Lesson

Introduction to Integers

Welcome to the world of integers! In mathematics, integers are whole numbers, but with a twist: they can be positive, negative, or zero. Understanding integers is a fundamental step in pre-algebra, as they form the basis for many mathematical concepts you'll encounter later on.

What are Integers?

Integers include all positive whole numbers (1, 2, 3, ...), all negative whole numbers (-1, -2, -3, ...), and zero (0). They do *not* include fractions, decimals, or any number that isn't a "whole" value. Think of it like counting steps on a staircase – you can take one step up (positive), one step down (negative), or stay on the same level (zero). You can't take half a step!

Positive Integers

Positive integers are simply the numbers you're most familiar with from basic counting. They are greater than zero. For example, 1, 2, 3, 4, 5, and so on are all positive integers. They represent quantities greater than nothing.

Negative Integers

Negative integers are numbers less than zero. They are written with a minus sign (-) in front of them, such as -1, -2, -3, -4, -5, and so on. Negative integers represent quantities less than nothing. You might think of them as representing debt or a temperature below zero.

Zero

Zero (0) is a special integer. It is neither positive nor negative. It represents the absence of quantity. It's the point where positive and negative numbers meet.

Representing Integers on a Number Line

A number line is a visual tool that helps us understand the order and relationship between numbers, including integers. On a number line, zero is in the middle. Positive integers are to the right of zero, and negative integers are to the left of zero. The further you move to the right, the larger the number, and the further you move to the left, the smaller the number.

Understanding Magnitude and Direction

Integers have two important properties: magnitude (or size) and direction (positive or negative). The magnitude is how far away the number is from zero, regardless of whether it is positive or negative. The direction tells us whether the number is greater than zero (positive) or less than zero (negative).

Examples of Integers in Real Life

Integers are used in many real-life situations:

• Temperature: Temperatures can be above zero (positive) or below zero (negative). For example, 25°C is a positive integer, while -5°C is a negative integer.
• Bank accounts: You can have money in your account (positive) or be overdrawn (negative). A balance of $100 is a positive integer, while owing $50 is a negative integer (-50).
• Altitude: Altitude can be above sea level (positive) or below sea level (negative).
• Sports: In some sports, like golf, scores can be above par (positive) or below par (negative).

Comparing Integers

Comparing integers is essential for understanding their relative values. Remember that numbers on the right side of the number line are always greater than numbers on the left side of the number line. Therefore, any positive integer is always greater than any negative integer.

Rules for Comparing Integers

1. Any positive integer is greater than zero. For example, 5 > 0.
2. Any negative integer is less than zero. For example, -3 < 0.
3. Any positive integer is greater than any negative integer. For example, 2 > -4.
4. When comparing two positive integers, the larger the number, the greater its value. For example, 7 > 3.
5. When comparing two negative integers, the smaller the number (closer to zero), the greater its value. For example, -2 > -6. This can be tricky, so visualize the number line!

Absolute Value

The absolute value of an integer is its distance from zero on the number line. The absolute value is always a non-negative number (positive or zero). We denote the absolute value of a number \( x \) using vertical bars: \( |x| \) .

Examples of Absolute Value

Here are some examples to illustrate the concept of absolute value:

• \( |5| = 5 \) (The absolute value of 5 is 5 because it is 5 units away from zero).
• \( |-3| = 3 \) (The absolute value of -3 is 3 because it is 3 units away from zero).
• \( |0| = 0 \) (The absolute value of 0 is 0 because it is 0 units away from zero).

Why is Absolute Value Important?

Absolute value is important because it allows us to focus on the magnitude of a number without considering its direction (positive or negative). This is useful in many situations, such as calculating distances or comparing the sizes of different quantities.

Practice with Integers

The best way to understand integers is to practice working with them. Try comparing different pairs of integers, finding their absolute values, and visualizing them on a number line. The more you practice, the more comfortable you'll become with these fundamental building blocks of mathematics.