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Introduction To Rational Numbers

What is a Rational Number?

  • A number that can be written in the form of \(\dfrac{a}{b}\), is called rational number, where \(a\) and \(b\) are integers and \(b\neq\) zero.

Note: 

An integer can be written in the form of \(\dfrac{a}{b}\) by putting it over one \((1)\).

Thus, integers are rational numbers.

For example:

\(2=\dfrac{2}{1}\)

\(-3=\dfrac{-3}{1}\)

\(0=\dfrac{0}{1}\)

Illustration Questions

Which one of the following options does NOT represent a rational number?

A \(5\)

B \(\dfrac{-3}{2}\)

C Zero

D None of these

×

Option (A):

\(5\) can be written in the form of \(\dfrac{a}{b}\) by putting it over \(1.\)

\(\Rightarrow\;5=\dfrac{5}{1}\)

\(\therefore\) It is a rational number.

Hence, option (A) is incorrect.

Option (B):

\(\dfrac{-3}{2}\) is written in the form of \(\dfrac{a}{b}\).

\(\therefore\) It is a rational number.

Hence, option (B) is incorrect.

Option (C):

Zero \((0)\) can be written in the form of \(\dfrac{a}{b}\) by putting it over \(1.\)

\(\Rightarrow\;0=\dfrac{0}{1}\)

\(\therefore\) It is a rational number.

Hence, option (C) is incorrect.

Hence, option (D) is correct.

Which one of the following options does NOT represent a rational number?

A

\(5\)

.

B

\(\dfrac{-3}{2}\)

C

Zero

D

None of these

Option D is Correct

Positive and Negative Rational Numbers

  • A rational number can be positive, negative or zero.
  • A rational number is positive if both numerator and the denominator have same signs, i.e. \((+)\) or \((-)\).

For example: \(\dfrac{3}{4},\;\dfrac{-3}{-4}\)

Numerator \((3)\) and denominator \((4)\) of \(\dfrac{3}{4}\) have same signs \((+)\).

Therefore, \(\dfrac{3}{4}\) is a positive rational number.

Similarly, numerator \((-3)\) and the denominator \((-4)\) of \(\dfrac{-3}{-4}\) have same signs\((-)\). Therefore, \(\dfrac{-3}{-4}\) is also a positive rational number.

  • A rational number is negative if both numerator and the denominator have opposite signs.

For example: \(\dfrac{-5}{8},\;\dfrac{5}{-8}\)

Numerator \((-5)\) and denominator \((8)\) of \(\dfrac{-5}{8}\) have opposite signs (– and + respectively).

Therefore, \(\dfrac{-5}{8}\) is a negative rational number.

Similarly, numerator \((5)\) and denominator \((-8)\) of \(\dfrac{5}{-8}\) have opposite signs (+ and – respectively).

Therefore, \(\dfrac{5}{-8}\) is a negative rational number.

Note:

If '\(a\)' and '\(b\)' are integers then

\(-\left(\dfrac{a}{b}\right)=\dfrac{(-a)}{b}=\dfrac{a}{(-b)}\)

where \(b\neq\) zero

Positive and negative rational numbers are used to describe quantities having opposite directions or values.

For example:

'\($500\) debited from the account', represents a negative value, i.e.

\(-500\)

While '\($500\) credited to the account', represents a positive value, i.e.

\(500\)

Illustration Questions

Which one of the following rational numbers does NOT have the same value as \(\dfrac{-4}{5}\)?

A \(\dfrac{4}{-5}\)

B \(-\dfrac{4}{5}\)

C \(\dfrac{-4}{-5}\)

D

×

We know that

If '\(a\)' and '\(b\)' are integers then

\(-\left(\dfrac{a}{b}\right)=\dfrac{(-a)}{b}=\dfrac{a}{(-b)}\)

where \(b\neq\) zero

So, \(\dfrac{-4}{5}\) is a negative rational number and it can be written as:

\(-\left(\dfrac{4}{5}\right)=\dfrac{-4}{5}=\dfrac{4}{-5}\)

In options (A) and (B), \(\dfrac{4}{-5}\) and \(-\dfrac{4}{5}\), both represent the same negative value as \(\dfrac{-4}{5}\).

Thus, option (A) and (B) are incorrect.

