Question
When a value is given in scientific notation, how can you tell if the number is very large or very small? Explain.
Answers
GPT-4o mini
In scientific notation, numbers are expressed in the form \( a \times 10^n \), where \( a \) is a number (the coefficient) typically between \( 1 \) and \( 10 \), and \( n \) is an integer (the exponent). To determine whether the number is very large or very small, you can look at the exponent \( n \).
- **Very Large Numbers:** If \( n \) is a positive integer (e.g., \( n = 1, 2, 3, \ldots \)), the number represents a large value. For example:
- \( 3 \times 10^5 = 300,000 \) (positive exponent indicates a large number).
- \( 1.2 \times 10^{12} = 1,200,000,000,000 \) (also a large number).
- **Very Small Numbers:** If \( n \) is a negative integer (e.g., \( n = -1, -2, -3, \ldots \)), the number represents a small value. For example:
- \( 4.5 \times 10^{-3} = 0.0045 \) (negative exponent indicates a small number).
- \( 2.1 \times 10^{-10} = 0.00000000021 \) (also a very small number).
In summary, the sign of the exponent in scientific notation is key: a positive exponent indicates a very large number, while a negative exponent indicates a very small number.
- **Very Large Numbers:** If \( n \) is a positive integer (e.g., \( n = 1, 2, 3, \ldots \)), the number represents a large value. For example:
- \( 3 \times 10^5 = 300,000 \) (positive exponent indicates a large number).
- \( 1.2 \times 10^{12} = 1,200,000,000,000 \) (also a large number).
- **Very Small Numbers:** If \( n \) is a negative integer (e.g., \( n = -1, -2, -3, \ldots \)), the number represents a small value. For example:
- \( 4.5 \times 10^{-3} = 0.0045 \) (negative exponent indicates a small number).
- \( 2.1 \times 10^{-10} = 0.00000000021 \) (also a very small number).
In summary, the sign of the exponent in scientific notation is key: a positive exponent indicates a very large number, while a negative exponent indicates a very small number.