Overview
This lecture covers how to convert numbers between standard (common) notation and scientific notation, with examples for both large and small numbers.
Purpose of Scientific Notation
- Scientific notation is used to simplify writing very large or very small numbers.
- It avoids the need to write lengthy strings of zeros.
Structure of Scientific Notation
- Scientific notation is written as N × 10ⁿ, where:
- N is a number between 1 and 10.
- n is an integer exponent (positive or negative).
Converting to Scientific Notation
- For large numbers, move the decimal left until one nonzero digit remains before the decimal; exponent is positive and equals number of moves.
- For small numbers, move the decimal right to the first nonzero digit; exponent is negative and equals number of moves.
Example: Large Number
- 6,022,000,000,000,000,000,000,000 becomes 6.022 × 10²³ (moved decimal 23 places left).
Example: Small Number
- 0.0000000000000000000000199 becomes 1.99 × 10⁻²³ (moved decimal 23 places right).
Example: 568.726
- Move decimal two places left: 5.68726 × 10².
Example: 0.00000722
- Move decimal six places right: 7.22 × 10⁻⁶.
Converting to Standard Notation
- For positive exponents, move the decimal right by the exponent value to expand the number.
- For negative exponents, move the decimal left by the exponent value to create a small number.
Example: 8.762 × 10⁴
- Move decimal four places right: 87,620.
Example: 3.02 × 10⁻⁶
- Move decimal six places left: 0.00000302.
Key Terms & Definitions
- Scientific Notation — A way to write numbers as N × 10ⁿ, where N is 1–10 and n is any integer.
- Exponent — Indicates how many times to multiply or divide by 10.
- Standard Notation — The usual way numbers are written, without exponents.
Action Items / Next Steps
- Practice expressing given numbers in both scientific and standard notation as assigned.