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- CBSE Class 6 Maths Notes
- CBSE Class 6 Maths Notes
- Chapter 1 - Knowing Our numbers
- Chapter 2 - Whole numbers
- Chapter 3 - Playing with numbers
- Chapter 4 - Basic Geometrical Ideas
- Chapter 5 - Understanding Elementary Shapes
- Chapter 6 - Integers
- Chapter 7 - Fractions
- Chapter 8 - Decimals
- Chapter 9 - Data Handling
- Chapter 10 - Mensuration
- Chapter 11 - Algebra
- Chapter 12 - Ratio and Proportion
- Chapter 13 - Symmetry
- Chapter 14 - Practical Geometry
Chapter 6 - Integers
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 on the number line](/cbse-class-6-maths-notes/images/integers_on_the_number_line.jpg)
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:
![Representing1](/cbse-class-6-maths-notes/images/representing_1.jpg)
When we move to the left of 1, we get 0, and this collection of numbers are called whole numbers.
![Representing2](/cbse-class-6-maths-notes/images/representing_2.jpg)
Moving further to the left, we get the negatives of natural numbers, which are −1, −2, −3, −4, ...
![Representing3](/cbse-class-6-maths-notes/images/representing_3.jpg)
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.
![Representing4](/cbse-class-6-maths-notes/images/representing_4.jpg)
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.
![Comparing1](/cbse-class-6-maths-notes/images/comparings_1.jpg)
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.
![Comparing2](/cbse-class-6-maths-notes/images/comparings_2.jpg)
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.
![Comparing3](/cbse-class-6-maths-notes/images/comparings_3.jpg)
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.
![Addition 1](/cbse-class-6-maths-notes/images/addition1.jpg)
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.
![Addition 4](/cbse-class-6-maths-notes/images/addition4.jpg)
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.
![Addition 2](/cbse-class-6-maths-notes/images/addition2.jpg)
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
![Subtraction1](/cbse-class-6-maths-notes/images/subtraction1.jpg)
Example: What is the result of (−3) + 5?
Solution: Start at −3. Since there is a plus sign, move 5 units to the right.
![Subtraction4](/cbse-class-6-maths-notes/images/subtraction4.jpg)
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
![Subtraction2](/cbse-class-6-maths-notes/images/subtraction2.jpg)
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,
![Subtraction3](/cbse-class-6-maths-notes/images/subtraction3.jpg)
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
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