Overview
This lecture explains how to write numbers in standard form, identify valid examples, and convert between standard form and ordinary numbers.
What is Standard Form?
- Standard form is a way to write very large or very small numbers using powers of ten.
- The general format is: ( a \times 10^n ) where ( a ) (front number) is ≥ 1 and < 10, and ( n ) (index) is a whole number (positive or negative).
Identifying Standard Form
- To be in standard form, the front number must be between 1 (inclusive) and 10 (exclusive).
- The index must be a whole number (no decimals or fractions, can be negative).
- Examples:
- ( 4.5 \times 10^4 ) is standard form.
- ( 0.7 \times 10^{-2} ) is not, because 0.7 < 1.
- ( 9.34 \times 10^{5.5} ) is not, because 5.5 is not a whole number.
- ( 1 \times 10^{-13} ) is standard form.
Understanding the Power (Index)
- A positive index means multiply the front number by 10 the specified number of times (( n ) times).
- Example: ( 2.7 \times 10^3 = 2,700 )
- A negative index means divide the front number by 10 the specified number of times.
- Example: ( 5 \times 10^{-2} = 0.05 )
- Positive indices produce large numbers; negative indices produce small numbers.
Moving the Decimal Point
- The power indicates how many places to move the decimal:
- Positive index: move decimal right (number gets bigger).
- Negative index: move decimal left (number gets smaller).
- Example: ( 2.7 \times 10^3 ): Move decimal 3 places right → 2,700.
- Example: ( 5 \times 10^{-2} ): Move decimal 2 places left → 0.05.
Key Terms & Definitions
- Standard Form — A way to write numbers as ( a \times 10^n ), where ( 1 \leq a < 10 ) and ( n ) is a whole number.
- Index (Power) — The exponent ( n ); shows how many times to multiply or divide by 10.
- Front Number — The ( a ) in standard form; must be ≥ 1 and < 10.
Action Items / Next Steps
- Practice writing numbers in standard form and converting them back to ordinary numbers.
- Review any assigned problems involving identifying and using standard form.