Chapter 2 · Measurement and Problem Solving
2.2Scientific Notation: Writing Large and Small Numbers
10 min · two checks
Predict
How do you write 0.000412 without a string of zeros?
The idea
- Convert between decimal form and scientific notation.
- Multiply and divide powers of ten by adding or subtracting exponents.
Scientific notation writes a number as a coefficient times a power of ten. The coefficient is at least 1 and less than 10. For a large number, the exponent is positive and equals how many places the decimal moved left: 602,000,000,000,000,000,000,000 is 6.02 × 10²³. For a small number, the exponent is negative and equals how many places the decimal moved right: 0.000412 is 4.12 × 10⁻⁴.
When you multiply, multiply the coefficients and add the exponents. When you divide, divide the coefficients and subtract the exponents. (3.0 × 10⁴) × (2.0 × 10⁻⁶) = 6.0 × 10⁻². If the coefficient wanders outside the range from 1 to 10, shift the decimal and adjust the exponent. Adding numbers in scientific notation requires the same power of ten first; otherwise you are adding different place values.
Keep these
- Coefficient a satisfies 1 ≤ a < 10.
- Positive exponents: large numbers. Negative exponents: small numbers.
- Multiply coefficients and add exponents; divide coefficients and subtract exponents.
Worked path
The radius of a hydrogen atom is about 5.3 × 10⁻¹¹ m. Write that in decimal form, then compute how many of those radii fit in 1.0 × 10⁻³ m.
Observe
5.3 × 10⁻¹¹ m means the decimal moves 11 places left: 0.000000000053 m.