CSEC Essential Skill: Scientific notation (also called standard form) is a mathematical tool you MUST master for CSEC Physics. It allows you to handle extremely large numbers (like astronomical distances) and extremely small numbers (like atomic sizes) efficiently and accurately. You’ll use it in calculations, when recording measurements, and when expressing final answers.
What is Scientific Notation (Standard Form)?
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It follows this format:
Where:
- A is a number between 1 and 10 (1 ≤ A < 10)
- × 10n means “multiplied by 10 raised to the power n”
- n is an integer (positive, negative, or zero)
- Speed of light: 300,000,000 m/s = 3.00 × 108 m/s
- Charge of an electron: 0.00000000000000000016 C = 1.6 × 10-19 C
- Diameter of a hydrogen atom: 0.0000000001 m = 1 × 10-10 m
- Earth’s mass: 5,970,000,000,000,000,000,000,000 kg = 5.97 × 1024 kg
Why CSEC Requires This: In physics exams, you’ll often work with measurements like “the wavelength of red light is 7 × 10⁻⁷ m” or “the charge is 1.6 × 10⁻¹⁹ C”. You need to be comfortable reading, writing, and calculating with numbers in this format.
Converting TO Scientific Notation: Step-by-Step
- If you moved the decimal left (for large numbers), n is positive
- If you moved the decimal right (for small numbers), n is negative
| Ordinary Number | Scientific Notation | Explanation |
|---|---|---|
| 4,200 | 4.2 × 103 | Decimal moved 3 places left → exponent = +3 |
| 0.00075 | 7.5 × 10-4 | Decimal moved 4 places right → exponent = -4 |
| 602,000,000,000,000,000,000,000 | 6.02 × 1023 | Avogadro’s number (chemistry/physics) |
| 0.000000001 | 1 × 10-9 | 1 nanometre (unit of wavelength) |
Converting FROM Scientific Notation
- 3.0 × 102 = 300 (decimal moved 2 places right)
- 5.6 × 10-3 = 0.0056 (decimal moved 3 places left)
- 1.67 × 10-27 = 0.00000000000000000000000000167 kg (mass of proton)
- 9.11 × 10-31 = 0.000000000000000000000000000000911 kg (mass of electron)
Physics Applications in CSEC
Why Scientific Notation is Essential in Physics
Physics deals with quantities that span an enormous range of magnitudes:
| Physics Quantity | Typical Value | Scientific Notation |
|---|---|---|
| Distance to Sun | 149,600,000,000 m | 1.496 × 1011 m |
| Wavelength of red light | 0.0000007 m | 7 × 10-7 m |
| Electron charge | 0.00000000000000000016 C | 1.6 × 10-19 C |
| Planck’s constant | 0.000000000000000000000000006626 J·s | 6.626 × 10-34 J·s |
| Atmospheric pressure | 101,300 Pa | 1.013 × 105 Pa |
⚠️ Common CSEC Mistakes to Avoid
- Incorrect A value: Writing 12.5 × 103 instead of 1.25 × 104 (A must be between 1 and 10)
- Wrong exponent sign: Confusing when to use positive vs negative exponents
- Forgetting units: Always include units after scientific notation: 3.00 × 108 m/s not just 3.00 × 108
- Calculation errors: Misplacing decimal points when multiplying or dividing numbers in scientific notation
- Significant figures: Not maintaining correct significant figures when converting to/from scientific notation
Calculations with Scientific Notation
Multiplication
Multiply the A values and ADD the exponents:
(A × 10m) × (B × 10n) = (A × B) × 10m+n
Calculate: (3.0 × 108) × (2.0 × 103)
Solution: Multiply 3.0 × 2.0 = 6.0, then add exponents: 8 + 3 = 11
Answer: 6.0 × 1011
Division
Divide the A values and SUBTRACT the exponents:
(A × 10m) ÷ (B × 10n) = (A ÷ B) × 10m-n
Calculate: (6.0 × 109) ÷ (2.0 × 103)
Solution: Divide 6.0 ÷ 2.0 = 3.0, then subtract exponents: 9 – 3 = 6
Answer: 3.0 × 106
Addition and Subtraction
First make the exponents the same, then add/subtract the A values:
Calculate: (4.2 × 103) + (1.5 × 102)
Solution: Convert to same exponent: 4.2 × 103 = 42 × 102
Now add: 42 × 102 + 1.5 × 102 = 43.5 × 102
Convert back: 4.35 × 103
Calculator Tip: Most scientific calculators have an “EXP” or “EE” button for entering numbers in scientific notation. For 3 × 108, you would press: 3 EXP 8 or 3 EE 8. Learn to use this feature for CSEC exam calculations!
CSEC Exam Practice
Explanation: Move the decimal point 5 places to the right to get 4.2. Since we moved right (making the number larger), the exponent is negative: -5.
Explanation: Move the decimal point 8 places to the left to get 3.00. Since we moved left (making the number smaller), the exponent is positive: +8. Note we keep three significant figures (3.00).
Explanation: Multiply the coefficients: 6.0 × 2.0 = 12.0 = 1.2 × 10¹. Add the exponents: 6 + (-3) = 3. So we have (1.2 × 10¹) × 10³ = 1.2 × 10⁴.
Explanation:
Total charge per second = (1.6 × 10⁻¹⁹ C/electron) × (2.0 × 10¹⁸ electrons/second)
= (1.6 × 2.0) × (10⁻¹⁹ × 10¹⁸)
= 3.2 × 10⁻¹⁹⁺¹⁸
= 3.2 × 10⁻¹ = 0.32 A
A) 12.5 × 10³ m
B) 0.75 × 10⁻⁵ s
C) 5.60 × 10² kg
D) 100 × 10⁻³ A
Explanation: In proper scientific notation, the coefficient (number before × 10ⁿ) must be between 1 and 10. Options A, B, and D have coefficients outside this range (12.5, 0.75, 100 respectively). Only option C has coefficient 5.60, which is between 1 and 10.
Explanation: The negative exponent (-4) means we move the decimal point 4 places to the left. Starting with 2.5, moving left 4 places gives 0.00025.
💡 Final CSEC Strategy: When you see very large or very small numbers in a physics problem, immediately convert them to scientific notation. This makes calculations easier and reduces errors. Always check that your final answer is in proper scientific notation (A between 1 and 10) unless the question specifies otherwise.
Quick Reference Guide
- Format: A × 10n where 1 ≤ A < 10
- Large numbers: Positive exponent (e.g., 106 for millions)
- Small numbers: Negative exponent (e.g., 10-9 for nanoseconds)
- Multiplication: Multiply coefficients, add exponents
- Division: Divide coefficients, subtract exponents
- +/- Operations: Make exponents the same first
- CSEC Requirement: Use for all extreme values, maintain significant figures