Understanding and Working with Integers

Introduction

Integers are a fundamental part of mathematics, forming the backbone of many mathematical operations and concepts. They include all whole numbers, their opposites, and zero. This unit will provide a comprehensive understanding of integers, their properties, and their operations.

What are Integers?

Integers are a set of numbers that include all whole numbers (both positive and negative) and zero. They are represented on the number line where each integer has a unique position. For example, -3, -2, -1, 0, 1, 2, 3, etc., are all integers.

Operations with Integers

Addition and Subtraction

When adding or subtracting integers, it's important to consider their signs.

• If the signs are the same, add the numbers and keep the sign. For example, -3 + (-2) = -5.
• If the signs are different, subtract the smaller number from the larger number and keep the sign of the larger number. For example, -3 + 5 = 2.

Multiplication and Division

The rules for multiplication and division are simpler:

• If the signs of the numbers are the same, the result is positive. For example, -3 * -2 = 6.
• If the signs are different, the result is negative. For example, -3 * 2 = -6.

Absolute Value and Opposites

The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always positive. For example, the absolute value of -3 and 3 is 3.

The opposite of an integer is the number that is the same distance from zero but in the opposite direction on the number line. For example, the opposite of -3 is 3, and the opposite of 3 is -3.

Integer Order of Operations

When performing operations with integers, follow the order of operations, often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Real-World Applications of Integers

Integers are used in various real-world scenarios:

• In banking, positive integers represent deposits and negative integers represent withdrawals.
• In temperature, positive integers represent temperatures above zero, and negative integers represent temperatures below zero.
• In sports, positive and negative integers are used to represent gains and losses.

Conclusion

Understanding integers and their operations is a fundamental skill in mathematics. This unit has provided a comprehensive overview of integers, their properties, and their operations. The next units will build on this foundation, exploring more complex mathematical concepts.