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Mathematics · Chapter 2: Real Numbers
Standard Form (Scientific Notation)
Writing very large and very small numbers as A × 10ⁿ, and calculating with them.
Key termRememberExampleWatch out
01Concept
Pakistan's population in the 2023 census was about 241,490,000. A dust particle may be 0.000002 m across. Counting all those zeros is slow and easy to get wrong. Standard form (also called scientific notation) fixes this.
A number is in standard form when it is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
| Ordinary number | Standard form | Why |
|---|---|---|
| 241,490,000 | 2.4149 × 10⁸ | Decimal point moves 8 places left |
| 5,600 | 5.6 × 10³ | 3 places left |
| 7 | 7 × 10⁰ | No move |
| 0.045 | 4.5 × 10⁻² | 2 places right |
| 0.000002 | 2 × 10⁻⁶ | 6 places right |
Multiplying and dividing
Deal with the numbers and the powers of ten separately, then tidy up.
- (3 × 10⁴) × (5 × 10³) = 15 × 10⁷ = 1.5 × 10⁸
- (8 × 10⁹) ÷ (2 × 10⁻³) = 4 × 10¹²
Adding and subtracting
First make the powers of ten the same (or change both to ordinary numbers).
- (4.2 × 10⁵) + (3 × 10⁴) = (4.2 × 10⁵) + (0.3 × 10⁵) = 4.5 × 10⁵
On a calculator
Use the ×10ˣ (or EXP) key. To enter 3.2 × 10⁻⁴, press 3.2, then ×10ˣ, then (−) 4. Do not type "× 10" as well.
02Virtual Lab
Try this topic for yourself in the virtual lab. It opens on its own page with steps, tasks and questions.
03Key Formulas
Standard form
04Real World Examples
Population
Pakistan's 2023 census counted about 2.41 × 10⁸ people. Census and economic reports use standard form and "million" and "billion" to keep numbers readable.
Space
The Earth is about 1.5 × 10¹¹ m from the Sun. SUPARCO, Pakistan's space agency, works with distances like these when it places satellites in orbit.
Very small things
A red blood cell is about 8 × 10⁻⁶ m across; a virus can be smaller than 1 × 10⁻⁷ m.
05Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Write in standard form: (a) 93,000,000 (b) 0.00071.Show answer
Answer
(a) 9.3 × 10⁷ (b) 7.1 × 10⁻⁴
Q2.Write 3.05 × 10⁴ and 8 × 10⁻³ as ordinary numbers.Show answer
Answer
3.05 × 10⁴ = 30,500; 8 × 10⁻³ = 0.008.
Q3.Work out (6 × 10⁵) × (4 × 10⁻²), giving the answer in standard form.Show answer
Answer
24 × 10³ = 2.4 × 10⁴.
Q4.Work out (9 × 10⁸) ÷ (3 × 10¹¹).Show answer
Answer
3 × 10⁻³
Q5.Add 5.1 × 10⁶ and 2 × 10⁵.Show answer
Answer
5.1 × 10⁶ + 0.2 × 10⁶ = 5.3 × 10⁶.
Q6.Which is larger: 3.9 × 10⁻⁵ or 4.1 × 10⁻⁶?Show answer
Answer
3.9 × 10⁻⁵, because 10⁻⁵ is ten times bigger than 10⁻⁶.
Long questions
Q7.Light travels 3 × 10⁸ m every second. The Sun is about 1.5 × 10¹¹ m from the Earth. How long does sunlight take to reach us?Show answer
Answer
- time = distance ÷ speed
- = (1.5 × 10¹¹) ÷ (3 × 10⁸)
- = 0.5 × 10³ = 500 s
- That is 8 minutes 20 seconds.