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Express 0.151515….. as a rational number

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  • Last Updated : 31 Dec, 2021

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 generally used 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 0.151515….. as a rational number

Solution:  

Given: 0.151515..  or 0.\bar{15}

Let’s assume x = 0.1515… ⇢ (1)

And there are two digits after decimal which are repeating, so we will multiply equation 1 both sides by 100.

So 100x = 15.\bar{15} ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 15.\bar{15} – 0.\bar{15}

99x = 15

x = 15/99

= 5/33

0.151515..  can be expressed 5/33 as rational number 

Similar Problems

Question 1: Rewrite the decimal as a rational number. 0.777777777…?

Solution: 

Given: 0.7777..  or 0.\bar{7}

Let’s assume x = 0.77777… ⇢ (1)

And there are one digits after decimal which are repeating, so we will multiply equation 1 both sides by 10.

So 10x = 7.\bar{7} ⇢ (2)

Now subtract equation (1) from equation (2)

10x – x = 7.\bar{7} – 0.\bar{7}

9x = 7

x = 7/9                      

0.7777777…  can be expressed 7/9 as rational number 

Question 2:  Express 3.927927927… as a rational number of the form p/q, where p and q have no common factors.

Solution: 

Given: 3.927927927 or 3.\bar{927}

Let’s assume x = 3.927927927… ⇢ (1)

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

So 1000 x = 3927.\bar{927} ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 3927.\bar{927} – 3.\bar{927}

999x = 3924

x = 3924/999

= 1308/333

3.927927927 can be expressed 1308/333 as rational number 

Question 3: Rewrite the decimal as a rational number 4.3232323232 …?

Solution: 

Given: 4.3232323232 or 4.\bar{32}

Let’s assume x = 4.3232323232… ⇢ (1)

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

So 100 x = 432.\bar{32} ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 432.\bar{32} – 4.\bar{32}

99x = 428

x = 428/99

= 428/99

4.323232323 can be expressed 428/99 as rational number 

Question 4: Rewrite the decimal as a rational number. 0.69696969…?

Solution: 

Given: 0.696969.. or 0.\bar{69}

Let’s assume x = 0.696969… ⇢ (1)

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

So 100x = 69.\bar{69} ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 69.\bar{69} – 0.\bar{69}

99x = 69

x = 69/99 

= 23/33                    

0.69696969… can be expressed 23/33 as rational number

Question 5: Express 4.8568568586… as a rational number of the form p/q, where p and q have no common factors ?

Solution

Given: 4.8568568586… or 4.\bar{856}

Let’s assume x = 4.8568568586… ⇢ (1)

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

So 1000 x = 4856.\bar{856} ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 4856.\bar{856} – 4.\bar{856}

999x = 4852

x = 4852/999                        

4.8568568586 can be expressed 4852/999 as rational number


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