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Physics · Chapter 2: Physical Quantities and Measurement
Scientific Notation
Writing very large and very small numbers in scientific notation (standard form), and multiplying and dividing with powers of ten.
Key termRememberExampleWatch out
01Concept
Physics deals with numbers that are very large or very small. The mass of the Earth is about 6,000,000,000,000,000,000,000,000 kg. Writing all these zeros is slow and easy to get wrong. Scientific notation solves this.
In scientific notation (also called standard form), a number is written as a number between 1 and 10 multiplied by a power of ten:
a × 10ⁿ, where 1 ≤ a < 10 and n is a whole number (positive or negative).
How to convert
- Large numbers: move the decimal point left until one non-zero digit remains in front. The number of places moved is a positive power. 384,000 km = 3.84 × 10⁵ km (moved 5 places).
- Small numbers: move the decimal point right until one non-zero digit is in front. The number of places moved is a negative power. 0.00052 m = 5.2 × 10⁻⁴ m (moved 4 places).
| Quantity | Ordinary form | Scientific notation |
|---|---|---|
| Speed of light | 300,000,000 m s⁻¹ | 3 × 10⁸ m s⁻¹ |
| Distance from Earth to Moon | 384,000 km | 3.84 × 10⁵ km |
| Mass of the Earth | about 6,000,000,000,000,000,000,000,000 kg | 6.0 × 10²⁴ kg |
| Thickness of a sheet of paper | 0.0001 m | 1 × 10⁻⁴ m |
| Charge of an electron | 0.00000000000000000016 C | 1.6 × 10⁻¹⁹ C |
Calculating with powers of ten
- Multiplying: multiply the numbers and add the powers. (2 × 10³) × (3 × 10⁴) = 6 × 10⁷
- Dividing: divide the numbers and subtract the powers. (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴
- If the answer's number is not between 1 and 10, adjust it. (5 × 10³) × (4 × 10²) = 20 × 10⁵ = 2.0 × 10⁶
02Virtual Lab
Try this topic for yourself in the virtual lab. It opens on its own page with steps, tasks and questions.
03Real World Examples
Population
Pakistan's population, about 241 million (2023 census), is 2.41 × 10⁸ in scientific notation.
Calculators
When a calculator shows 3.2E-05, it means 3.2 × 10⁻⁵. The E stands for "exponent" (power of ten).
Space news
News reports about space often give distances in scientific notation, because the Sun is about 1.5 × 10¹¹ m away.
04Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.What is scientific notation?Show answer
Answer
Scientific notation is a way of writing a number as a × 10ⁿ, where a is between 1 and 10 and n is a whole number.
Q2.Write in scientific notation: (a) 45,000 (b) 0.0063 (c) 6,371 kmShow answer
Answer
- (a) 45,000 = 4.5 × 10⁴
- (b) 0.0063 = 6.3 × 10⁻³
- (c) 6,371 km = 6.371 × 10³ km
Q3.Write in ordinary form: (a) 7.2 × 10³ (b) 3.5 × 10⁻²Show answer
Answer
- (a) 7,200
- (b) 0.035
Q4.Calculate (3 × 10⁵) × (4 × 10⁻²). Give the answer in scientific notation.Show answer
Answer
- = (3 × 4) × 10⁽⁵⁻²⁾
- = 12 × 10³
- = 1.2 × 10⁴
Q5.Light travels at 3 × 10⁸ m s⁻¹. How far does it travel in 500 s (about the time from the Sun to the Earth)?Show answer
Answer
- distance = speed × time
- = (3 × 10⁸) × (5 × 10²)
- = 15 × 10¹⁰ m = 1.5 × 10¹¹ m
Long questions
Q6.Why is scientific notation used in physics? Write the rules for converting large and small numbers, with two examples of each.Show answer
Answer
It is used because physics deals with very large and very small numbers. Scientific notation makes them short, easy to read and easy to compare and calculate with, and it shows the significant figures clearly.
- Large number: move the decimal point left; the power is positive. 150,000,000 km = 1.5 × 10⁸ km; 6,000 m = 6 × 10³ m.
- Small number: move the decimal point right; the power is negative. 0.000 01 m = 1 × 10⁻⁵ m; 0.0029 kg = 2.9 × 10⁻³ kg.