Rational Numbers and Roots

Understanding Rational Numbers and Roots

Welcome to this comprehensive module on rational numbers, fractions, decimals, and roots. In this lesson you will master essential concepts that appear in everyday calculations, from depth measurements to discounts, and learn how to reason with negative numbers, convert fractions to percentages, work with square roots, and identify irrational numbers.

1. Working with Negative Numbers

Negative numbers represent quantities below a chosen reference point, such as sea level or a zero‑balance account. Adding a positive value to a negative number moves the result toward zero, but the sign remains negative until the positive amount exceeds the magnitude of the negative.

• Key idea: Negative plus smaller positive stays negative.
• Example: A submarine at -6 m rises 2.5 m. The new depth is -6 + 2.5 = -3.5 m. The submarine is still below the reference level.

2. Converting Fractions to Percentages

Percent means “per hundred.” To turn a fraction into a percent, divide the numerator by the denominator to get a decimal, then multiply by 100 (move the decimal two places to the right).

• Step‑by‑step:
  1. Compute the decimal: 3 ÷ 8 = 0.375.
  2. Shift the decimal two places: 0.375 → 37.5%.
• Mnemonic: “3‑8 becomes 37‑5.”

3. Principal Square Roots

The square root of a non‑negative number has two solutions, one positive and one negative. The principal square root is defined as the non‑negative (positive) solution and is denoted by the radical sign √.

• Example: √49 = 7. The value -7 is also a square root of 49, but it is not the principal root.
• Tip: Principal = positive.

4. Ordering Decimals on the Number Line

When arranging numbers from smallest to largest, remember that more negative values are smaller, and numbers increase as you move right on the number line.

• Set: -0.75, -0.40, 0.10, 0.30.
• Correct order: -0.75, -0.40, 0.10, 0.30.
• Visual cue: Think left‑to‑right on a ruler.

5. Multiplying Rational Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together, then simplifying the result.

• Problem: (-3/5) × (10/9).
• Calculation:
  1. Numerators: -3 × 10 = -30.
  2. Denominators: 5 × 9 = 45.
  3. Fraction: -30/45 simplifies by dividing both terms by 15 → -2/3.
• Rule of thumb: Multiply, then simplify.

6. Estimating Square Roots Using Nearby Perfect Squares

Exact roots of non‑perfect squares are often irrational, but we can estimate their size by locating the nearest perfect squares.

• Example: √20.
  • Identify perfect squares: 4² = 16 and 5² = 25.
  • Since 20 lies between 16 and 25, √20 is between 4 and 5, closer to 5 because 20 is nearer to 25.
• Memory aid: Think of a square between 16 and 25.

7. Applying Percentage Discounts

Discount calculations are a practical use of percentages. Subtract the discount amount from the original price to find the sale price.

• Scenario: Original price ₱600, discount 20%.
• Steps:
  1. Find 20% of ₱600: 0.20 × 600 = ₱120.
  2. Subtract: 600 – 120 = ₱480.
• Shortcut: 20% off = subtract one‑fifth of the price.

8. Recognizing Irrational Numbers

An irrational number cannot be expressed as a fraction of two integers. Typically, the square root of a non‑perfect square is irrational.

• Examples:
  • ∛125 = 5 (rational).
  • √36 = 6 (rational).
  • √13 – not a perfect square, so it is irrational.
  • 5/8 – a simple rational fraction.
• Quick test: Non‑square → irrational.

9. Summary of Core Concepts

Below is a concise checklist you can use to verify your understanding after completing the module.

• Adding a positive number to a negative number moves the result toward zero but keeps the sign negative unless the positive exceeds the magnitude.
• To convert a fraction to a percent, divide then multiply by 100 (move the decimal two places right).
• The principal square root is always the non‑negative root.
• When ordering decimals, more negative values come first; move left‑to‑right on the number line.
• Multiply fractions by multiplying numerators and denominators, then simplify.
• Estimate √n by locating the nearest perfect squares n lies between.
• Calculate discounts by finding the percentage of the original price and subtracting it.
• A number is irrational if it cannot be written as a fraction, such as the square root of a non‑perfect square.

10. Practice Problems

Test your knowledge with these additional questions. Write your answers on paper before checking the solutions.

• 1. A diver is at -12 m and descends another 3 m. What is the new depth?
• 2. Convert 5/16 to a percentage.
• 3. What is the principal square root of 144?
• 4. Order: -1.2, 0.5, -0.3, 0.0.
• 5. Multiply (2/7) × (-9/4) and simplify.
• 6. Estimate √50 using perfect squares.
• 7. A jacket costs $80 with a 15% discount. Find the sale price.
• 8. Identify which of the following is irrational: √81, √2, 7/9, ∛27.

Answers:

• 1. -15 m
• 2. 31.25%
• 3. 12
• 4. -1.2, -0.3, 0.0, 0.5
• 5. -9/14
• 6. Between 7 and 8 (since 7²=49, 8²=64)
• 7. $68 (15% of 80 = $12; 80‑12 = 68)
• 8. √2

By mastering these concepts, you will be equipped to solve a wide range of mathematical problems that involve rational numbers, percentages, and roots. Keep practicing, and the logic will become second nature.