What are rational numbers and how do you add them as decimals?

📚 What are Rational Numbers?

A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and $q$ is not zero. Basically, if you can write a number as a ratio of two whole numbers, it's rational. This includes integers, fractions, terminating decimals, and repeating decimals.

• ➕Definition: A number expressible in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$.
• 📜History: The concept of rational numbers dates back to ancient civilizations, used for measuring and dividing quantities.
• 🔑Key Principle: The ability to represent numbers as ratios allows for precise calculations and comparisons.

➕ Rational Numbers as Decimals

When a rational number is expressed as a decimal, it either terminates (ends) or repeats. Terminating decimals can be written as a fraction with a denominator that is a power of 10 (e.g., 0.25 = $\frac{25}{100}$ = $\frac{1}{4}$). Repeating decimals also represent rational numbers (e.g., 0.333... = $\frac{1}{3}$).

• ♾️Repeating Decimals: Decimals with a repeating pattern.
• 🛑Terminating Decimals: Decimals that end after a finite number of digits.
• ↔️Conversion: Any rational number can be converted to a decimal through long division.

➕ Adding Rational Numbers as Decimals

Adding rational numbers in decimal form is straightforward. You simply align the decimal points and add the numbers as you would with whole numbers.

1. 📏Alignment: Line up the decimal points.
2. ➕Addition: Add each column, starting from the right, carrying over if necessary.
3. 📍Placement: Keep the decimal point in the same position in the sum.

➕ Real-World Examples

• 🏦Finance: Calculating interest rates (e.g., 2.5% interest).
• 🌡️Science: Measuring temperature (e.g., 25.5°C).
• 🍳Cooking: Adjusting recipes (e.g., using 0.75 cups of flour).

➕ Examples of Adding Decimals

Example 1: Add 3.25 and 1.75

3.25 + 1.75 = 5.00

Example 2: Add 0.666... and 0.333...

0.666... + 0.333... = 1.000...

Example 3: Add 1.5 and 2.75

1.50 + 2.75 = 4.25

➕ Practice Problems

Solve each problem by adding the decimals

1. What is the result of $2.5 + 3.75$?
2. Solve $1.333... + 0.666...$.
3. Calculate $4.2 + 1.8$.
4. Add $5.6 + 2.4$.
5. Add $7.333... + 2.666...$.
6. Evaluate $8.5 + 3.25$.
7. Evaluate $9.1 + 2.9$.

💡 Conclusion

Rational numbers are fundamental in mathematics and everyday life. Understanding how to add them, especially in decimal form, is a crucial skill for various applications. Whether you're calculating finances or measuring ingredients, a solid grasp of rational numbers will serve you well. Keep practicing, and you'll master it in no time!