What is Integers Class 7: Definition and Key Concepts Explained

Understanding Integers: Definition and Examples

Integers are a set of numbers that include all whole numbers and their negatives, along with zero. Formally, integers are written as:

$$ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots $$

Here are some examples:

• Positive integers: $1, 2, 3, 100$
• Negative integers: $-1, -5, -20$
• Zero: $0$

Integers do not include fractions or decimals. They are important because they allow us to represent quantities that can increase or decrease, such as temperatures below zero or debts in money.

In Class 7 NCERT Mathematics, understanding integers forms the base for learning more advanced math topics like algebra and coordinate geometry.

Properties of Integers: What You Must Know

Integers follow certain properties that make calculations easier. Here are key properties:

• Closure Property: Adding or multiplying two integers always gives an integer.
• Commutative Property: $a + b = b + a$ and $a \times b = b \times a$
• Associative Property: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$
• Additive Identity: $a + 0 = a$
• Additive Inverse: For every integer $a$, there is an integer $-a$ such that $a + (-a) = 0$

These properties help in simplifying expressions and solving problems involving integers.

Representing Integers on the Number Line

A number line is a straight line with numbers placed at equal intervals. It helps visualize integers and their order.

• Zero is at the center.
• Positive integers are placed to the right of zero.
• Negative integers are placed to the left of zero.

For example:

$$ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots $$

Using the number line, you can easily compare integers, add or subtract them by moving right or left.

Example:

Find $3 + (-5)$ using the number line:

Start at 3, move 5 steps left (because of $-5$), you reach $-2$. So, $3 + (-5) = -2$.

Addition and Subtraction of Integers: Rules and Examples

Adding and subtracting integers follow specific rules based on their signs:

• Adding two positive integers: Add normally, result is positive.
• Adding two negative integers: Add their absolute values, result is negative.
• Adding a positive and a negative integer: Subtract smaller absolute value from larger, sign of the larger absolute value.

Examples:

1. $5 + 3 = 8$ 2. $-4 + (-6) = -10$ 3. $7 + (-10) = -3$

For subtraction, convert it into addition:

$$ a - b = a + (-b) $$

Example:

Calculate $6 - 9$:

$$ 6 - 9 = 6 + (-9) = -3 $$

Understanding these rules is crucial to solve integer problems in exams.

Multiplication and Division of Integers: Sign Rules

Multiplying and dividing integers involves sign rules:

Examples:

• Multiplication:
• $4 \times 3 = 12$
• $-4 \times -3 = 12$
• $-4 \times 3 = -12$

• Division:
• $12 \div 3 = 4$
• $-12 \div -3 = 4$
• $-12 \div 3 = -4$

Remember, zero multiplied or divided by any integer (except division by zero) is zero.

Real-Life Applications of Integers

Integers are used in many daily life situations:

• Temperature: Representing temperatures above and below zero.
• Banking: Deposits (positive) and withdrawals or debts (negative).
• Elevation: Heights above sea level (positive) and depths below sea level (negative).
• Games: Scoring points gained or lost.

Understanding integers helps students relate math to real-world problems and prepares them for advanced topics in Class 8 and beyond.