🔢 Working with Integers and Rational Numbers
Master positive and negative numbers, fractions, and decimals! These building blocks are essential for all of algebra.
🔢 What are Integers?
Integers are whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}
Types of Integers
- Positive integers: $1, 2, 3, 4, ...$ (numbers greater than zero)
- Negative integers: $-1, -2, -3, -4, ...$ (numbers less than zero)
- Zero: $0$ (neither positive nor negative)
🌡️ Real Example: Temperature can be $20°$ (positive) or $-5°$ (negative, below zero!)
➕➖ Adding and Subtracting Integers
Think of a number line—positive numbers go right, negative numbers go left!
Rules for Addition
- Same signs: Add and keep the sign
$5 + 3 = 8$ and $(-5) + (-3) = -8$ - Different signs: Subtract and use the sign of the number with the larger absolute value
$7 + (-3) = 4$ and $(-7) + 3 = -4$
Rules for Subtraction
Key idea: Subtracting is the same as adding the opposite!
$5 - 3 = 5 + (-3) = 2$
$5 - (-3) = 5 + 3 = 8$ ✨ (Two negatives make a positive!)
✖️➗ Multiplying and Dividing Integers
The rules are simple: just remember the sign patterns!
Sign Rules
- Same signs → Positive result
$(+) \times (+) = (+)$ → $3 \times 4 = 12$
$(-) \times (-) = (+)$ → $(-3) \times (-4) = 12$ - Different signs → Negative result
$(+) \times (-) = (-)$ → $3 \times (-4) = -12$
$(-) \times (+) = (-)$ → $(-3) \times 4 = -12$
Memory Trick: "Friends with same signs = happy (positive)! 😊"
🍰 Rational Numbers (Fractions & Decimals)
A rational number can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.
Examples of Rational Numbers
- Fractions: $\frac{1}{2}$, $\frac{3}{4}$, $-\frac{2}{5}$
- Decimals: $0.5$ (which is $\frac{1}{2}$), $0.75$ (which is $\frac{3}{4}$)
- Integers: $5$ (which is $\frac{5}{1}$), $-3$ (which is $\frac{-3}{1}$)
💡 Fun fact: All integers are rational numbers!
🔢 Operations with Fractions
Working with fractions follows special rules!
Adding/Subtracting Fractions
Step 1: Find a common denominator
Step 2: Add/subtract the numerators
$$\frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}$$
Multiplying Fractions
Multiply numerators together, multiply denominators together!
$$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$$
Dividing Fractions
Flip the second fraction and multiply!
$$\frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3}$$
🌟 Real-Life Applications
- 💰 Banking: Positive balance (+$100) vs. Overdrawn (-$50)
- 🏔️ Elevation: Above sea level (+500m) vs. Below sea level (-200m)
- 🍕 Cooking: Using $\frac{1}{2}$ cup of flour or $\frac{3}{4}$ teaspoon of salt
- 📊 Sports: Golf scores can be positive or negative