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Working with Integers and Rational Numbers

1. Understanding Integers

Integers are whole numbers that can be positive, negative, or zero.

  • 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-Life Example:

Bank Account: A deposit of $10 is +10, a withdrawal of $10 is -10.
Temperature: If today is -5 °C, it's colder than 0 °C.

2. Adding and Subtracting Integers

Rule 1: Adding Integers

  • Same signs: Add the numbers and keep the sign.
  • Different signs: Subtract and keep the sign of the larger number.

Example 1: -4 + (-3) = -7

Example 2: 8 + (-12) = -4

Example 3: -5 + 7 = 2

Rule 2: Subtracting Integers

Example 1: 5 - (-3) = 5 + 3 = 8

Example 2: -6 - (-2) = -6 + 2 = -4

Example 3: -9 - 4 = -13

3. Multiplying and Dividing Integers

Rule 1: Multiplication Rules

  • Same signs: The answer is positive.
  • Different signs: The answer is negative.

Example 1: (-3) × (-2) = 6

Example 2: 10 × (-5) = -50

Example 3: (-4) × 6 = -24

Rule 2: Division Rules

Example 1: (-12) ÷ (-3) = 4

Example 2: 15 ÷ (-5) = -3

Example 3: (-30) ÷ 6 = -5

4. Understanding Rational Numbers

A rational number is any number that can be written as a fraction a/b, where a and b are integers and b ≠ 0.

  • Examples of Rational Numbers:
    • Fractions: 1/2, -3/4, 7/12
    • Decimals that terminate: 0.5, -3.75, 2.0
    • Repeating decimals: 1.333..., 0.666...
  • Examples of Non-Rational (Irrational) Numbers:
    • Square roots of non-perfect squares (e.g., √2, √3)
    • Non-repeating, non-terminating decimals (e.g., π, e)

5. Comparing and Ordering Rational Numbers

To compare rational numbers:

  1. Convert fractions to decimals if needed.
  2. Place them on a number line.
  3. Compare the values directly.

Example: Compare 3/4 and 0.7.
Convert 3/4 = 0.75 and since 0.75 > 0.7, we conclude: 3/4 > 0.7.

6. Converting Between Decimals and Fractions

To convert:

  • Fraction → Decimal: Divide the numerator by the denominator.
  • Decimal → Fraction: Write the decimal over a power of 10 and simplify.

Example 1: Convert 5/8 to a decimal: 5 ÷ 8 = 0.625
Example 2: Convert 0.75 to a fraction: 75/100 simplifies to 3/4.

Practice Questions (Basic Level) 🎯

  1. Solve: -7 + 5 = ?
  2. Solve: -4 × 6 = ?
  3. Solve: 15 ÷ (-3) = ?
  4. Convert 3/4 to a decimal.
  5. Is 0.75 a rational number? Why?
  6. Solve: 5 - (-2)
  7. Compare 0.6 and 2/3.
  8. Convert 1.25 to a fraction.
  9. Order from smallest to largest: -1, 1/2, -2.5, 0.25.
  10. Solve: (-3) × (-5) + 2.

Want a Bigger Challenge? 🤔🔥

  1. Solve: -8 + (-4) - (-2).
  2. Find the missing number: _ × (-6) = 24.
  3. If a temperature drops 4°C per hour from an initial 20°C, what is the temperature after 5 hours?
  4. Convert the repeating decimal 0.3‾ to a fraction.
  5. A bank account starts with $500 and withdraws $25 per day. Write and solve an equation to determine how many days until the account reaches $100.

9. Summary

  • ✅ Integers include positive and negative whole numbers and zero.
  • ✅ Adding integers: Same signs add; different signs subtract and take the larger sign.
  • ✅ Multiplying & dividing integers: Same signs produce positive; different signs produce negative.
  • ✅ Rational numbers can be expressed as fractions and include decimals that terminate or repeat.
  • ✅ Comparing rational numbers can be done by converting to decimals and using a number line.
  • ✅ Converting between fractions and decimals involves division and expressing decimals over powers of 10.

🎉 Awesome work! Now that you understand Integers and Rational Numbers, you're ready for the next topic: Basic Exponents! 🚀

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