👋 Welcome to the World of Integers!

Hello future mathematician! This chapter, Integers, is fundamental to everything you will do in Mathematics. Don't worry if numbers seem tricky sometimes—we're going to break down positive and negative whole numbers into simple, manageable steps. By the end of this session, you'll be adding, subtracting, multiplying, and dividing integers like a pro!

Why are Integers important? They help us describe things in the real world where we need to count below zero, such as:

  • Temperature (e.g., \( -5^{\circ}\mathrm{C} \))
  • Bank balances (e.g., an overdraft of \(\$100\) is \( -\$100 \))
  • Altitude (e.g., a submarine at \( -30\) meters)

Chapter 1: Defining and Visualizing Integers

What is an Integer?

An Integer is simply a whole number. It includes all positive whole numbers, all negative whole numbers, and zero.

Integers DO NOT include fractions, decimals, or mixed numbers.

Examples of Integers: \( \dots, -3, -2, -1, 0, 1, 2, 3, \dots \)
Examples of NON-Integers: \( 1.5, \frac{1}{2}, -3.75 \)

Key Divisions of Integers:

  • Positive Integers: \( 1, 2, 3, 4, \dots \) (The numbers greater than zero)
  • Negative Integers: \( -1, -2, -3, -4, \dots \) (The numbers less than zero)
  • Zero: \( 0 \) (Zero is neither positive nor negative)

Visualizing Integers: The Number Line

The easiest way to understand integers, especially negative numbers, is to use a Number Line.

Remember this rule:

  • Moving RIGHT increases the value (gets larger).
  • Moving LEFT decreases the value (gets smaller).

Imagine zero as the entrance to a building. Positive numbers are the floors above ground, and negative numbers are the basement levels below ground.

🔑 Quick Review: Integers

Integers are whole numbers only, including negatives and zero. Visualize them on a number line where numbers get smaller as you move left.

Chapter 2: Ordering and Comparing Integers

Comparing integers means deciding which one is larger or smaller. We use the inequality symbols:

  • \( > \): Greater than
  • \( < \): Less than

Comparing Positive and Negative Numbers

This is where the number line really helps!

Example 1: Compare \( 3 \) and \( -5 \).
\( 3 \) is to the right of \( -5 \) on the number line, so \( 3 \) is greater.
\( 3 > -5 \)

Example 2: Compare \( -1 \) and \( -8 \).
Don't be fooled by the '8'! Since both are negative, the number closest to zero is the largest. \( -1 \) is much closer to zero than \( -8 \).
\( -1 > -8 \)

Think about temperature: \( -1^{\circ}\mathrm{C} \) is warmer than \( -8^{\circ}\mathrm{C} \).

Understanding Magnitude (Absolute Value)

Sometimes, we care about the size of the number, regardless of whether it's positive or negative. This is called the Absolute Value, or Magnitude.

The absolute value of a number is its distance from zero. Since distance cannot be negative, the absolute value is always positive (or zero). We use vertical bars \( | \dots | \).

Example:

  • \( | 7 | = 7 \)
  • \( | -7 | = 7 \)
Both \( 7 \) and \( -7 \) have the same magnitude (they are the same distance from zero).

Step-by-Step: Ordering Integers
  1. Identify the positive integers. These are the largest.
  2. Identify zero.
  3. Identify the negative integers.
  4. Among the negative numbers, the one with the smallest magnitude (closest to zero) is the largest negative number.

Task: Order the following from smallest to largest: \( 5, -2, 0, -10, 3 \)
Answer: \( -10, -2, 0, 3, 5 \)

Chapter 3: Arithmetic Operations with Integers (Adding and Subtracting)

This is the most common area for errors, but with clear rules, you'll master it!

Adding Integers (Same Signs)

If the signs are the same, treat it like regular addition and keep the sign.

  • Positive + Positive: Add the numbers, answer is positive.
    Example: \( 5 + 3 = 8 \)
  • Negative + Negative: Add the magnitudes of the numbers, answer is negative.
    Example: \( -5 + (-3) = -8 \)
    Analogy: If you owe \(\$5\) (debt) and then you owe \(\$3\) more (more debt), your total debt is \(\$8\).
Adding Integers (Different Signs)

If the signs are different, treat it like subtraction, and use the sign of the number with the largest magnitude (the 'winner').