In option (C), numerator \((-4)\) and denominator \((-5)\) of \(\dfrac{-4}{-5}\) have same signs \((-)\).

Therefore, \(\dfrac{-4}{-5}\) is a positive rational number.

Thus, values of \(\dfrac{-4}{5}\) and \(\dfrac{-4}{-5}\) are not same.

Hence, option (C) is correct.

Which one of the following rational numbers does NOT have the same value as \(\dfrac{-4}{5}\)?

A

\(\dfrac{4}{-5}\)

.

B

\(-\dfrac{4}{5}\)

C

\(\dfrac{-4}{-5}\)

D

Option C is Correct

Equivalent Rational Numbers

  • Two or more rational numbers are equivalent to each other if their simplest forms are equal.
  • When the numerator and denominator of a rational number do not have any common factor other than one \((1)\), it is called the simplest form of a rational number.

For example: \(\dfrac{3}{6}\) and \(\dfrac{9}{18}\)

Simplest form of \(\dfrac{3}{6}=\dfrac{3\div3}{6\div3}\)

\(=\dfrac{1}{2}\)

Simplest form of \(\dfrac{9}{18}=\dfrac{9\div9}{18\div9}\)

\(=\dfrac{1}{2}\)

Here, the simplest form of \(\dfrac{3}{6}\) and \(\dfrac{9}{18}\) are equal, i.e. \(\dfrac{1}{2}\).

\(\therefore\;\dfrac{3}{6}\) and \(\dfrac{9}{18}\) are equivalent rational numbers.

\(\dfrac{3}{6}=\dfrac{9}{18}\)

  • We can obtain an equivalent of a rational number by dividing or multiplying both numerator and the denominator by same non-zero number.

For example: \(-\dfrac{4}{6}\)

\(-\dfrac{4}{6}=-\dfrac{4\div2}{6\div2}=-\dfrac{2}{3}\)

\(-\dfrac{4}{6}=-\dfrac{4×3}{6×3}=-\dfrac{12}{18}\)

Thus, \(-\dfrac{4}{6},\;-\dfrac{2}{3}\) and \(-\dfrac{12}{18}\) are equivalent rational numbers, i.e.

\(-\dfrac{4}{6}=-\dfrac{2}{3}=-\dfrac{12}{18}\)

Illustration Questions

Which of the following is true?

A \(\dfrac{-3}{4}=\dfrac{-6}{2}\)

B \(\dfrac{8}{7}=\dfrac{4}{-21}\)

C \(\dfrac{16}{28}=\dfrac{4}{7}\)

D \(\dfrac{13}{53}=\dfrac{1}{4}\)

×

Two or more rational numbers are equivalent to each other if their simplest forms are equal.

Option (A): \(\dfrac{-3}{4}=\dfrac{-6}{2}\)

\(\dfrac{-3}{4}\) is in its simplest form.

But \(\dfrac{-6}{2}\) can further be simplified by dividing numerator and denominator by \(2.\)

\(=\dfrac{-(6\div2)}{2\div2}\)

\(=\dfrac{-3}{1}\)

Both the simplest forms \(\left(\dfrac{-3}{4}\;\text{and}\;\dfrac{-3}{1}\right)\) are not equal.

\(\therefore\;\dfrac{-3}{4}\neq\dfrac{-6}{2}\)

Hence, option (A) is false.

Option (B): \(\dfrac{8}{7}=\dfrac{4}{-21}\)

Here, \(\dfrac{8}{7}\) and \(\dfrac{4}{-21 }\) both the rational numbers are already in their simplest forms.

Both the simplest forms \(\left(\dfrac{8}{7}\;\text{and}\;\dfrac{4}{-21}\right)\) are not equal.

\(\therefore\;\dfrac{8}{7}\neq\dfrac{4}{-21}\)

Hence, option (B) is false.

Option (C): \(\dfrac{16}{28}=\dfrac{4}{7}\)

\(\dfrac{4}{7}\) is in its simplest form.

But \(\dfrac{16}{28}\) can further be simplified by dividing numerator and denominator by \(4.\)

\(=\dfrac{16\div4}{28\div4}\)

\(=\dfrac{4}{7}\)

Both the simplest forms \(\left(\dfrac{4}{7}\;\text{and}\;\dfrac{4}{7}\right)\) are equal.

