Chapter 6 - Integers

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Introduction to Integers

Positive Numbers

Positive numbers are greater than 0. For example, numbers like 5, 11, 87 are to the right of 0 and are positive numbers.

Negative Numbers

The numbers with a negative or minus sign before them are called negative numbers. Numbers like −4, −19, −112 are to the left of 0 and are negative numbers.

0 is neither a positive nor a negative number.

Natural Numbers

All the positive numbers excluding the fractions are called natural numbers. The natural numbers, zero, and the negative numbers are collectively known as Integers.

Integers in Real-Life Applications

Temperature is measured in degree Celsius, which can be positive or negative. The freezing point of water is 0°C and anything less than that becomes negative. So, we can have temperatures like −5°C, −21°C, etc.

In elevator panels, we get to see negative numbers like −1, −2, −3, etc. Here we treat the ground floor as 0 and the floors beneath that are shown using negative numbers.

On thermometers, temperatures above 0 degrees are marked positive, while those below 0 are marked negative temperatures.

We use positive numbers to show profit and negative numbers to show loss. For example, a loss of ₹20 can be thought of as −₹20.

Representation of Integers on Number Line

Natural numbers are 1, 2, 3, 4, and so on. On the number line, they are represented as:

When we move to the left of 1, we get 0, and this collection of numbers are called whole numbers.

Moving further to the left, we get the negatives of natural numbers, which are −1, −2, −3, −4, ...

Natural numbers, their negatives, and 0 together form the set of integers.

Vertical Number Lines

Number lines can also be vertical with 0 in the middle and positive and negative integers above and below zero.

Vertical number lines are used to measure altitudes and depths. Here, the mean sea level is taken as the reference point or 0.

• The height or altitude of Mt. Everest is 8,848 m above mean sea level.
• The depth of Mariana Trench in the Pacific is nearly − 11,000 m.

The thermometer scale is like a vertical number line. 0 degrees is the reference point in the middle. The temperatures above 0 are hot temperatures, while those below 0 are colder temperatures.

Comparing Integers

We know how to compare positive integers. For example,

9 > 6 or 2 < 5

Let's learn how to compare two negative numbers.

Example: Compare 2 and 5 using inequality signs like <, >.

Solution: The best way to compare numbers is by imagining their position on a number line.

On a number line,

• the number that lies to the right is greater, and
• the one that lies to the left is smaller.

For example, 2 lies to the left of 5, so 2 < 5.

Also 5 lies to the right of 2, so, 5 > 2.

Example: Compare −2 and −5.

Solution: When we go to the left of 0 on the number line, we see that −2 lies to the right of −5.

So, −2 > −5.

Alternatively, we can write, −5 < −2.

Ascending and Descending Order

When more than two numbers are involved, we mark the points on the number line and organize the numbers from left to right.

• If the numbers are from the smallest to the largest, it's called the ascending or increasing order.
• If the numbers are from the largest to the smallest, it's called the descending or decreasing order.

Example: Arrange the numbers −5, −3, 0 and 1, in increasing and decreasing order.

Solution: First mark the numbers on the number line as follows.

The leftmost number −5 is the smallest, the other numbers in order are −3, 0, and 1.

Ascending order:

−5 < −3 < 0 < 1

Descending order:

1 > 0 > −3 > −5

Addition of Integers

Moving to the right of 0 on the number line increases the value. It's like adding two numbers.

Moving on the number line to the left of 0 decreases the value of numbers and it's like subtracting two numbers.

Adding Two Positive Integers

Example: Add 3 + 5.

Solution: Start from 3 on the number line.

Move 5 unit distances to the right to land on 8 on the number line.

So, we have,

3 + 5 = 8

Adding Two Negative Integers

Example: Add (−3) + (−5)

Solution: Start from −3 on the number line.

Move 5 unit distances to the left to land on −8 on the number line.

Result of the addition,

−3 + (−5) = −8

Adding a Positive and a Negative Integer

Example: Add 3 + (−5)

Solution: Start from 3 on the number line.

