Arithmetic
Scientific Notation
The mass of an electron (CODATA 2018) is 9.1093837 × 10^-31 kg. The distance from the Sun to Proxima Centauri is 4.0 × 10^16 m. NASA telemeters the Perseverance rover in the same notation. Without scientific notation, physics simply cannot be written down: the digit count in plain decimal makes comparison and arithmetic impossible.
- **Astronomy:** distances to stars and galaxies
- **Particle physics:** sizes of atoms and elementary particles
- **Biology:** number of cells, molecules, bacteria
What is Scientific Notation
The distance to the Sun is 150,000,000,000 m. The size of an atom is 0.0000000001 m. Counting zeros is tedious and error-prone. **Scientific notation** solves this problem.
**Scientific notation (standard form):** a × 10ⁿ, where 1 ≤ a < 10 150,000,000,000 = 1.5 × 10¹¹ 0.0000000001 = 1 × 10⁻¹⁰
In scientific notation the mantissa (the number before 10) must be from 1 to 10 (not including 10). This is called 'standard form'.
How is 0.00034 written in scientific notation?
Order of Magnitude
Order of magnitude is the exponent of ten in scientific notation. It shows the 'scale' of a number and enables quick comparison of quantities.
**Estimating order:** Often the exact value doesn't matter, only the order does. 'How many people in the city?', 10⁶ (a million) 'How many stars in the galaxy?', 10¹¹ (hundreds of billions)
Order of magnitude is useful for checking answers: if a calculation gives an elephant's mass as 10⁵ kg instead of 10³ kg, there's an error of two orders of magnitude somewhere.
By how many orders of magnitude is 1,000,000 greater than 100?
Operations in Scientific Notation
Scientific notation simplifies calculations with very large and very small numbers. Multiplication and division are especially convenient!
**Converting to standard form:** • If mantissa ≥ 10: divide by 10, increase the exponent 20 × 10⁵ = 2 × 10⁶ • If mantissa < 1: multiply by 10, decrease the exponent 0.5 × 10⁴ = 5 × 10³
On a calculator, scientific notation is shown with the letter E: 3E8 means 3 × 10⁸. This is handy for entering large numbers.
Calculate (4 × 10⁶) × (5 × 10⁻²):
Scales of the Universe
Scientific notation opens a window into scales from subatomic particles to galaxy clusters, 40 orders of magnitude!
**Interesting ratios:** • Atoms in the human body: ~10²⁸ • Stars in the observable Universe: ~10²⁴ • A human body contains more atoms than the Universe has stars!
Understanding orders of magnitude is an important skill for scientific thinking. It enables assessment of the plausibility of claims and results.
10⁵ × 10³ = 10¹⁵
10⁵ × 10³ = 10⁸ (exponents are added, not multiplied)
When multiplying powers with the same base, exponents are added: 10⁵ × 10³ = 10^(5+3) = 10⁸. Note: 10² × 10² = 10⁴, which is 100 × 100 = 10000 = 10⁴, this confirms addition (2+2=4), not multiplication (2×2=4 happens to equal the same here, but only by coincidence for this specific case).
Light travels from the Sun to the Earth (1.5 × 10¹¹ m) in 500 seconds. What is the speed of light?
Key Ideas
- a × 10ⁿ, scientific notation, where 1 ≤ a < 10
- Order of magnitude, the exponent of ten
- Multiplication: exponents add; division: exponents subtract
- For addition: same order required
Related Topics
Scientific notation connects to powers and measurement:
- Integer Exponents — We use 10ⁿ
- Approximate Calculations — Estimating order of magnitude
- Units of Measurement — SI prefixes (mega-, nano-)
Вопросы для размышления
- Why do scientists prefer scientific notation to ordinary notation?
- How does order of magnitude help check the plausibility of answers?
- By how many orders of magnitude does the size of an atom differ from the size of a galaxy?