What is Integers Class 7: Definition and Key Concepts Explained

Definition of Integers in Class 7 Mathematics

In Class 7 NCERT Mathematics, integers are defined as the set of whole numbers that include:

• All positive whole numbers (1, 2, 3, ...)
• All negative whole numbers (-1, -2, -3, ...)
• Zero (0)

This means integers extend beyond just counting numbers to include their negative counterparts and zero. The symbol for the set of integers is $\mathbb{Z}$.

Example:

• $5$ is an integer
• $-3$ is an integer
• $0$ is an integer

Integers do not include fractions or decimals. They are used to represent quantities that can increase or decrease, such as temperature changes, elevations, and bank balances.

Understanding the Number Line for Integers

A number line is a straight line where integers are represented as points spaced equally. It helps visualize the position and order of integers.

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

Why use a number line?

• It shows the relative size of integers.
• Helps in understanding addition and subtraction.

Example:

If you start at $-2$ on the number line and move 4 steps to the right, you land at $2$.

The number line is a visual tool to compare and operate with integers effectively.

Properties of Integers Every Class 7 Student Should Know

Integers follow several important properties that help in calculations:

• Closure Property: The sum or product of two integers is always 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: Adding zero to any integer leaves it unchanged, $a + 0 = a$
• Additive Inverse: For every integer $a$, there exists $-a$ such that $a + (-a) = 0$

These properties make integers easy to work with in algebra and arithmetic.

Worked Example:

Calculate $(-4) + 7$ using additive inverse:

$$ (-4) + 7 = 3 $$

Here, adding a positive number to a negative number results in a positive integer.

Addition and Subtraction Rules for Integers

Adding and subtracting integers follow specific rules:

• 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 absolute value; sign of the larger absolute value remains.

Subtraction is adding the additive inverse:

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

Example 1: $5 + (-3) = 2$

Example 2: $-7 - 4 = -7 + (-4) = -11$

Using these rules helps solve integer problems accurately.

Comparing and Ordering Integers

To compare integers, remember:

• Any positive integer is always greater than any negative integer.
• Among positive integers, the larger number is greater.
• Among negative integers, the number with the smaller absolute value is greater.

Example:

• $3 > -2$ because positive is greater than negative.
• $-4 > -7$ because $4 < 7$ but on the negative side, $-4$ is closer to zero.

Ordering integers: Arrange numbers from least to greatest or vice versa using the number line as a guide.

Practical Applications of Integers in Daily Life

Integers are used in many real-life situations:

• Temperature: Negative integers represent temperatures below zero.
• Banking: Deposits and withdrawals are shown as positive and negative integers.
• Elevation: Heights above sea level are positive; depths below are negative.
• Sports: Scores can increase or decrease, represented by integers.

Understanding integers helps solve problems related to these scenarios.

Example:

If the temperature is $-5^ ext{o}C$ in the morning and rises by $7^ ext{o}C$, the new temperature is:

$$ -5 + 7 = 2^ ext{o}C $$

This shows how integers model everyday changes effectively.