\(\therefore\;\dfrac{16}{28}=\dfrac{4}{7}\)

Hence, option (C) is true.

Option (D): \(\dfrac{13}{53}=\dfrac{1}{4}\)

Here, \(\dfrac{13}{53}\) and \(\dfrac{1}{4}\) both the rational numbers are already in their simplest forms.

Both the simplest forms \(\left(\dfrac{13}{53}\;\text{and}\;\dfrac{1}{4}\right)\) are not equal.

\(\therefore\;\dfrac{13}{53}\neq\dfrac{1}{4}\)

Hence, option (D) is false.

Which of the following is true?

A

\(\dfrac{-3}{4}=\dfrac{-6}{2}\)

.

B

\(\dfrac{8}{7}=\dfrac{4}{-21}\)

C

\(\dfrac{16}{28}=\dfrac{4}{7}\)

D

\(\dfrac{13}{53}=\dfrac{1}{4}\)

Option C is Correct

Plotting a Rational Number on the Number Line

  • Plotting a rational number is similar to plotting a fraction on the number line.
  • Consider the following two cases:

Case 1: For positive rational numbers:

Let's plot \(\dfrac{5}{8}\) on the number line.

Both numerator and the denominator have the same signs.

\(\therefore\;\dfrac{5}{8}\) is a positive number.

First draw a number line and divide the interval \(0\) to \(1\) into \(8\) parts.

Now, count \(5\) units from zero to its right. It is also called moving up or moving forward from zero up to \(5\) units.

Note: All positive numbers are at the right side of zero on the number line.

Case 2: For negative rational numbers:

Let's plot \(\dfrac{-5}{8}\) on the number line.

Both numerator and the denominator have the opposite signs.

\(\therefore\;\dfrac{-5}{8}\) is a negative number.

First, draw a number line and divide the interval \(0\) to \(-1\) into \(8\) parts.

Now, count \(5\) units from zero to its left. It is also called moving down or moving backward from zero up to \(5\) units.

Note: All negative numbers are at the left side of zero on the number line.

Note : All negative numbers are at the left side of zero on the number line.

Illustration Questions

Which one of the number lines represents \(2\dfrac{1}{3}\) as point \(G\) ?

A

B

C

D

×

Given: \(2\dfrac{1}{3}\)

\(2\dfrac{1}{3}\) can be written as:

\(2+\dfrac{1}{3}\)

\(=\dfrac{(2×3)+1}{3}\)

\(=\dfrac{6+1}{3}\)

\(=\dfrac{7}{3}\)

Both numerator and the denominator have the same signs.

\(\therefore\;\dfrac{7}{3}\) is a positive number.

Draw a number line and divide each interval or segment (like \(0\) to \(1\)) into \(3\) parts.

image

Now, count \(7\) units from zero to its right. It is also called moving up or forward from zero up to \(7\) units.

image

Hence, option (A) is correct.

Which one of the number lines represents \(2\dfrac{1}{3}\) as point \(G\) ?

A image
B image
C image
D image

Option A is Correct

Distance between Two Rational Numbers

  • When two points are plotted on a number line, they are known as endpoints.
  • The distance between two endpoints is always positive whether they are positive or negative.

Distance can be calculated by using two methods

Method I: By using distance formula

  • If '\(a\)' and '\(b\)' are two rational numbers, then the distance between them on the number line is:

\(|a-b|\;\text{or}\;|b-a|\)

For example: Distance between \(\dfrac{4}{5}\) and \(\dfrac{6}{5}=\left|\dfrac{4}{5}-\dfrac{6}{5}\right|=\left|-\dfrac{2}{5}\right|=\dfrac{2}{5}\) units

or  \(\left|\dfrac{6}{5}-\dfrac{4}{5}\right|=\left|\dfrac{2}{5}\right|=\dfrac{2}{5}\) units

Method II: By using number line

Consider an example to understand it.

Here, \(\dfrac{-4}{3}\) and \(2\) are rational numbers that are representing the endpoints on the number line.

Step 1: Count the number of units from one endpoint to another.

