Exercise 1.3 (Number Systems)

Shubham Vision

Academic

2025-01-11

Introduction

This exercise focuses on understanding the decimal representation of rational numbers and identifying the types of decimal expansions they exhibit. It introduces students to two main types of decimal expansions:

Terminating Decimals: Decimals that come to an end after a finite number of digits. For example, and
Non-Terminating Repeating Decimals: Decimals that never end but have a repeating block of digits. For instance, (where repeats infinitely).

Exercise 1.3

Question 1

Write the following in decimal form and say what kind of decimal expansion each has:
(i)
(ii)
(iii)
(iv)
(v)
(vi)

Solution:

Let’s solve each part step by step.

Divide by
(Exact division with no remainder).
Type: Terminating decimal.
:
Divide by :
$= 0. 0999..$ ( repeats forever).
Type: Non-terminating repeating decimal.
:
Divide by :
(Exact division with no remainder).
Type: Terminating decimal.
:
Divide 3 by :
( repeats forever).
Type: Non-terminating repeating decimal.
:
Divide by :
( repeats forever).
Type: Non-terminating repeating decimal.
:
Divide by :
(Exact division with no remainder).
Type: Terminating decimal.

Question 2

You know that . Can you predict what the decimal expansions of are, without actually dividing? If so, how?

Solution:

The decimal expansion of .
For multiples of , the repeating sequence will simply shift depending on the numerator.

$0. 428571$ .... (Repeats 428571)

Explanation: The decimal expansion shifts but repeats in the same order.

Question 3

Express the following in the form , where and are integers and :
(i)
(ii)
(iii)

Solution:

(i)

Let
Multiply both sides by to shift the repeating part:
.
Subtract
.
Solve for $x$ :

Answer:

(ii)

Let
Multiply both sides by to shift the repeating part:
Subtract
Solve for $x$ :

Answer:

(iii) :

Let
Multiply both sides by to shift the repeating part:
Subtract
Solve for $x$ :

Answer:

Question 4

Express (i.e., ) in the form . Are you surprised by your answer? Discuss.

Solution:

Let
Multiply both sides by to shift the repeating part:
Subtract
Solve for $x$ :

Answer:

Discussion: It might seem surprising, but mathematically, is equal to . This is because the difference between and $1$ is infinitely small (practically zero).

Question 5

What can the maximum number of digits be in the repeating block of the decimal expansion of ? Perform the division to check your answer.

Solution:

When dividing $1$ by , perform long division:

Observe the repeating block:

Answer:
The repeating block has 16 digits.

Question 6

Look at several examples of rational numbers in the form (where ). Can you guess what property $q$ must satisfy for the decimal expansion of to terminate?

Solution:

A rational number has a terminating decimal expansion if, in its simplest form, the denominator has only 2 and/or 5 as its prime factors.

Examples:

(Terminating: ).
(Non-terminating: ).

Answer:
For a rational number to have a terminating decimal, $q$ (in simplest form) must be of the form , where $m$ and $n$ are non-negative integers.

Question 7

Write three numbers whose decimal expansions are non-terminating and non-repeating.

Solution:

Examples of non-terminating and non-repeating numbers:

Explanation:

Non-terminating: These decimals go on forever without stopping.
Non-repeating: There is no repeating pattern in their decimal expansion.

Answer:
Three numbers are √

Question 8

Find three different irrational numbers between and .

Solution:

Irrational numbers between and $3$ :

Explanation:

Irrational numbers are non-terminating and non-repeating decimals.
They cannot be expressed as $q$ , where $p$ and $q$ are integers.

Answer:
Three examples are √ $.$

Question 9

Classify the following numbers as rational or irrational:

Solution:

: Irrational — √ is non-terminating and non-repeating.
: Rational — √, which is a whole number.
Rational — This decimal terminates after 4 digits.
: Rational — This decimal repeats the block $478$ .
: Irrational — There is no repeating pattern in the decimal.

Answer:

Irrational
Rational
Rational
Rational
Irrational

Important Questions Based On Ex. 1.3
( Ex. 1.3 पर आधारित महत्वपूर्ण प्रश्न )

Define Real Numbers. Explain their classification.
Prove that √is an irrational number.
Show that the sum of a rational number and an irrational number is always irrational.
Write five rational and five irrational numbers between $1$ and $2$ .
Prove that is an irrational number.
Find five rational numbers between and
Represent the following on the number line: √, √, √.
Classify the following numbers as rational or irrational: √, √ ,
Rationalize the denominator of the following: , .
State whether the following statements are true or false. Justify your answers: Every terminating decimal is a rational number. The product of two irrational numbers is always irrational. Every real number is a rational number.

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Chapters

Exercise 1.3 (Number Systems)

Introduction