Overview
This lecture explains scientific notation: what it is, why it's useful, how to convert between scientific and decimal notation, and how to perform arithmetic operations—including multiplication, division, addition, subtraction, roots, and powers—using scientific notation. The lecture also covers tips for quick mental conversions and emphasizes the importance of proper scientific notation.
Introduction to Scientific Notation
- Scientific notation is a way to write very large or very small numbers using powers of 10.
- Numbers are expressed as a product of a coefficient (between 1 and 10) and a power of ten, simplifying calculations and representation.
- Examples:
- 1,000,000,000 = 1 × 10⁹
- 0.000064 = 6.4 × 10⁻⁵
- Scientific notation is especially useful in science and engineering, where extremely large or small values are common.
Converting Scientific Notation to Decimal Notation
- For positive exponents, move the decimal point to the right by the value of the exponent.
- Example: 4.5 × 10¹ = 45
- 2.3 × 10² = 230
- 7.4 × 10³ = 7,400
- 4.28 × 10⁴ = 42,800 (move decimal 4 places right)
- For negative exponents, move the decimal point to the left by the value of the exponent.
- Example: 3.4 × 10⁻² = 0.034
- 4.5 × 10⁻³ = 0.0045
- 3.746 × 10⁻⁴ = 0.0003746
- The result for negative exponents is always a number between 0 and 1 (but greater than 0).
- Tip: Count the number of places you move the decimal to ensure accuracy. Add zeros as needed.
Converting Decimal Numbers to Scientific Notation
- Move the decimal point to create a number between 1 and 10; count the number of moves to determine the exponent.
- Example: 4,680 → 4.68 × 10³ (moved 3 places left)
- 32,500 → 3.25 × 10⁴ (4 places left)
- 476 → 4.76 × 10² (2 places left)
- For numbers less than 1, move the decimal to the right; the exponent will be negative.
- Example: 0.024 → 2.4 × 10⁻² (2 places right)
- 0.036 → 3.6 × 10⁻² (2 places right)
- 0.00071 → 7.1 × 10⁻⁴ (4 places right)
- Moving the decimal left increases the exponent (positive); moving right decreases it (negative).
- The coefficient should always be between 1 and 10.
- If the decimal is moved to the right, the exponent decreases; if moved to the left, the exponent increases.
Proper Scientific Notation
- The coefficient (the number before × 10) must be at least 1 and less than 10.
- If the coefficient is not in this range, adjust by moving the decimal and changing the exponent:
- Move decimal left: increase exponent.
- Move decimal right: decrease exponent.
- Example: 42.8 × 10⁵ is not proper; move decimal left to get 4.28 × 10⁶.
- Always ensure the final answer is in proper scientific notation, with the decimal point between the first two nonzero digits.
Multiplication and Division in Scientific Notation
- Multiplication:
- Multiply the coefficients.
- Add the exponents.
- Example: (4 × 10³) × (2 × 10⁵) = 8 × 10⁸
- If the result is not in proper form, adjust the decimal and exponent accordingly.
- Example: (5 × 10⁴) × (7 × 10⁻⁸) = 35 × 10⁻⁴ → 3.5 × 10⁻³ (move decimal left, increase exponent by 1)
- Division:
- Divide the coefficients.
- Subtract the exponents (numerator exponent minus denominator exponent).
- Example: (12 × 10⁶) ÷ (3 × 10⁻⁴) = 4 × 10¹⁰
- If the result is not in proper form, adjust as needed.
- Example: (96 × 10⁻⁵) ÷ (6 × 10²) = 16 × 10⁻⁷ → 1.6 × 10⁻⁶ (move decimal left, increase exponent by 1)
- Always check if the coefficient needs to be adjusted to stay between 1 and 10.
Addition and Subtraction in Scientific Notation
- Only combine numbers with the same exponent.
- If exponents differ, adjust one number so both exponents match (usually convert the smaller exponent to match the larger).
- Move the decimal point and adjust the exponent accordingly.
- Example: (8 × 10⁴) + (2 × 10³) → convert 2 × 10³ to 0.2 × 10⁴, then add: (8 + 0.2) × 10⁴ = 8.2 × 10⁴
- Once exponents match, add or subtract the coefficients.
- Example: (5 × 10³) + (4 × 10³) = 9 × 10³
- Example: (8 × 10⁴) - (3 × 10⁴) = 5 × 10⁴
- Always express the final answer in proper scientific notation.
- If the result is not in proper form, adjust the coefficient and exponent as needed.
Roots and Powers in Scientific Notation
- Roots:
- Take the root of the coefficient.
- Divide the exponent by the root's degree.
- Example: √(4 × 10⁶) = 2 × 10³
- √(9 × 10⁸) = 3 × 10⁴
- For negative exponents: √(36 × 10⁻⁶) = 6 × 10⁻³
- For cube roots: ∛(8 × 10⁹) = 2 × 10³
- If the coefficient is not a perfect square or cube, adjust it to make calculation easier (e.g., move decimal and adjust exponent).
- Powers:
- Raise the coefficient to the power.
- Multiply the exponent by the power.
- Example: (5 × 10⁴)³ = 125 × 10¹² → 1.25 × 10¹⁴ (move decimal left, increase exponent by 2)
- Example: (4 × 10⁻³)² = 16 × 10⁻⁶ → 1.6 × 10⁻⁵ (move decimal left, increase exponent by 1)
- If the coefficient is not between 1 and 10 after the operation, adjust and update the exponent accordingly.
Quick Mental Conversions
- Recognize common powers of ten for fast calculations:
- 10³ = 1,000 (thousand)
- 10⁶ = 1,000,000 (million)
- 10⁹ = 1,000,000,000 (billion)
- Use these benchmarks to quickly estimate or convert numbers in scientific notation to standard form.
- Example: 4 × 10⁴ = 40,000
- 8 × 10⁷ = 80,000,000
- 3 × 10¹⁰ = 30,000,000,000
- For quick division or multiplication, remember to add or subtract exponents and adjust the coefficient as needed.
Key Terms & Definitions
- Scientific Notation: A way to write numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
- Decimal Notation: The standard way of writing numbers without exponents.
- Exponent: The power to which 10 is raised, indicating how many places to move the decimal.
- Coefficient: The number (1 ≤ a < 10) in scientific notation.
- Proper Scientific Notation: The form where the coefficient is between 1 and 10.
Action Items / Next Steps
- Practice converting numbers between scientific and decimal notation for both large and small numbers.
- Complete exercises on multiplying, dividing, adding, and subtracting numbers in scientific notation, ensuring answers are in proper form.
- Try finding roots (square, cube) and raising numbers to powers using scientific notation, adjusting the coefficient and exponent as needed.
- Use mental shortcuts for quick conversions and to check your work.
- Double-check that all answers are in proper scientific notation, with the coefficient between 1 and 10 and the correct exponent.