Overview
This lecture covers standard form, a method for writing very large or small numbers using powers of ten, including its rules, examples, and how to convert numbers.
What Is Standard Form?
- Standard form writes numbers as ( a \times 10^n ), where ( 1 \leq a < 10 ) and ( n ) is a whole number.
- It is used mainly for very large or very small numbers to make them easier to write and understand.
Rules for Standard Form
- The front number ( a ) must be greater than or equal to 1 but less than 10.
- The index ( n ) (also called the power) must be a whole number (can be positive or negative).
- Examples:
- ( 4.5 \times 10^4 ) is valid (4.5 is between 1 and 10, 4 is a whole number).
- ( 0.7 \times 10^{-2} ) is invalid (0.7 is less than 1).
- ( 9.34 \times 10^{5.5} ) is invalid (5.5 is not a whole number).
- ( 1 \times 10^{-13} ) is valid (1 is allowed, -13 is a whole number).
Understanding the Index
- A positive index means multiply the front number by 10 that many times (number gets bigger).
- A negative index means divide the front number by 10 that many times (number gets smaller).
- Example: ( 2.7 \times 10^3 = 2,700 )
- Example: ( 5 \times 10^{-2} = 0.05 )
Moving the Decimal Point
- Positive indices move the decimal point to the right.
- Negative indices move the decimal point to the left.
- Fill empty places with zeros after moving the decimal, and remove unnecessary decimal points or zeros as needed.
Key Terms & Definitions
- Standard Form — a way to write numbers as ( a \times 10^n ), where ( 1 \leq a < 10 ) and ( n ) is an integer.
- Front Number (a) — the number between 1 and 10 in standard form.
- Index/Power (n) — the exponent showing how many times to multiply or divide by 10.
- Positive Index — indicates multiplication by powers of ten.
- Negative Index — indicates division by powers of ten.
Action Items / Next Steps
- Practice writing numbers in and out of standard form.
- Be able to identify if a number is correctly written in standard form.