Understanding Significant Digits
* Non-zero digits: All non-zero digits are significant. (e.g., 123.45 has 5 significant digits)
* Zeros between non-zero digits: Zeros between non-zero digits are significant. (e.g., 100.5 has 4 significant digits)
* Leading zeros: Zeros to the left of the first non-zero digit are *not* significant. (e.g., 0.0025 has 2 significant digits)
* Trailing zeros: Trailing zeros are significant *only* if there is a decimal point. (e.g., 100 has 1 significant digit, but 100. has 3 significant digits)
Multiplication Rule
* The result of a multiplication can have no more significant digits than the factor with the *least* number of significant digits.
Example:
Let's say you want to multiply 2.54 (3 significant digits) by 12.0 (3 significant digits):
1. Perform the multiplication: 2.54 * 12.0 = 30.48
2. Determine the least significant digits: Both factors have 3 significant digits.
3. Round the result: Since we need to keep 3 significant digits, we round 30.48 to 30.5.
More Examples:
* 1.234 * 5.0 = 6.17: Both factors have 4 and 2 significant digits respectively. The result is rounded to 6.2, reflecting the least number of significant digits (2).
* 0.005 * 23.1 = 0.1155: The factors have 1 and 3 significant digits respectively. The result is rounded to 0.12, reflecting the least number of significant digits (1).
* 1000 * 3.14 = 3140: The factors have 1 and 3 significant digits respectively. The result is rounded to 3140, reflecting the least number of significant digits (1).
Important Notes:
* Significant digits are about precision. They tell us how reliably we know the value of a measurement.
* Rounding rules: When rounding, look at the digit immediately to the right of the last significant digit. Round up if it's 5 or greater, and round down if it's less than 5.
* Scientific notation: Scientific notation can be helpful for expressing numbers with many significant digits.
Let me know if you'd like to work through more examples!
