IB Mathematics AI – Topic 1

Decimal Places (d.p.):

The number of digits after the decimal point.

Rounding Rules:

1. Identify the digit at the required decimal place
2. Look at the digit immediately to the right (the deciding digit)
3. If deciding digit ≥ 5: Round up (increase the required digit by 1)
4. If deciding digit < 5: Round down (keep the required digit the same)
5. Remove all digits to the right (after decimal point)

Examples:

• 3.14159 rounded to 2 d.p. = 3.14 (deciding digit is 1 < 5)
• 3.14159 rounded to 3 d.p. = 3.142 (deciding digit is 5 ≥ 5)
• 7.8965 rounded to 1 d.p. = 7.9 (deciding digit is 9 ≥ 5)

⚠️ Common Pitfalls & Tips:

• Don't round in stages – always round from the original number
• When rounding 2.995 to 2 d.p., get 3.00, not 2.99
• Keep trailing zeros when specified: 5.00 (2 d.p.) is different from 5
• IB exams typically ask for 3 significant figures or specific decimal places
• Read question carefully – "correct to 2 d.p." means show 2 decimal places

Rules for Identifying Significant Figures:

1. All non-zero digits are significant:

567 has 3 s.f.

2. Zeros between non-zero digits are significant:

5007 has 4 s.f.

40.08 has 4 s.f.

3. Leading zeros (before first non-zero digit) are NOT significant:

0.0045 has 2 s.f. (4 and 5)

0.000702 has 3 s.f. (7, 0, 2)

4. Trailing zeros after decimal point ARE significant:

5.00 has 3 s.f.

0.0450 has 3 s.f.

5. Trailing zeros in whole numbers without decimal are ambiguous:

5000 could have 1, 2, 3, or 4 s.f. (use scientific notation for clarity)

⚠️ Common Pitfalls & Tips:

• Most common mistake: Forgetting that leading zeros are NOT significant
• 0.0045 has 2 s.f., NOT 4 s.f.
• When rounding large numbers, use zeros as placeholders: 45,678 → 46,000 (2 s.f.)
• Scientific notation is clearer for trailing zeros
• IB default: 3 significant figures unless otherwise stated

Solution:

(a) Significant figures in 0.0045678:

Leading zeros (0.00) are NOT significant.

Count starts from first non-zero digit: 4, 5, 6, 7, 8

Answer: 5 significant figures

(b) Round to 2 s.f.:

First 2 significant figures are: 4 and 5

Next digit is 6 (≥ 5), so round up

Answer: 0.0046

(c) Round to 3 d.p.:

Fourth decimal place is 5 (≥ 5), so round up

Answer: 0.005

Standard Form:

\[ a \times 10^k \quad \text{where } 1 \leq a < 10 \text{ and } k \in \mathbb{Z} \]

Examples:

• 5,600 = \(5.6 \times 10^3\)
• 0.0078 = \(7.8 \times 10^{-3}\)
• 345,000,000 = \(3.45 \times 10^8\)

Operations:

Multiplication: \((a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}\)

Division: \(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)

⚠️ Common Pitfalls & Tips:

• The coefficient must be between 1 and 10
• \(45 \times 10^3\) is NOT standard form; should be \(4.5 \times 10^4\)
• GDC shows as 5.6E3 for \(5.6 \times 10^3\)

Formula:

\[ \text{Percentage Error} = \frac{|\text{Approximate Value} - \text{Exact Value}|}{|\text{Exact Value}|} \times 100\% \]

Alternative notation:

\[ \% \text{ Error} = \frac{|v_A - v_E|}{|v_E|} \times 100\% \]

Key Points:

• Always use absolute values (no negative errors)
• Exact value goes in the denominator
• Result is expressed as a percentage
• Smaller percentage error = more accurate measurement

⚠️ Common Pitfalls & Tips:

• Don't forget absolute value signs
• Exact value (not approximate) goes in denominator
• Don't forget to multiply by 100 for percentage
• Round final answer appropriately (usually 3 s.f.)

Solution:

(a) Calculate percentage error:

Approximate (measured) value = 8.2 m

Exact (actual) value = 8.5 m

Using the formula:

\[ \% \text{ Error} = \frac{|8.2 - 8.5|}{|8.5|} \times 100\% \]

\[ = \frac{|-0.3|}{8.5} \times 100\% = \frac{0.3}{8.5} \times 100\% \]

\[ = 0.0353 \times 100\% = 3.53\% \]

Answer: 3.53% (3 s.f.)

(b) Percentage error in drawing:

The percentage error remains the same regardless of scale, as both measurements would be scaled proportionally.

Answer: 3.53%

Definition: Bounds represent the range of possible original values that would round to give a specific number.

For a measurement to n decimal places:

Subtract/add half of the place value of the last digit

Examples:

• 7.2 (1 d.p.): Bounds = 7.15 ≤ x < 7.25
• 34 (2 s.f.): Bounds = 33.5 ≤ x < 34.5

Solution:

Find bounds:

Length: 12.45 ≤ L < 12.55

Width: 8.25 ≤ W < 8.35

Maximum area:

Use upper bounds for both dimensions

\[ \text{Max Area} = 12.55 \times 8.35 = 104.7925 \text{ cm}^2 \]

Answer: 104.8 cm² (4 s.f.)

Significant Figures

• All non-zero digits count
• Leading zeros DON'T count
• Trailing zeros after decimal DO count

Scientific Notation

• \(a \times 10^k\) where \(1 \leq a < 10\)
• Big numbers: positive k
• Small numbers: negative k

Percentage Error

• \(\frac{|v_A - v_E|}{|v_E|} \times 100\%\)
• Always use absolute values
• Exact value in denominator

Bounds

• LB ≤ value < UB
• Add ±½ last digit value
• Max area: UB × UB