IB MYP 4-5 Maths- Standard form (scientific notation) - Study Notes - New Syllabus
IB MYP 4-5 Maths- Standard form (scientific notation) – Study Notes
Standard
- Standard form (scientific notation)
Extended
- Types of Numbers
IB MYP 4-5 Maths- Standard form (scientific notation) – Study Notes – All topics
Absolute Values
Standard Form (Scientific Notation)
Standard form (also called scientific notation) is a way of writing very large or very small numbers using powers of 10.
\( a \times 10^n \), where:
- \( 1 \leq a < 10 \)
- \( n \) is an integer (positive or negative)
This form is especially useful in science, engineering, and large-scale calculations.
Examples:
How to Convert a Number to Standard Form:
- Move the decimal point to make the number between 1 and 10.
- Count how many places you moved the decimal:
- Move left → positive power of 10
- Move right → negative power of 10
Example:
Write 452000 in standard form.
▶️ Answer/Explanation
Step 1: Move decimal after the first digit: \( 4.52 \)
Step 2: Count places moved: 5 places left → power of 5
\(\boxed{4.52 \times 10^5}\)
Example:
Write 0.00081 in standard form.
▶️ Answer/Explanation
Step 1: Move decimal to make number between 1 and 10: \( 8.1 \)
Step 2: Count places moved: 4 places right → negative power
\(\boxed{8.1 \times 10^{-4}}\)
Multiplying and Dividing in Standard Form
Rule:
- Multiply/divide the decimal parts
- Add/subtract the powers of 10
Example:
Calculate \( (3 \times 10^4) \times (2 \times 10^6) \)
▶️ Answer/Explanation
Step 1: Multiply decimal parts: \( 3 \times 2 = 6 \)
Step 2: Add powers of 10: \( 10^4 \cdot 10^6 = 10^{10} \)
\(\boxed{6 \times 10^{10}}\)
Example:
Calculate \( \frac{6 \times 10^{-3}}{2 \times 10^2} \)
▶️ Answer/Explanation
Step 1: Divide decimal parts: \( 6 \div 2 = 3 \)
Step 2: Subtract powers: \( 10^{-3} \div 10^2 = 10^{-5} \)
Final Answer:
\(\boxed{3 \times 10^{-5}}\)
Example :
The mass of the Earth is approximately \( 5.97 \times 10^{24} \) kg and the mass of the Moon is approximately \( 7.35 \times 10^{22} \) kg.
Calculate how many times heavier the Earth is than the Moon. Give your answer in standard form.
▶️ Answer/Explanation
Write the ratio of Earth to Moon
\( \frac{5.97 \times 10^{24}}{7.35 \times 10^{22}} \)
Divide the decimal parts
\( 5.97 \div 7.35 \approx 0.812 \)
Subtract the powers of 10
\( 10^{24} \div 10^{22} = 10^{2} \)
Combine
\( 0.812 \times 10^2 = 8.12 \times 10^1 \)
Significant Figures (sig. figs)
Significant figures are the digits in a number that carry meaning and contribute to its precision. They are used when rounding or reporting measurements.
Rules for Counting Significant Figures:
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
e.g. \( 104 \) has 3 significant figures. - Leading zeros (before non-zero digits) are not significant.
e.g. \( 0.0025 \) has 2 significant figures. - Trailing zeros (after decimal) are significant.
e.g. \( 3.400 \) has 4 significant figures.
Examples:
- \( 230.45 \) → 5 significant figures
- \( 0.0060 \) → 2 significant figures
- \( 900 \) (no decimal) → 1 or 2 sig figs (ambiguous unless stated)
- \( 9.00 \) → 3 significant figures
Rounding to Significant Figures:
To round a number to a certain number of significant figures:
- Count from the first non-zero digit.
- Use rounding rules (if next digit is 5 or more, round up).
Example:
Round \( 6.437 \) to 2 significant figures.
▶️ Answer/Explanation
The first two digits are \( 6.4 \), next digit is 3 → round down.
\(\boxed{6.4}\)
Example:
Round \( 0.009786 \) to 2 significant figures.
▶️ Answer/Explanation
Ignore leading zeros. First two significant digits are 9 and 7. Third digit is 8 → round up.
\(\boxed{0.0098}\)
Example:
A bottle contains 5.678 litres of water. A measuring cylinder can only show values to 2 significant figures.
What volume should be recorded from the measuring cylinder?
▶️ Answer/Explanation
Round \( 5.678 \) to 2 significant figures. First two digits: \( 5.6 \), next digit is 7 → round up.
\(\boxed{5.7 \, \text{litres}}\)
Important Note: Using Scientific Notation on Calculators
Calculators often display numbers in scientific notation using the letter E, which stands for “Exponent of 10”. For example:
- \( 2.95 \times 10^8 \) may appear as 2.95E8 on your calculator.
Most calculators have a button for entering scientific notation. This button may be labeled as:
- EXP (common on scientific calculators, as shown below)
- Or written as \( \times 10^x \)
Make sure you’re familiar with how your calculator enters and displays scientific notation.
Example: To enter \( 1.5 \times 10^9 \) on your calculator:
- You would press:
Always check your calculator’s manual if the notation looks different or if you’re unsure about the syntax.