Step 2: There are \(10\) units of \(\dfrac{1}{3}\;\left(10\:\text{times}\;\dfrac{1}{3}\right)\) between endpoints, i.e. \(\dfrac{-4}{3}\) and \(2.\)

Thus, distance between \(\dfrac{-4}{3}\) and \(2\)

\(=10×\dfrac{1}{3}\)

\(=\dfrac{10}{3}\) units

Illustration Questions

Two rational numbers, \(P\) and \(Q\) are plotted on the given number line. Find the values of \(P\) and \(Q.\) Also find the distance between them.

A \(P=\dfrac{-3}{4},\;Q=\dfrac{-4}{3},\;\text{Distance}=\dfrac{-2}{4}\text{units}\)

B \(P=\dfrac{-2}{3},\;Q=\dfrac{-7}{3},\;\text{Distance}=\dfrac{5}{3}\text{units}\)

C \(P=\dfrac{-7}{3},\;Q=\dfrac{-2}{3},\;\text{Distance}=\dfrac{-5}{3}\text{units}\)

D \(P=\dfrac{-4}{3},\;Q=\dfrac{-3}{4},\;\text{Distance}=\dfrac{1}{3}\text{units}\)

×

Given:

image

Since \(P\) and \(Q\) both are plotted on the left side of zero, therefore both are negative rational numbers.

Here, each unit \(=\dfrac{1}{3}\)

\(\therefore\;P=\dfrac{-2}{3}\) and \(Q=\dfrac{-7}{3}\)

Here, \(\dfrac{-2}{3}\) and \(\dfrac{-7}{3}\) are rational numbers that are representing the endpoints on the number line.

Count the number of units from one endpoint to another.

image

There are \(5\) units of \(\dfrac{1}{3}\;\left(5\;\text{times}\;\dfrac{1}{3}\right)\) between the end points.

Thus, distance between \(\dfrac{-7}{3}\;(Q)\) and \(\dfrac{-2}{3}\;(P)\)

\(=5×\dfrac{1}{3}\)

\(=\dfrac{5}{3}\) units

Hence, option (B) is correct.

Two rational numbers, \(P\) and \(Q\) are plotted on the given number line. Find the values of \(P\) and \(Q.\) Also find the distance between them.

image
A

\(P=\dfrac{-3}{4},\;Q=\dfrac{-4}{3},\;\text{Distance}=\dfrac{-2}{4}\text{units}\)

.

B

\(P=\dfrac{-2}{3},\;Q=\dfrac{-7}{3},\;\text{Distance}=\dfrac{5}{3}\text{units}\)

C

\(P=\dfrac{-7}{3},\;Q=\dfrac{-2}{3},\;\text{Distance}=\dfrac{-5}{3}\text{units}\)

D

\(P=\dfrac{-4}{3},\;Q=\dfrac{-3}{4},\;\text{Distance}=\dfrac{1}{3}\text{units}\)

Option B is Correct

Absolute Value of a Rational Number

  • An absolute value of a rational number is the distance of that number from zero on the number line.
  • We can show this distance by a sign of arrow.
  • An arrow points to right for positive rational numbers and to left for negative rational numbers.

For example: We will find the absolute value of \(\dfrac{-8}{5}\).

Since, distance is always positive, therefore the absolute value of a rational number is always positive.

Hence, absolute value of \(\dfrac{-8}{5}\) is \(\dfrac{8}{5}\).

or absolute value of \(\dfrac{-8}{5}=\left|\dfrac{-8}{5}\right|=\dfrac{8}{5}\)

Here, '\(|\;\;\;|\)' sign is used to represent the absolute value.

Note: Since, distance is always positive, therefore the absolute value of a rational number is always positive.

Hence, absolute value of \(\dfrac{-8}{5}\) is \(\dfrac{8}{5}\)

or absolute value of \(\dfrac{-8}{5}=\left|\dfrac{-8}{5}\right|=\dfrac{8}{5}\)

Here, '\(|\;\;\;|\)' sign is used to represent the absolute value.

Illustration Questions

Which is true?

A Absolute value of \(\dfrac{13}{15}\) is \(\dfrac{13}{-15}\).

B \(\left|\dfrac{-31}{42}\right|\)  \(=\dfrac{42}{31}\)

C Absolute value of \(\dfrac{11}{-2}\) is \(\dfrac{-2}{11}\).

D Absolute value of \(\dfrac{-14}{25}\) is \(\dfrac{14}{25}\).

