Scientific Notation & Significant Figures
IMAT Interactive Study Tool
Scientific Notation
Scientific notation is a way of writing very large or very small numbers in a compact form. It makes calculations easier and clearly shows the number's magnitude.
Format: \(a \times 10^b\)
Where \(1 \le |a| < 10\) and \(b\) is an integer.
- Large Numbers: The decimal point moves to the left, and the exponent \(b\) is positive. Example: \(5,972,000 = 5.972 \times 10^6\).
- Small Numbers: The decimal point moves to the right, and the exponent \(b\) is negative. Example: \(0.000025 = 2.5 \times 10^{-5}\).
Significant Figures (Sig Figs)
Significant figures are the digits in a number that are reliable and necessary to indicate the precision of a measurement. They include all certain digits plus one estimated (uncertain) digit.
The Rules for Identifying Sig Figs:
- Non-zero digits are always significant. (e.g., 123 has 3 sig figs).
- "Captive" zeros between non-zero digits are significant. (e.g., 101 has 3 sig figs).
- Leading zeros are NOT significant. They are just placeholders. (e.g., 0.052 has 2 sig figs).
- Trailing zeros are significant ONLY if the number contains a decimal point. (e.g., 25.0 has 3 sig figs, but 2500 has only 2).
Calculations with Significant Figures
The result of a calculation can only be as precise as the least precise measurement used.
Addition/Subtraction: The answer must have the same number of decimal places as the measurement with the fewest decimal places.
Example: \(12.11 + 18.0 + 1.013 = 31.123 \to 31.1\) (one decimal place)
Multiplication/Division: The answer must have the same number of significant figures as the measurement with the fewest significant figures.
Example: \(4.56 \times 1.4 = 6.384 \to 6.4\) (two significant figures)
1. True or False: The number 0.0050 has only one significant figure.
2. True or False: Scientific notation is primarily used to make numbers look more complex.
Example 1: Calculation and Rounding
Calculate the area of a rectangle with a length of 12.5 cm and a width of 2.3 cm. Express the answer in scientific notation with the correct number of significant figures.
Step 1: Perform the raw calculation.
\(Area = length \times width = 12.5 \, cm \times 2.3 \, cm = 28.75 \, cm^2\)
Step 2: Determine the correct number of significant figures.
This is a multiplication problem. The length (12.5 cm) has 3 significant figures. The width (2.3 cm) has 2 significant figures. The answer must be rounded to the fewest number of significant figures, which is 2.
Step 3: Round the answer.
We need to round 28.75 to 2 significant figures. This gives us 29.
Step 4: Convert to scientific notation.
\(29 \, cm^2 = 2.9 \times 10^1 \, cm^2\)
Conclusion: The area is \(2.9 \times 10^1 \, cm^2\).
Example 2: Identifying Significant Figures
How many significant figures are in each of the following numbers: a) 10.05, b) 0.003, c) 500, d) 500.?
- a) 10.05: The two zeros are "captive" between non-zero digits, so they are significant. Result: 4 significant figures.
- b) 0.003: The three leading zeros are placeholders and are not significant. Only the 3 is significant. Result: 1 significant figure.
- c) 500: The two trailing zeros are not significant because there is no decimal point. They are ambiguous placeholders. Result: 1 significant figure.
- d) 500.: The trailing zeros are significant because the decimal point indicates they were measured. Result: 3 significant figures.
| Rule | Example | # of Sig Figs | Is it Significant? |
|---|---|---|---|
| Non-zero digits | 1, 2, 3, 4, 5, 6, 7, 8, 9 | N/A | Always |
| Leading Zeros | 0.05 | 1 | Never |
| Captive Zeros | 505 | 3 | Always |
| Trailing Zeros (No decimal) | 500 | 1 | Never |
| Trailing Zeros (With decimal) | 5.00 | 3 | Always |