Overview
This lecture explains standard form, a method for writing very large or very small numbers using powers of ten, and provides examples for identifying and converting numbers to standard form.
What is Standard Form?
- Standard form is a way to write numbers as ( a \times 10^n ), where ( a ) is between 1 (inclusive) and 10 (exclusive), and ( n ) is any whole number.
- Standard form is useful for expressing extremely large or small numbers more simply.
Identifying Standard Form
- ( 4.5 \times 10^4 ): correct, as 4.5 is between 1 and 10 and 4 is a whole number.
- ( 0.7 \times 10^{-2} ): incorrect, since 0.7 is less than 1.
- ( 9.34 \times 10^{5.5} ): incorrect, because the index 5.5 is not a whole number.
- ( 1 \times 10^{-13} ): correct, as 1 is allowed and -13 is a whole number.
How Standard Form Works
- If ( n ) (the index) is positive, multiply the front number by 10 ( n ) times.
- Example: ( 2.7 \times 10^3 = 2,700 )
- If ( n ) is negative, divide the front number by 10 (|n|) times.
- Example: ( 5 \times 10^{-2} = 0.05 )
- Positive ( n ) means a large number; negative ( n ) means a small number.
Moving the Decimal Point
- A positive index tells you to move the decimal point to the right by ( n ) places.
- Example: ( 2.7 \times 10^3 ) move the decimal 3 places right → 2,700.
- A negative index tells you to move the decimal point to the left by (|n|) places.
- Example: ( 5 \times 10^{-2} ) move the decimal 2 places left → 0.05.
Key Terms & Definitions
- Standard Form — A number written as ( a \times 10^n ), with ( 1 \leq a < 10 ) and ( n ) as a whole number.
- Index/Power — The exponent ( n ) in standard form indicating how many times to multiply or divide by 10.
- Whole Number — An integer, positive, negative, or zero (no fractions or decimals).
Action Items / Next Steps
- Practice converting numbers to and from standard form.
- Review how to move decimal points according to the index.