Lecture on Scientific Notation

Definition and Purpose

• Scientific Notation: A method to express very large or very small numbers in a simplified format.
  • Example for large number: 1 billion as 1 * 10^9
  • Example for small number: 0.000064 as 6.4 * 10^-6
• Usage: Useful for concise representation of numbers in scientific and engineering contexts.

Converting from Scientific Notation to Decimal Notation

• General Rule:
  • Positive exponent: Move the decimal point to the right.
  • Negative exponent: Move the decimal point to the left.

Examples:

• 4.5 * 10^1: 4.5 * 10 = 45
• 2.3 * 10^2: 2.3 * 100 = 230
• 7.4 * 10^3: 7.4 * 1000 = 7400
• Process: Move decimal according to the exponent.

Practice Problems:

• 2.7 * 10^5: Move decimal 5 places right = 270,000
• 8.36 * 10^7: Move decimal 7 places right = 83,600,000

Converting from Decimal to Scientific Notation

• Steps:
  1. Move the decimal so the number is between 1 and 10.
  2. Count the places moved; this determines the exponent.

Examples:

• 4680: Move decimal 3 places left = 4.68 * 10^3
• 32,500: Move decimal 4 places left = 3.25 * 10^4

Dealing with Negative Exponents

• Represent numbers less than one.
• Shift the decimal to the left.

Examples:

• 3.4 * 10^-2: Move decimal 2 places left = 0.034
• 4.5 * 10^-3: Move decimal 3 places left = 0.0045

Addition and Subtraction of Scientific Notation

• Align exponents by adjusting one term to match the other.
• Combine coefficients when exponents are the same.

Examples:

• 5 * 10^3 + 4 * 10^3 = 9 * 10^3
• 8 * 10^4 - 3 * 10^4 = 5 * 10^4

Multiplication and Division of Scientific Notation

• Multiplication: Multiply coefficients, add exponents.
• Division: Divide coefficients, subtract exponents.

Examples:

• 4 * 10^3 * 2 * 10^5: (4*2) * 10^(3+5)
• 12 * 10^6 / 3 * 10^-4: (12/3) * 10^(6-(-4))

Square and Square Roots in Scientific Notation

• Square Roots: Divide exponent by 2.
• Squares: Multiply exponent by 2.

Examples:

• √(4 * 10^6) = 2 * 10^3
• (5 * 10^4)^2 = 25 * 10^8 (adjust to proper scientific notation)

Common Mistakes and Tips

• Decimal Movement:
  • Right for positive exponents, left for negative.
• Exponent Adjustment:
  • Increase when moving decimal left.
  • Decrease when moving decimal right.

Mental Math Tricks

• Recognize 10^3 as 1,000, 10^6 as 1 million, etc., for quick conversions.

This lecture covers techniques and examples for converting between scientific and decimal notation, as well as performing arithmetic operations using scientific notation. Use these notes to review key concepts and practice problems effectively.