Informative line

# Absolute Value

• The absolute value is the numerical value of any number.
• On a number line, the distance of any number from zero is the absolute value of the number.
• Since distance is always positive, so the absolute value of any number is always positive.
• We use modulus sign (||) for an absolute value.

For example: The absolute value of $$\dfrac {-3}{4}$$ is represented as $$\left | {\dfrac {-3}{4}}\right|$$ which is $$\dfrac {3}{4}$$.

#### Which one of the following is the absolute value of $$\dfrac {-3}{7}$$?

A $$\dfrac {3}{7}$$

B $$\dfrac {-3}{7}$$

C $$\dfrac {7}{3}$$

D $$\dfrac {3}{-7}$$

×

The absolute value of any number is the numerical value of that number and is always positive.

So, the numerical value of $$\dfrac {-3}{7}$$ is $$\dfrac {3}{7}$$  i.e.

$$\left |\dfrac {-3}{7}\right|=\dfrac {3}{7}$$

Hence, option (A) is correct.

### Which one of the following is the absolute value of $$\dfrac {-3}{7}$$?

A

$$\dfrac {3}{7}$$

.

B

$$\dfrac {-3}{7}$$

C

$$\dfrac {7}{3}$$

D

$$\dfrac {3}{-7}$$

Option A is Correct

# Rational Number

• A rational number is represented in the form of $$\dfrac {p}{q}$$, where $$p$$ and $$q$$ are integers and $$q\neq0$$.
• A rational number can be an integer, a fraction or a decimal.
• To understand it, consider an example:

$$2,\;\dfrac {3}{5}$$ and $$0.25$$ are rational numbers. Check whether the statement is true or not?

• Now, we have to find out whether the given numbers are rational or not.
• Let's consider the $$2$$ first.

$$2$$ is an integer. It can be written in the form of $$\dfrac {p}{q}$$ by putting it over $$1$$.

So, $$2=\dfrac {2}{1}$$

Thus, $$2$$ is a rational number.

• Now, consider $$\dfrac {3}{5}$$.
• $$\dfrac {3}{5}$$ is a fraction which is already in the form of $$\dfrac {p}{q}$$. So, $$\dfrac {3}{5}$$ is a rational number.
• Lastly, $$0.25$$ is a decimal number which can also be simplified into a fraction i.e.

$$0.25=\dfrac {25}{100}=\dfrac {1}{4}$$

So, $$0.25$$ is a rational number.

Note: All perfect square roots are rational numbers.

Example: $$\sqrt 4,\;\sqrt {16},\;\sqrt 9,\;\sqrt {25}.......$$

#### Which one of the following is NOT a rational number?

A $$-2$$

B $$\sqrt 3$$

C $$0.4$$

D $$\dfrac {-3}{7}$$

×

In option (A),

$$-2$$ is an integer and can be converted in the form of  $$\dfrac {p}{q}$$  by putting it over $$1$$.

So, $$-2=\dfrac {-2}{1}$$

Thus, $$-2$$ is a rational number.

Hence, option (A) is incorrect.

In option (B),

$$\sqrt 3$$ is not a perfect square root.

So, $$\sqrt 3$$ is not a rational number.

Hence, option (B) is correct.

In option (C),

$$0.4$$ is a decimal number which can be written in the form of  $$\dfrac {p}{q}$$ as

$$0.4=\dfrac {4}{10}$$ or $$\dfrac {2}{5}$$

So, $$0.4$$ is a rational number.

Hence, option (C) is incorrect.

In option (D),

$$\dfrac {-3}{7}$$ is a fraction which is already in the form of $$\dfrac {p}{q}$$.

So, $$\dfrac {-3}{7}$$ is a rational number.

Hence, option (D) is incorrect.

### Which one of the following is NOT a rational number?

A

$$-2$$

.

B

$$\sqrt 3$$

C

$$0.4$$

D

$$\dfrac {-3}{7}$$

Option B is Correct

# Representation of Rational Numbers (Decimal Form) on Number line

• Rational numbers (decimal form) can be represented on the number line.
• To do so, first, mark the wholes and then the parts.
• For rational (negative decimal) numbers, move left from zero on the number line.

For example:

• Mark $$-1.20$$ on the given number line.  • There are $$5$$ equal segments between each whole, which means each segment has a scale of $$0.20$$  • First, find the whole of $$-1.20$$, i.e. $$-1$$.
• To find the '$$-1$$', start from $$0$$, move in backward direction and mark '$$-1$$'.  • Next, we find parts of $$-1.20$$, i.e. $$0.20$$.
• To find the '$$.20$$', start from '$$-1$$',  move in backward direction and mark $$-1.20$$
• Hence, the number line represents $$-1.20$$  Hence, the above number line represents $$-1.20$$

#### Which one of the following options represents the location of point $$P$$ on the given number line?

A $$-1.75$$

B $$-0.75$$

C $$-0.80$$

D $$-1.80$$

×

On the given number line, there are four equal segments between each whole, which means each segment has a scale of $$0.25$$. To find the location of point $$P$$ on the given number line, start from $$0$$ by moving in the backward direction and continue moving till we reach point $$P$$.