Move 5 unit distances to the left to land on −2 on the number line.

The result is,

3 + (−5) = −2

Adding Three or More Integers

Example: Add (13) + (−7) + (−9) using the number line

Solution: Start at the position of 13 on the number line.

Since −7 is to be added, move seven units to the left.

13 + (−7) = 6

Next, since −9 is to be further added, move nine units from 6 to the left.

6 + (−9) = −3

Or,

13 + (−7) + (−9) = −3

Subtraction of Integers

Moving to the left on the number line is like doing subtraction.

Positive (minus) Positive Integer

Example: Subtract 5 from 3.

Solution: Start from 3. Move 5 positions to the left. So,

3 − 5 = −2

Example: What is the result of (−3) + 5?

Solution: Start at −3. Since there is a plus sign, move 5 units to the right.

The result is,

−3 + 5 = 2

Positive (minus) Negative Integer

Example: Consider the subtraction 3 − (−5).

Solution: Subtracting a negative integer from a positive integer is equivalent to adding the two numbers.

Two minus signs together make a plus sign.

(−) (−) = +

3 − (−5) = 3 + 5

Start at 3 and move 5 units to the right. The result is,

3 + 5 = 8

Negative (minus) Negative Integer

Example: Consider the subtraction (−3) − (5).

Solution: Subtracting a negative integer from another negative integer is equivalent to adding the two numbers and putting a minus sign before the sum.

Start at (−3) and then move 5 units to the left. Thus,

Thus,

−3 − (5) = −3 − 5 = −8

Subtracting Three or More Integers

Example

Question: Simplify −30 + 5 − (−17) − (−11)
Solution: It is known that
      (−) (−) = +
Rewriting the expression,
      −30 + 5 − (−17) − (−11)
      = −30 + 5 + 17 + 11
Adding all the positive integers,
      −30 + (5 + 17 + 11)
      = (−30) + 33
      = 3
The result is,
      −30 + 5 − (−17) − (−11) = 3

Multiplication and Division of Integers

Multiplication and division of integers is similar to multiplication and division of whole numbers, with a few extra steps.

The steps are as follows:

• Count the number of negative signs.
• Ignore the negative signs and perform multiplication or division of the numbers.
• If the number of negative signs is odd, the result will be negative.
• If the number of negative signs is even, the result will be positive.

Example

Question: Solve −4 × 3
Solution: Number of negative integers = 1
Multiply the numbers, ignoring the negative sign,
      4 × 3 = 12
There are odd number of negative signs, so the product will be negative.
Thus, the answer is −12.

Example

Question: Solve (−4) × (−3)
Solution: The number of negative integers = 2
Ignore the negative signs and perform multiplication,
      4 × 3 = 12
There are even number of negative signs, so the product will be positive.
Thus, the answer is 12.

Example

Question: Solve (−9) ÷ 3
Solution: The number of negative integers = 1
Ignore the negative signs and perform division,
      9 ÷ 3 = 3
There are odd number of negative signs, so the quotient will be negative.
Thus, the answer is −3.

Example

Question: Solve (−9) ÷ (−3)
Solution: Count the number of negative signs: 2
Ignore the negative signs and perform the division of the numbers: 9 ÷ 3 = 3
There are even number of negative signs, so the quotient will be positive.
Thus, the answer is 3.

Example

Question: Solve (5 × 7) − (3 × 4 × −7)
Solution: Count the number negative signs in (3 × 4 × −7).
There is only 1 negative sign and it is odd
Ignore the sign and multiply to get,
      3 × 4 × 7 = 84
Since the number of negative signs is odd, the product is negative.
Rewriting the expression,
      (5 × 7) − (3 × 4 × −7)
      = 35 − (−84)
As (−) (−) = +,
      (5 × 7) − (3 × 4 × −7)
      = 35 − (−84)
      = 35 + 84
      = 119