1. Introduction to Integers
Integers are numbers that include positive numbers, negative numbers, and zero. They help us describe situations that go below zero, such as temperatures, elevations, or money owed.
If you think about a number line, integers are all the numbers you can point to — going left (negative), right (positive), and including zero in the center.
2. Definition of Integers
Integers are the set of whole numbers and their negative counterparts.
In mathematical notation:
\( \mathbb{Z} = \{ ..., -3, -2, -1, 0, 1, 2, 3, ... \} \)
They do not include fractions or decimals.
3. Understanding Integers with a Number Line
A number line is the easiest way to visualize integers.
- Numbers to the right of 0 are positive
- Numbers to the left of 0 are negative
- 0 is in the middle
Example:
... -3, -2, -1, 0, 1, 2, 3 ...
4. Types of Integers
Integers are grouped into three main types:
4.1. Positive Integers
These are numbers greater than zero.
Examples: 1, 2, 3, 10...
4.2. Negative Integers
These are numbers less than zero.
Examples: -1, -5, -12...
4.3. Zero
Zero is neither positive nor negative. It acts as a neutral point on the number line.
5. Properties of Integers
Integers follow several important mathematical properties.
5.1. Closure Property
Integers are closed under addition, subtraction, and multiplication.
- \( 3 + (-5) = -2 \)
- \( -4 \times 2 = -8 \)
- \( 6 - 10 = -4 \)
5.2. Commutative Property
This applies to addition and multiplication.
5.2.1. Addition
\( a + b = b + a \)
Example: \( -2 + 5 = 5 + (-2) \)
5.2.2. Multiplication
\( a \times b = b \times a \)
Example: \( (-3) \times 4 = 4 \times (-3) \)
5.3. Identity Element
There are two identity elements related to integers:
5.3.1. Additive Identity
Zero is the additive identity.
\( a + 0 = a \)
5.3.2. Multiplicative Identity
One is the multiplicative identity.
\( a \times 1 = a \)
5.4. Opposites
Every integer has an opposite (also called additive inverse).
Example:
- Opposite of 5 is -5
- Opposite of -3 is 3
- Opposite of 0 is 0
6. Examples of Integers
- -20
- -1
- 0
- 5
- 100
All of the above are integers since they do not involve fractions or decimals.
7. Integers in Real Life
Integers are used in many real-life situations where values go above or below zero.
7.1. Temperature
Temperature can be 10°C or -5°C. Negative temperatures are integers.
7.2. Bank Balance
A negative balance means you owe money: -₹500.
7.3. Altitude
Mountain peaks are above sea level (positive integers), while ocean depths are below sea level (negative integers).
8. Difference Between Integers and Whole Numbers
Many students mix these two sets, but the difference is simple.
8.1. Comparison Table
| Integers | Whole Numbers |
|---|---|
| Include negatives | No negatives |
| ..., -3, -2, -1, 0, 1, 2, ... | 0, 1, 2, 3, ... |
| Broader set | Subset of integers |
9. Practice Questions
- Is -8 an integer?
- Is 0 an integer?
- Write three negative integers.
- Write the opposite of -12.
- Is 3.5 an integer?
10. Summary
Integers include all positive numbers, negative numbers, and zero. They are used to describe real-life situations that involve values going above and below zero. Integers do not include fractions or decimals.