Integers – Definition, Rules, Properties, and Examples Explained Clearly

What Are Integers?

Integers are a set of numbers that include all whole numbers (both positive and negative) along with zero. They do not include fractions or decimals. Mathematically, integers are represented as:

…,-3, -2, -1, 0, 1, 2, 3,…

In simple terms, integers are numbers that can be written without a fractional or decimal component.

Types of Integers

1. Positive Integers: Numbers greater than zero (1, 2, 3, …).
2. Negative Integers: Numbers less than zero (-1, -2, -3, …).
3. Zero: Neither positive nor negative, zero (0) is a neutral integer.

Rules of Integers

Understanding the basic rules of integers is essential for solving mathematical problems efficiently. Here are the key rules:

1. Addition of Integers:

• When adding two positive integers, the result is always positive. (5 + 3 = 8)
• When adding two negative integers, add their absolute values and place a negative sign. (-5 + -3 = -8)
• When adding a positive and a negative integer, subtract the smaller absolute value from the larger absolute value and take the sign of the larger absolute value. (-7 + 3 = -4)

2. Subtraction of Integers:

• Convert subtraction into addition by changing the sign of the second number. (-4 – (-7) = -4 + 7 = 3)
• Follow the rules of integer addition.

3. Multiplication of Integers:

• Positive × Positive = Positive (3 × 4 = 12)
• Negative × Negative = Positive (-3 × -4 = 12)
• Positive × Negative = Negative (3 × -4 = -12)

4. Division of Integers:

• Positive ÷ Positive = Positive (12 ÷ 3 = 4)
• Negative ÷ Negative = Positive (-12 ÷ -3 = 4)
• Positive ÷ Negative = Negative (12 ÷ -3 = -4)

Properties of Integers

Integers follow certain fundamental properties that help in simplifying calculations. These include:

1. Closure Property:

• Integers are closed under addition, subtraction, and multiplication, meaning that the result of any of these operations is always an integer.

2. Commutative Property:

• Addition: a + b = b + a (e.g., 4 + (-2) = -2 + 4)
• Multiplication: a × b = b × a (e.g., 5 × (-3) = -3 × 5)

3. Associative Property:

• Addition: (a + b) + c = a + (b + c) (e.g., (2 + -3) + 4 = 2 + (-3 + 4))
• Multiplication: (a × b) × c = a × (b × c)

4. Distributive Property:

• a × (b + c) = (a × b) + (a × c)

5. Identity Property:

• Additive Identity: Any number plus zero remains the same (a + 0 = a).
• Multiplicative Identity: Any number multiplied by one remains the same (a × 1 = a).

Examples of Integer Operations

Example 1: Adding Integers

(-6) + 9 = 3 (Take the difference and apply the sign of the larger absolute value.)

Example 2: Multiplying Integers

(-4) × (-5) = 20 (Negative × Negative = Positive)

Example 3: Dividing Integers

(-20) ÷ (5) = -4 (Negative ÷ Positive = Negative)

Real-Life Applications of Integers

1. Temperature Measurement: Negative values represent cold temperatures (e.g., -10°C in winter).
2. Banking and Finance: Deposits (positive integers) and withdrawals (negative integers).
3. Stock Market: Rise and fall in stock prices are expressed using positive and negative integers.
4. Elevation and Depth: Altitude is positive above sea level and negative below sea level.

Conclusion

Integers form a crucial foundation in mathematics and have widespread applications in real life. Understanding their properties, rules, and operations helps students grasp mathematical concepts more efficiently. By mastering integers, you build a strong base for solving algebraic expressions, equations, and real-world problems.

For more math concepts and in-depth learning, visit Guwahati Maths Academy and enhance your mathematical skills today!