Mathematics 4024 · Syllabus tracker
Standard Form
Writing numbers as A × 10ⁿ, converting in and out of standard form, ordering and calculating with it.
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Quick revision
Standard Form on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Writing very large and very small numbers as A × 10n, ordering them, and calculating with them.
1What standard form is
- A number in standard form is A × 10n with 1 ⩽ A < 10 and n a whole number.
- Large numbers have a positive power: 4 560 000 = 4.56 × 106.
- Numbers less than 1 have a negative power: 0.00072 = 7.2 × 10−4.
- The power is how many places the decimal point moves to get A.
How to do it
- Write the digits with the point after the first non-zero digit (that is A).
- Count the places the point moved.
- Moved left (big number) → positive power; moved right (small number) → negative power.
| Ordinary | Standard form |
|---|---|
| 300 000 | 3 × 105 |
| 45 600 | 4.56 × 104 |
| 0.05 | 5 × 10−2 |
| 0.000 307 | 3.07 × 10−4 |
Worked example
Write 0.000 307 in standard form.
- First non-zero digit is 3: A = 3.07.
- The point moves 4 places to the right.
3.07 × 10−4
2Ordering numbers in standard form
- Compare the powers first: the bigger power is the bigger number (for positive numbers).
- If the powers are equal, compare the A values.
- Negative powers: 10−2 is bigger than 10−3.
Worked example
Order, smallest first: 3.1 × 10−2, 9.8 × 10−3, 1.2 × 10−2.
- 10−3 is the smallest power, so 9.8 × 10−3 is smallest.
- Then 1.2 × 10−2 before 3.1 × 10−2.
9.8 × 10−3, 1.2 × 10−2, 3.1 × 10−2
3Calculating without a calculator
- Multiply: multiply the A values and add the powers.
- Divide: divide the A values and subtract the powers.
- Then fix the answer if A is not between 1 and 10: 12 × 109 = 1.2 × 1010; 0.5 × 103 = 5 × 102.
- Add or subtract: change both numbers to ordinary numbers (or to the same power), then add.
Formulas
Worked example
Work out (6 × 105) ÷ (1.5 × 10−2).
- 6 ÷ 1.5 = 4.
- 105 − (−2) = 107.
4 × 107
Exam tip: Adding: 5 × 103 + 7 × 102 = 5000 + 700 = 5700 = 5.7 × 103.
4Using a calculator and real problems
- Use the ×10x (EXP) key: type 3.2 ×10x 5, not 3.2 × 10 ^ 5 (which can go wrong with division).
- Give the final answer in standard form if the question uses it, rounded to 3 significant figures.
- Typical contexts: distances in space, masses of atoms, populations, speed of light (3 × 108 m/s).
Worked example
Light travels 3 × 108 m each second. How long does it take to travel 1.5 × 1011 m?
- time = distance ÷ speed.
- 1.5 ÷ 3 = 0.5 and 1011−8 = 103.
- 0.5 × 103 = 500.
500 seconds
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Leaving 45.6 × 105 as the answer | A must be between 1 and 10: 4.56 × 106 |
| 0.00072 = 7.2 × 104 | Small numbers need a negative power: 7.2 × 10−4 |
| Adding the powers when adding numbers | Only multiplication adds the powers |
| Calculator answer 2.4E7 copied as 2.47 | It means 2.4 × 107 |
★In the exam
- Read whether the answer must be in standard form — then give it in standard form.
- On the non-calculator paper, show the separate steps for the numbers and the powers.
- Check size: a speed or distance should be sensible after you divide.