Scientific Notation and Significant Figures

Learning Objectives

Convert numbers between standard form and scientific notation.

Identify the number of significant figures in a measurement.

Perform basic calculations and round the answer to the correct number of significant figures.

Handling Very Big and Very Small Numbers

In science, we often deal with numbers that are incredibly large (like the distance to a star) or incredibly small (like the size of an atom). Writing all the zeros is tedious and can lead to mistakes. To solve this, scientists use scientific notation.

Scientific Notation

Scientific notation expresses a number as the product of a coefficient and a power of 10. The format is:

c x 10ⁿ

Where 'c' is the coefficient (a number greater than or equal to 1 and less than 10) and 'n' is the exponent.

Positive Exponent: For large numbers. The exponent tells you how many places to move the decimal to the right.

Example: The speed of light is about 300,000,000 m/s.

To convert: Move the decimal to the left until you have a number between 1 and 10. We moved it 8 places.

Scientific Notation: 3.0 x 10⁸ m/s

Negative Exponent: For small numbers. The exponent tells you how many places to move the decimal to the left.

Example: The diameter of a red blood cell is about 0.000007 m.

To convert: Move the decimal to the right until you have a number between 1 and 10. We moved it 6 places.

Scientific Notation: 7.0 x 10⁻⁶ m

Significant Figures: The Digits That Matter

When we measure something, the result is never perfectly exact. Significant figures (or 'sig figs') are the digits in a number that are reliable and necessary to indicate the quantity of something. They tell us about the precision of a measurement.

Rules for Identifying Significant Figures:

1.Non-zero digits are always significant. (e.g., 281 cm has 3 sig figs)

2.Zeros between non-zero digits are significant. (e.g., 50.4 m has 3 sig figs)

3.Leading zeros (at the beginning) are never significant. (e.g., 0.045 g has 2 sig figs)

4.Trailing zeros (at the end) are significant ONLY if the number contains a decimal point.

2500 L has 2 sig figs (no decimal).

2500. L has 4 sig figs (decimal present).

2.500 L has 4 sig figs (decimal present).

Calculations with Significant Figures

When you multiply or divide, your answer should have the same number of significant figures as the measurement with the fewest significant figures.

Example: Calculate the area of a rectangle with sides 4.5 cm and 2.13 cm.

Calculation: 4.5 cm (2 sig figs) * 2.13 cm (3 sig figs) = 9.585 cm²

Rounding: Your answer must have only 2 sig figs. You round 9.585 to 9.6 cm².

Key Terms

**Scientific Notation: A method of writing or displaying numbers in terms of a decimal number between 1 and 10 multiplied by a power of 10.~|~Coefficient: The numerical part of a term in scientific notation (the 'c' in c x 10ⁿ).~|~Exponent: The power to which a number is raised (the 'n' in c x 10ⁿ).~|~Significant Figures: All the digits in a measurement that are known with certainty plus the first digit that is uncertain or estimated. They indicate the precision of a measurement.