The Standard Form & Powers of 10
Section: Number and Calculation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is Standard Form?
Standard form (also called scientific notation) writes any number as a × 10n, where a is between 1 and 10 (1 ≤ a < 10) and n is an integer. It exists to make very large and very small numbers manageable.
- Write and read extremely large numbers, e.g. distance to the Sun: 150,000,000,000 m = 1.5 × 1011 m
- Write and read extremely small numbers, e.g. size of an atom: 0.00000000001 m = 1 × 10-11 m
- Perform calculations with these numbers without losing track of the zeros
- Compare the size of different quantities quickly
Converting to Standard Form
Large Numbers (Positive Powers)
- Move the decimal point to get a number between 1 and 10.
- Count how many places you moved the decimal point to the LEFT.
- This count becomes the positive power of 10.
The number of places the decimal point moves becomes the power of 10
-
Question: Write 45,000 in standard form.
- Step 1: Move the decimal point until one non-zero digit remains before it: 45,000.0 → 4.5000 (moved 4 places left)
- Answer: 4.5 × 104
-
Question: Write 3,200,000 in standard form.
- Step 1: Move the decimal point: 3,200,000.0 → 3.200000 (moved 6 places left)
- Answer: 3.2 × 106
Small Numbers (Negative Powers)
- Move the decimal point to get a number between 1 and 10.
- Count how many places you moved the decimal point to the RIGHT.
- This count becomes the negative power of 10.
-
Question: Write 0.0007 in standard form.
- Step 1: Move the decimal point: 0.0007 → 7.0 (moved 4 places right)
- Answer: 7 × 10-4
-
Question: Write 0.00000256 in standard form.
- Step 1: Move the decimal point: 0.00000256 → 2.56 (moved 6 places right)
- Answer: 2.56 × 10-6
Converting from Standard Form to Ordinary Numbers
- Positive power: move the decimal point to the RIGHT.
- Negative power: move the decimal point to the LEFT.
-
Question: Write 6.7 × 105 as an ordinary number.
- Step 1: The power is +5, so move the decimal point 5 places RIGHT: 6.7 → 670000
- Answer: 670,000
-
Question: Write 3.8 × 10-3 as an ordinary number.
- Step 1: The power is −3, so move the decimal point 3 places LEFT: 3.8 → 0.0038
- Answer: 0.0038
Operations with Standard Form
Multiplication
(a × 10m) × (b × 10n) = (a × b) × 10m+n
Multiply the number parts and add the powers.
-
Question: Calculate (3 × 104) × (2 × 105).
- Step 1: Multiply the numbers: 3 × 2 = 6
- Step 2: Add the powers: 104+5 = 109
- Answer: 6 × 109
-
Question: Calculate (4 × 103) × (5 × 10-2).
- Step 1: Multiply the numbers: 4 × 5 = 20
- Step 2: Add the powers: 103+(-2) = 101, giving 20 × 101
- Step 3: 20 is not between 1 and 10, so adjust: 20 × 101 = 2.0 × 101+1
- Answer: 2 × 102
Division
(a × 10m) ÷ (b × 10n) = (a ÷ b) × 10m-n
Divide the number parts and subtract the powers.
-
Question: Calculate (8 × 107) ÷ (4 × 103).
- Step 1: Divide the numbers: 8 ÷ 4 = 2
- Step 2: Subtract the powers: 107-3 = 104
- Answer: 2 × 104
-
Question: Calculate (6 × 10-2) ÷ (3 × 104).
- Step 1: Divide the numbers: 6 ÷ 3 = 2
- Step 2: Subtract the powers: 10-2-4 = 10-6
- Answer: 2 × 10-6
Addition and Subtraction
To add or subtract numbers in standard form, the powers of 10 must first be made the same.
-
Question: Calculate (5 × 104) + (3 × 104).
- Step 1: The powers already match (both 104), so add the numbers directly: 5 + 3 = 8
- Answer: 8 × 104
-
Question: Calculate (7 × 105) − (2 × 104).
- Step 1: The powers differ, so rewrite one to match: 7 × 105 = 70 × 104
- Step 2: Subtract: (70 × 104) − (2 × 104) = 68 × 104
- Step 3: Adjust to standard form: 68 × 104 = 6.8 × 105
- Answer: 6.8 × 105
Powers of 10 - Quick Reference
| Power of 10 | Value | Name |
|---|---|---|
| 106 | 1,000,000 | Million |
| 105 | 100,000 | Hundred thousand |
| 104 | 10,000 | Ten thousand |
| 103 | 1,000 | Thousand |
| 102 | 100 | Hundred |
| 101 | 10 | Ten |
| 100 | 1 | One |
| 10-1 | 0.1 | Tenth |
| 10-2 | 0.01 | Hundredth |
| 10-3 | 0.001 | Thousandth |
| 10-4 | 0.0001 | Ten-thousandth |
| 10-6 | 0.000001 | Millionth |
Real-World Applications
| Quantity | Ordinary Form | Standard Form |
|---|---|---|
| Speed of light | 300,000,000 m/s | 3 × 108 m/s |
| Distance Earth to Sun | 150,000,000,000 m | 1.5 × 1011 m |
| Mass of Earth | 6,000,000,000,000,000,000,000,000 kg | 6 × 1024 kg |
| Diameter of atom | 0.0000000001 m | 1 × 10-10 m |
| Mass of electron | 0.000000000000000000000000000000911 kg | 9.11 × 10-31 kg |
| Thickness of paper | 0.0001 m | 1 × 10-4 m |
Exam Tips
Common Mistakes
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