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# What is the 5280th digit in the decimal expansion of 9/17?

In mathematics, a numeral or number system is a writing system for expressing numbers in various forms. It is a mathematical notation for expressing numbers consistently using digits or other symbols. A number is a mathematical value used to count or measure objects and perform various arithmetic calculations. We have different types of numbers, like natural numbers, integers, decimals, rational numbers, irrational numbers, etc. Similarly, we also have different types of number systems, such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

## Rational Numbers

A number that can be expressed in the form of p/q, where p and q are integers, and q is a non-integer, is called a rational number. Here, p is known as a numerator, and q is known as a denominator. So, a number is said to be rational if it has a non-zero denominator. A rational number is a real number and all natural numbers, whole numbers, integers, fractions of integers, and decimals are subsets of rational numbers.

“Q” represents the set of rational numbers. The term “rational” is derived from the word “ratio”. Hence, we can deduce that a number is said to be rational if it can be expressed as a fraction where the numerator is an integer, and the denominator is a non-zero integer. -3/8, -0.123, 0, 1/3, 4/5, 9/3, etc., are some examples of rational numbers. All terminating and all recurring decimals are rational numbers, whereas non-terminating and non-recurring decimals are irrational numbers.

## Method to Represent Recurring Decimals as Rational Numbers

Now, let us discuss a method to represent recurring decimals or repeating decimals as rational numbers.

Example: Represent 0.67676767… as a Rational Number.

Solution:

Let X = 0.67676767…

Now, we have to identify, after how many places the pattern is repeated. We can notice that the pattern repeats after 2 places. Now, multiply the given number by 100 such that exactly one cycle of the pattern moves to the left.

⇒100X = 67. Now subtract 67 on both sides, and we get,

⇒ 100X − 67 = − 67

⇒ 100X − 67 = We know that, So, 100X − 67 = X

100X − X = 67

99X = 67

X = 67/99

So, 67/99 is the rational representation of the decimal 0.67676767…

## What is the 5280th digit in the decimal expansion of 9/17?

Solution:

From the definition of a rational number, 9/17 is a rational number.

So, the decimal value of 9/17 must be either a terminating decimal or a recurring decimal.

The value of 9/17 is approximately equal to 0.52941176470588235294117647058824……..

We can observe that the decimal is non-terminating. So, it must be a recurring decimal. Now, we have to identify, after how many places the pattern is repeated. We can notice that the repeated pattern is very long. So, let us multiply the given number by a large power of ten so that the place of the decimal point shifts to the right. Here, we multiplying the number by 1010.

0.52941176470588235294117647058824… × 10000000000 = 5294117647.0588235294117647058824…

Now, let us subtract the whole number part from the obtained number.

5294117647.05882352941176470588235……. − 5294117647 = 0.05882352941176470588235294..……

We can notice a repeating pattern a bit farther in the obtained decimal expansion. Now, repeat the same technique that we applied above, i.e., multiplying the given number by a large power of ten and then subtracting.

0.0582352941176470588235294…….. × 1000000 = 0.52941176470588235294117647058824……..

Now, we can observe the repeat pattern in full. By counting carefully, we can observe that the pattern is repeating after 16 digits (5294117647058823).

Now, we need to find the 5280th digit in the decimal expansion of 9/17. Divide 5280 by 16.

5280 ÷ 16 = 330

So, 5280 is a multiple of 16. That means the pattern will be completed for the 330th time on the 5280th digit. As we know, the end digit of the pattern is 3, so the 5280th digit will be 3.

Hence, the 5280th digit in the decimal expansion of 9/17 is 3.

## Solved Examples

Example 1: Is √13 a rational number?

Solution:

No, √13 is not a rational number. The value of √13 is 3.6055512754639… It is a non-terminating and non-recurring decimal. So, √13 is an irrational number.

Example 2: What is the decimal form of the rational number 7/21?

Solution:

The given rational number, 7/21, can be converted into a decimal number by dividing the numerator by the denominator. So, by dividing 7 by 21, we get 0.3333333…, which is a recurring decimal. Hence, the decimal value of 6/24 is 0.3333333334.

Example 3: Determine the rational numbers among the following numbers:

a) √(16/4)

b) 7/5

c) 2/√3

d) -6.7082039324993690…

Solution:

Remember that, when a rational number is simplified, it should either be a terminating or non-terminating and recurring decimal.

a) √(16/4) = √4 = 2

2 is a rational number. So, √(16/4) is a rational number.

b) 7/5 = 1.4

The obtained number is a terminating decimal. So, 7/5 is a rational number.

c) 2/√3 = 1.1547005383792515….

The obtained number is a non-terminating and non-recurring decimal. So, 2/√3 is an irrational number.

d) -6.7082039324993690…

The given number is a non-terminating and non-recurring decimal. So, the given number is irrational.

Example 4: Is “e” a rational number?

Solution:

No, e is not a rational number. The value of e is 2.718281828459045… It is a non-terminating and non-recurring decimal. So, “e” is an irrational number.

Example 5: From the given statements choose the correct statement concerning rational numbers.

a) All non-terminating decimals are rational numbers.

b) 1/8 is a rational number.

Solution:

a) No, not all non-terminating decimals are rational numbers. Rational numbers can either be terminating decimals or non-terminating and repeating decimals. So, the given statement is false.

b) 1/8 = 0.125. The obtained number is a terminating decimal. So, 1/8 is a rational number. So, the given statement is true.

## FAQs on Number System

Question 1:  What is meant by a rational number? Give some examples.

A number that can be expressed in the form of p/q, where p and q are integers, and q is a non-integer, is called a rational number. -3/8, -0.123, 0, 1/3, 4/5, 9/3, etc., are some examples of rational numbers.

Question 2: What is meant by a number? What are the different types of numbers?

A number is a mathematical value used to count or measure objects and perform various arithmetic calculations. We have different types of numbers, like natural numbers, integers, decimals, rational numbers, irrational numbers, etc.

Question 3: What is a recurring decimal?

A recurring decimal or a repeating decimal is a decimal representation of a number whose digits repeat its values at regular intervals, i.e., the digits are periodic, and the infinitely repeated portion must not be zero. For example, 0.123512351235… is a recurring decimal, but 13.0000… is not a recurring decimal.

Question 4: Define an Irrational Number.