Overview
This lesson introduces scientific notation, explains how to convert between standard and scientific notation for large and small numbers, and provides practice examples.
Scientific Notation Basics
- Scientific notation expresses very large or very small numbers using powers of ten.
- To convert a number to scientific notation, move the decimal after the first non-zero digit.
- The number is written as a decimal (between 1 and 10) times 10 raised to an exponent.
- A positive exponent indicates a large number; a negative exponent indicates a small number (between 0 and 1).
Converting Standard Numbers to Scientific Notation
- Move the decimal to just after the first non-zero digit; count the spaces moved.
- Example: 45,000 becomes 4.5 × 10⁴ (moved 4 places left).
- Examples:
- 3,750 → 3.75 × 10³ (3 places left)
- 580,000 → 5.8 × 10⁵ (5 places left)
- 72,000,000 → 7.2 × 10⁷ (7 places left)
- 9,300,000,000 → 9.3 × 10⁹ (9 places left)
Converting Small Numbers to Scientific Notation
- Move the decimal to just after the first non-zero digit; count spaces right for negative exponent.
- Example: 0.0023 → 2.3 × 10⁻³ (3 places right)
- Examples:
- 0.00076 → 7.6 × 10⁻⁴
- 0.049 → 4.9 × 10⁻²
- 0.00000541 → 5.41 × 10⁻⁶
- 0.000000000835 → 8.35 × 10⁻¹⁰
Converting Scientific Notation to Standard Notation
- For positive exponents, move the decimal right.
- 2.4 × 10² → 240
- 3.56 × 10³ → 3,560
- 4.27 × 10⁵ → 427,000
- 3.96 × 10⁷ → 39,600,000
- For negative exponents, move the decimal left.
- 3.7 × 10⁻³ → 0.0037
- 4.16 × 10⁻⁵ → 0.0000416
Mixed Practice Examples
- 0.00735 → 7.35 × 10⁻³
- 364,000 → 3.64 × 10⁵
- 0.015 → 1.5 × 10⁻²
- 2,800 → 2.8 × 10³
Key Terms & Definitions
- Scientific Notation — A way to write numbers as a product of a decimal and a power of ten.
- Exponent — Indicates how many times the base (10) is multiplied (positive: large number; negative: small number).
- Standard Notation — The usual way of writing numbers.
Action Items / Next Steps
- Practice converting numbers between standard and scientific notation.
- Review the rules for moving the decimal and determining the exponent’s sign.