set can be represented as W = 0, 1, 2, 3, 4, 5,…
  • Integers: Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is denoted by ‘Z’. The set of integers can be represented as Z = …,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…
  • Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. It can be expressed as 2.5, 0.567, etc.
  • Real number: Real numbers are the set numbers that do not include any imaginary value. It includes all the positive integers, negative integers, fractions, and decimal values. It is generally denoted by ‘R’.
  • Complex number: Complex numbers are a set of numbers that include imaginary numbers. It can be expressed as a + bi where “a” and “b” are real numbers. It is denoted by ‘C’.
  • Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals. It is denoted by ‘Q’.
  • Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. It is denoted by ‘P’.

Express 2.927927927… as a rational number

Solution: 

Given: 2.927927927 or 2.\bar{927}

lets assume x = 2.927927927… ⇢ (1)

And there are three digits after decimal which are repeating,

So, multiply equation (1) both sides by 1000,

So 1000 x = 2927.\bar{927}  ⇢ (2) 

Now subtract equation (1) from equation (2) 

1000x – x   =  2927.\bar{927}  – 2.\bar{927}

999x = 2925

x = 2925/999

= 325/111

2.927927927 can be expressed 325/111 as rational number 

Similar Problems

Question 1: Express 7.765765765… as a rational number of the form p/q, where p and q have no common factors.

Solution: 

Given: 7.765765765 or 7.\bar{765}

Let’s assume x = 7.765765765… ⇢ (1)

And, there are three digits after decimal which are repeating

So multiply equation (1) both sides by 1000

So, 1000x = 7765.\bar{765}  ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 7765.\bar{765}  – 7.\bar{765}

999x = 7758

x = 7758/999

7.765765765 can be expressed 7758/999 as rational number 

Question 2: Express 10.927927927… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 10.927927927… or 10.\bar{927}

Let’s assume x = 10.927927927… ⇢ 1

And there are three digits after decimal which are repeating

So multiply equation 1 both sides by 1000

So 1000 x = 10927.\bar{927}  ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 10927.\bar{927}  – 10.\bar{927}

999x = 10917

x = 10917/999

= 1213/111

10.927927927 can be expressed 1213/111 in form of p/q as rational number 

Question 3: Express 1.272727… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 1.272727… or 1.\bar{27}

Let’s assume x = 1.272727…. ⇢  (1)

And there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = 127.\bar{27}  ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 127.\bar{27}  – 1.\bar{27}

99x = 126

x = 126/99                     

1.272727…. can be expressed 126/99 in form of p/q as rational number 

Question 4:  Express 2.37373737… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given : 2.37373737… or 2.\bar{37}

Let’s assume x = 2.37373737…. ⇢  (1)

And, there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So, 100 x = 237.\bar{37}  ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 237.\bar{37}  – 2.\bar{37}

99x = 235

x = 235/99                    

2.37373737… can be expressed 235/99 in form of p/q as rational number 

Question 5: Express 15.827827827… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 15.827827827… or 15.\bar{827}

Let’s assume x = 15.827827827… ⇢ (1)

And there are three digits after decimal which are repeating, so multiply equation (1) both sides by 1000,

So 1000 x = 15827.\bar{827}  ⇢ (2)

Now, subtract equation (1) from equation (2)

1000x – x = 15827.\bar{827}  – 15.\bar{827}

999x = 15812

x = 15812/999

= 15812/999

15.827827827 can be expressed 15812/999 in form of p/q as rational number 


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