Count the number of segments we moved to reach the point $$P$$.

Here, we moved $$3$$ segments, each of length $$0.25$$

Thus, point $$P$$ represents the point  $$0.75(=3\times0.25)$$ on the number line. Hence, option (B) is correct.

### Which one of the following options represents the location of point $$P$$ on the given number line? A

$$-1.75$$

.

B

$$-0.75$$

C

$$-0.80$$

D

$$-1.80$$

Option B is Correct

# Representation of Rational Numbers (Fraction Form) on Number Line

• Rational numbers (fraction form) can be represented on the number line.
• To represent the rational number $$\dfrac {a}{b}$$ on a number line, follow the given steps:
• Define the intervals starting from zero.
• Divide each interval on the right side of zero for the positive rational number and left side for the negative rational number.
• The size of each part is $$\dfrac {1}{b}$$.
• The rational number $$\dfrac {a}{b}$$ represents the combined length of '$$a$$' parts of size $$\dfrac {1}{b}$$.

For example: Represent $$\dfrac {4}{5}$$ on the number line.

Step 1: Define the interval $$0$$ to $$1$$.  Step 2: Divide the interval into $$5$$ equal parts.  Step 3: Size of each part is $$\dfrac {1}{5}$$.  Step 4: Thus, $$\dfrac {4}{5}$$ represents the combined length of $$4$$ parts.  Step 5: The resulting number line representation is  #### Which one of the following options represents the location of point $$S$$ on the given number line?

A $$\dfrac {-1}{2}$$

B $$\dfrac {1}{2}$$

C $$\dfrac {-1}{4}$$

D $$\dfrac {1}{4}$$

×

On the given number line, point $$S$$ lies in between $$0$$ and $$-1$$. Since the interval between $$0$$ and  $$-1$$ is divided into four equal parts, so the size of each part represents $$\dfrac {1}{4}$$. Point $$S$$ represents the combined length of $$2$$ parts from $$0$$ in the leftward direction.

So, point $$S$$ represents $$-\dfrac {2}{4}$$ or $$-\dfrac {1}{2}$$. Hence, option (A) is correct.

### Which one of the following options represents the location of point $$S$$ on the given number line? A

$$\dfrac {-1}{2}$$

.

B

$$\dfrac {1}{2}$$

C

$$\dfrac {-1}{4}$$

D

$$\dfrac {1}{4}$$

Option A is Correct

# Representation of Rational Numbers (Mixed Numbers) on Number line

• Rational numbers (mixed number form) can be represented on the number line.
• To represent the rational number '$$a\dfrac {b}{c}$$' on a number line, follow the given steps:
• First, find the whole number.
• Define the intervals starting from '$$a$$'.
• Divide each interval into '$$c$$' equal parts.
• The size of each part is $$\dfrac {1}{c}$$.
• The rational number $$a\dfrac {b}{c}$$ represents the combined length of '$$b$$' parts of size $$\dfrac {1}{c}$$ to the right of $$a$$.

For example: Represent $$2\dfrac {3}{4}$$ on the number line.

In the rational number $$2\dfrac {3}{4}$$$$2$$ is the whole and $$\dfrac {3}{4}$$ is the fraction.

Step 1: Find the whole number, i.e. $$2$$.  Step 2: Divide the interval between  $$2$$ and $$3$$ into $$4$$ equal parts.  Step 3: The size of each part is $$\dfrac {1}{4}$$.  Step 4: Thus, $$2\dfrac {3}{4}$$ represents the combined length of $$3$$ parts.  Step 5: The resulting number line representation is  #### Which one of the following options represents the location of point $$T$$ on the given number line?

A $$-4\dfrac {1}{3}$$

B $$-3\dfrac {1}{3}$$

C $$-4\dfrac {2}{3}$$

D $$-3\dfrac {2}{3}$$

×

On the given number line, point $$T$$ lies in between $$-3$$ and $$-4$$.

So, the whole is $$-3$$, as point $$T$$ lies to the left of $$-3$$ The size of each part between $$-3$$ and $$-4$$ represents $$\dfrac {1}{3}$$ as it is divided into $$3$$ equal parts. Point $$T$$ represents the combined length of $$2$$ parts. Thus, $$-3\dfrac {2}{3}$$ represents the location of point $$T$$ on the given number line.

Hence, option (D) is correct.

### Which one of the following options represents the location of point $$T$$ on the given number line? A

$$-4\dfrac {1}{3}$$

.

B

$$-3\dfrac {1}{3}$$

C

$$-4\dfrac {2}{3}$$

D

$$-3\dfrac {2}{3}$$

Option D is Correct