 ## Absolute Value

Subject: Compulsory Maths

#### Overview

Absolute Value is the distance the number is from zero on the number line. It is always a positive value.

##### Absolute Value

Absolute value describes the distance of a number line from 0 without considering which direction from the number lies. A whole number which is positive or negative is known as integers. They can be displayed on the number line while representing the number in the number line. The number in the right side of number line is always greater.

- 3> - 4, 0> - 10, -13< 1 etc

The distance of a number from a zero point is known as absolute values. For example:

| -3 | = 3; | 4 | = 4, - | 3 | = -3 etc

In calculations on the absolute value of an integer can be considered as a positive integer. For example:

2 x |-6| = 2 x 6 =12

5 x |7| = 5 + 7 =12

### Multiplication and division of two integers

Rules of signs  1. (+)*(+) = +
2. (-)*(-) = +
3. (+)*(-) = -
4. (-)*(+) = -
5. (+)÷(+)= +
6. (-)÷(-) = +
7. (+)÷(-) = -
8. (-)÷(+) = -

In case of dividing or multiplying two integers of a same sign then there is always positive and in case of two differents sign then the reasult sign is negative.

So, to multiply or divide two integers we multiply or divide their absolute value and then select a sign using the rule of signs.

Examples:

(i) -12×3

Multiply the absolute values:12×3=36, then signs are the opposite: -36

Thus, -12×3= -36

(ii) (-8)×(-6)

Multiply the absolute values: 8×6= 48

The signs are the same:48

Thus,(-8)×-6)=48

(iii)10×4

Multiply the absolute values: 10×4= 40

The signs are the same:40

Thus,10×4=40

(iv) $\frac{45}{-5}$

Divide the absolute values:$\frac{45}{-5}$= 9

The signs are opposite: -9

Thus,$\frac{45}{-5}$= -9

(v)$\frac{-18}{-6}$

Divide the absolute values:$\frac{18}{6}$= 3

The signs are the same: 3

Thus, $\frac{-18}{-6}$= 3

(vi) $\frac{-56}{7}$

Divide the absolute values: $\frac{-56}{7}$= 8

The signs are the opposite: -8

Thus, $\frac{-56}{7}$= -8

(vii) $\frac{32}{8}$

Divide the absolute values: $\frac{32}{8}$= 4

The signs are the same: 4

Thus,$\frac{32}{8}$

= 4

### Multiplication of several integers

(-2)(-4)(-3)= 8(-3)= -24

(-3)(-2)(-4)(-5)= 6×20= 120

While multiplying odd numbers of negative integers we multiply their values and select a negative sign(-) and while multiplying even numbers of negative integers we multiply their values and select a positive sign(+).

Following are the examples;

(i) (-2)(-5)(-2)

Multiply the absolute values: 2×5×2= 20

but(-2)(-5)(-2) = 20

(ii) (-6)(-2)(-1)(-2)

Multiply the absolute values: 6× 2× 1× 2 = 24

but, (-6)(-2)(-1)(-2)= -24

##### Things to remember
• Zero Absolute value describes the distance of a number on the number line from 0 without considering which direction from the number lies.
• The absolute value of a number is never negative.
• The absolute value of 0 is 0.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

Solution:

$\frac{(- 3) (- 5) 2(-8)}{6 (-10)}$

= $\frac{- 240}{-60}$

= 4

Solution:

$\frac{15(-4)(-3)7}{6(-21)}$

= $\frac{1260}{-126}$

= -10

Solution:

( -10) + (-2)

= -10 -2

= -12

Solution:

$\frac{960?-5}{2400}$

= $\frac{-4800}{2400}$

= -2

Solution:

a)(-10) + (-2) = -10 -2 = -12

b) -23 + 8= -15

Solution:

Multiplying the Absolute values:

3 $\times$ 7 $\times$ 2 = 42

Thus, (-3) (-7) (-2)

= 21 $\times$ -2

= -42 ( Since the signs are negative)

Solution:

Multiply the absolute values: 8 $\times$ 6 = 48

The signs are the same: 48

Thus, (-8) $\times$ (-6)

= 48

Solution:

Divide the absolute values.

$\frac{18}{6}$

The signs are the same.

Thus, $\frac{-18}{-6}$

=3

Solution:

(-4) (-2) (-1) (-3)

Multiply the absolute values: 4? 2? 1?3 = 24

Thus, (-4) (-2) (-1) (-3) = 24

Solution:

Divide the absolute values: $\frac{35}{5}$ = 7

The signs are opposite: -9

Thus, $\frac{35}{-5}$

= -7

Solution:

Multiply the absolute values: 25× 4 = 100

The signs are the same: 100

Thus, 25× 4 = 100

Solution:

Multiply the absolute values: 24× 8 = 192

The signs are the same. 192

Thus, ( -24)× (-8) = 192

Solution:

Multiply the absolute values:

3× 2× 4×5 = 120

The signs are the same: 120

Thus, (-3)×(-2)× (-4) ×(-5)

= 120

Solution:

$\frac{ -4(-5)\times3(-6)}{10 (-18)}$

= $\frac{20×(-18)}{-180}$

= $\frac{-360}{-180}$

= 2

Solution:

$\frac{ 16 (-4)\times (-5) 7}{10 (-28)}$

= $\frac{-64 \times -35}{-280}$

= $\frac{2240}{-280}$

= -8

Solution:

$\frac{6 (-8) \times- 4(-5)}{(-24) 10}$

= $\frac{-48 \times -20}{-240}$

= $\frac{960}{-240}$

= -4

Solution:

( -1) (-6) (- 5) (-8) (-3)(-7)

= 1 $\times$ 6 $\times$ 5 $\times$ 8 $\times$ 3 $\times$ 7

= 5040

The signs are the same

Thus, (-1) (-6) (-5) (-8) (-3) (-7) = 5040

Solution:

$\frac{ 30 (-5) (-2) 5}{5(-20)}$

= $\frac{-150 \times -10}{-100}$

= $\frac{1500}{-100}$

= - 15

Solution:

(-5) (8) (-5) (4) (-3) (2)

= -5 $\times$ 8 $\times$ -5 $\times$ 4 $\times$ -3 $\times$ 2

= -40 $\times$ -20 $\times$ -6

= 800 $\times$ -6

= - 4800

The signs are different

Thus, - 40 $\times$ -20 $\times$ -6 = - 4800

Solution:

$\frac{-8(-10) \times3(5)}{10(-20)}$

= $\frac{-80 \times 15}{-200}$

= $\frac{-1200}{-200}$

= 6

Solution:

( -2) (-4) (-3) (-5)(-4)

= 8 $\times$ 15 $\times$ -4

= 120 $\times$ -4

The signs are different

Thus, (-2) (-4) (-3) (-5) ( -4) = -480