×

An absolute value of a number is the number itself but always positive.

\(|a|=a\;\text{or}\;|-a|=a\)

Option (A):

Absolute value of \(\dfrac{13}{15}=\left|\dfrac{13}{15}\right|=\dfrac{13}{15}\)

Hence, option (A) is false.

Option (B):

Absolute value of \(\dfrac{-31}{42}=\left|\dfrac{-31}{42}\right|=\dfrac{31}{42}\)

Hence, option (B) is false.

Option (C):

Absolute value of \(\dfrac{11}{-2}=\left|\dfrac{11}{-2}\right|=\dfrac{11}{2}\)

Hence, option (C) is false.

Option (D):

Absolute value of \(\dfrac{-14}{25}=\left|\dfrac{-14}{25}\right|=\dfrac{14}{25}\)

Hence, option (D) is true.

Which is true?

A

Absolute value of \(\dfrac{13}{15}\) is \(\dfrac{13}{-15}\).

.

B

\(\left|\dfrac{-31}{42}\right|\)  \(=\dfrac{42}{31}\)

C

Absolute value of \(\dfrac{11}{-2}\) is \(\dfrac{-2}{11}\).

D

Absolute value of \(\dfrac{-14}{25}\) is \(\dfrac{14}{25}\).

Option D is Correct

Opposite Rational Numbers

  • An opposite of a rational number is the number itself but with a different sign.

For example: Opposite of \(3=-3\)

Opposite of \(\dfrac{-4}{3}=\dfrac{4}{3}\)

  • All positive numbers are opposites of their negatives and all negative numbers are opposites of their positives.
  • On the number line, the distance of opposite numbers from zero is same although they are on the opposite sides.

Example: Are \(-3\) and \(3\) opposite numbers?

Distance between \(0\) and \(3=|0-3|=3\)

Distance between \(0\) and \(-3=|0-(-3)|=3\)

Distance of both the numbers from zero is same, i.e. \(3\).

\(\therefore\) They are opposite numbers.

  • Zero doesn't have an opposite as it is neither negative nor positive.
  • The opposite of an opposite of a number is the number itself, i.e.

\(-(-9)=9\)

Here, \(-9\) is the opposite of \(9.\)

Opposite of \(-9=9\)

Other examples of opposites:

  1. The temperature above sea level is opposite to the temperature below sea level and vice versa.
  2. The amount debited from an account is opposite to the amount credited to the account and vice versa.
  3. Loss is opposite to gain and vice versa.
  4. Giving something is opposite to taking something and vice versa.

Illustration Questions

Which one of the following options represents the opposite pair of rational numbers?

A Marc has a loss of \($25.00\) and gain of \($50\) on purchasing some items.

B The temperature above sea level is \(40°C\) and temperature below sea level is \(38°C\).

C Kevin gives \(5\) chocolates and take \(6\) chocolates.

D \($40.00\) is debited from an account and \($40.00\) is credited to an account.

×

An opposite of a rational number is the number itself but with a different sign.

Option (A):

Loss of \($25.00\) represents \(-$25.00\) and gain of \($50\) represents \(+$50\).

Though both the numbers have opposite signs but they are different numbers.

Hence, option (A) is incorrect.

Option (B):

The temperature above sea level represents \(+40°C\) and temperature below sea level represents \(-38°C\).

Though both the numbers have opposite signs but they are different numbers.

Hence, option (B) is incorrect.

Option (C):

Kevin gives \(5\) chocolates means \(-5\) and takes \(6\) chocolates means \(+6\).

Though both the numbers have opposite signs but they are different numbers.

Hence, option (C) is incorrect.

Option (D):

\($40.00\) is debited from an account represents \(-$40.00\) and \($40.00\) is credited to an account represents \(+$40.00\).

\(\therefore\) Both the numbers have opposite signs and are same.

Hence, option (D) is correct.

Which one of the following options represents the opposite pair of rational numbers?

A

Marc has a loss of \($25.00\) and gain of \($50\) on purchasing some items.

.

B

The temperature above sea level is \(40°C\) and temperature below sea level is \(38°C\).

C

Kevin gives \(5\) chocolates and take \(6\) chocolates.

D

\($40.00\) is debited from an account and \($40.00\) is credited to an account.

Option D is Correct

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