  • Rule: Subtract the smaller magnitude from the larger magnitude.
  • Sign: Take the sign of the number that is furthest from zero.

Example 1: \( -7 + 3 \)
Step 1: Find the difference between \( 7 \) and \( 3 \). \( 7 - 3 = 4 \).
Step 2: The number with the largest magnitude is \( -7 \). So the answer is negative.
Answer: \( -4 \)

Example 2: \( 12 + (-5) \)
Step 1: Find the difference: \( 12 - 5 = 7 \).
Step 2: The number with the largest magnitude is \( 12 \) (positive). So the answer is positive.
Answer: \( 7 \)

Subtracting Integers (The Double Sign Rule)

The simplest way to handle subtraction is to change the problem into an addition problem using the rule:
"Subtracting a negative is the same as adding a positive."

Rule: A minus sign followed immediately by a negative sign turns into a plus sign.

\( - (-) = + \)

Example 1: \( 5 - (-3) \)
Change the double negative: \( 5 + 3 = 8 \)

Example 2: \( -10 - (-6) \)
Change the double negative: \( -10 + 6 \)
(Now use the 'Different Signs' addition rule): \( 10 - 6 = 4 \). The 'winner' is \( -10 \).
Answer: \( -4 \)

🛑 Common Mistake to Avoid!

Students often confuse subtraction (e.g., \( 5 - 3 \)) with a negative sign (e.g., \( -5 \)). Always pay close attention to whether the sign is operating on the number or operating between two numbers.

🔑 Quick Review: Addition & Subtraction

  1. Same Signs: Add and keep the sign.
  2. Different Signs: Subtract and take the sign of the bigger number (magnitude).
  3. Double Negative: Change \( -(-) \) to \( + \).

Chapter 4: Arithmetic Operations with Integers (Multiplication and Division)

The rules for multiplying and dividing integers are identical. They depend only on the signs of the numbers you are using.

The Golden Rules for Multiplying and Dividing

You only need to remember two simple rules:

Rule 1: Same Signs = Positive Answer

If both integers have the same sign (both positive OR both negative), the result is always positive.

  • \( (+) \times (+) = (+) \)
    Example: \( 4 \times 5 = 20 \)
  • \( (-) \times (-) = (+) \)
    Example: \( -4 \times (-5) = 20 \)
Rule 2: Different Signs = Negative Answer

If the integers have different signs (one positive and one negative), the result is always negative.

  • \( (+) \times (-) = (-) \)
    Example: \( 4 \times (-5) = -20 \)
  • \( (-) \times (+) = (-) \)
    Example: \( -4 \times 5 = -20 \)
Memory Aid: The Social Analogy

Think of positive as "Friend" (\( + \)) and negative as "Enemy" (\( - \)).

  • A friend of a friend is a friend. (\( + \times + = + \))
  • An enemy of an enemy is a friend. (\( - \times - = + \))
  • A friend of an enemy is an enemy. (\( + \times - = - \))

Division Examples

The exact same rules apply! First, ignore the signs and calculate the normal division. Then apply the sign rule.

Example 1: \( -24 \div 6 \)
Signs are different (Negative and Positive). Result is Negative.
\( 24 \div 6 = 4 \).
Answer: \( -4 \)

Example 2: \( -40 \div (-8) \)
Signs are the same (Negative and Negative). Result is Positive.
\( 40 \div 8 = 5 \).
Answer: \( 5 \)

Did you know?

The word "integer" comes from the Latin word integer, meaning "whole" or "untouched"!

Final Key Takeaways

You have successfully covered the core concepts of integers! Remember these essentials:

🧠 Comprehensive Quick Check
  • Definition: Integers are whole numbers (\( \dots, -2, -1, 0, 1, 2, \dots \)).
  • Comparison: On the number line, numbers get larger as you move to the right. \( -1 \) is larger than \( -10 \).
  • Addition/Subtraction Trick: Change subtraction of a negative to addition of a positive (\( - (-) = + \)).
  • Multiplication/Division:
    Same Signs = Positive
    Different Signs = Negative

Keep practicing these rules, especially with the addition and subtraction of negative numbers, and you will find the rest of your number work much simpler! You got this!