👋 Hey there, future Mathematician! Welcome to Limits of Accuracy!
This chapter is all about understanding that in the real world, measurements are never absolutely perfect. Whether you measure the height of a table or the volume of water, there is always a tiny bit of error, or "inaccuracy."
Understanding Limits of Accuracy teaches you how to handle these imperfections by rounding numbers and calculating the possible range of a value. This is crucial for making sure your calculations are realistic and reliable!
🎯 What You Will Learn in This Chapter:
- How to round numbers using Decimal Places (DP) and Significant Figures (SF).
- How to perform quick estimations for complex calculations.
- How to find the Lower Bound (LB) and Upper Bound (UB) of a measurement.
- How to use these bounds to find the maximum and minimum values of combined calculations (like area or speed).
Section 1: Rounding – Making Numbers Tidy (C1.9.1 / E1.9.1)
Rounding is a fundamental skill. You need to be able to round a value to a specified number of Decimal Places (DP) or Significant Figures (SF).
1.1 Rounding to Decimal Places (DP)
This method focuses only on the digits after the decimal point.
Step-by-Step Guide:
- Identify the position of the last decimal place you need.
- Look at the digit immediately following it (the check digit).
- If the check digit is 5 or more (5, 6, 7, 8, 9), round up the last digit.
- If the check digit is less than 5 (0, 1, 2, 3, 4), keep the last digit the same.
Example: Round 4.7382 to 2 decimal places.
The second decimal place is 3. The check digit is 8. Since 8 is 5 or more, we round up.
Result: 4.74 (2 d.p.)
1.2 Rounding to Significant Figures (SF)
This method focuses on the most important, or "significant," digits in the entire number.
Rule: The counting starts from the first non-zero digit.
- Starting Digit: The first significant figure is the first non-zero digit you meet, moving from left to right.
- Leading Zeros: Zeros at the start of a number (like in 0.0051) are NOT significant.
- Trailing Zeros: Zeros between non-zero digits (like in 4007) or at the end of a whole number (like in 2500) ARE usually significant (depending on context, but generally count them unless they are placeholders after rounding).
Example 1: Round 50921 to 3 significant figures (s.f.).
The first significant figure is 5. The third is 9. The check digit is 2. Since 2 is less than 5, we keep 9 the same, and replace the following digits with zeros to maintain place value.
Result: 50900 (3 s.f.)
Example 2: Round 0.00476 to 1 significant figure.
The first non-zero digit is 4.
Result: 0.005 (1 s.f.)
⚠️ Common Mistake: Losing Place Value!
When rounding large numbers to s.f., remember to replace lost digits with zeros. Rounding 4567 to 2 s.f. is 4600, not 46. (46 is too small!)
Quick Review: Rounding
| Value | Round to | Result |
| 19.451 | 1 d.p. | 19.5 |
| 0.0307 | 2 s.f. | 0.031 |
| 999 | Nearest Ten | 1000 |
Section 2: Estimation (C1.9.2)
Estimation means finding a rough, sensible answer quickly. In IGCSE Maths, estimations are almost always done by rounding every number in the calculation to 1 significant figure first.
Why estimate? It helps you check if your detailed calculator answer is reasonable. If your estimate is 50, and your calculator says 5000, you know you made a mistake!
Step-by-Step: Estimating Calculations
- Round every single number in the expression to 1 significant figure.
- Perform the calculation using the rounded numbers.
Example: Estimate the value of \( \frac{41.3}{9.79 \times 0.765} \)
Step 1: Round each number to 1 s.f.
- \(41.3 \approx 40\)
- \(9.79 \approx 10\)
- \(0.765 \approx 0.8\)
Step 2: Calculate using rounded numbers.
Estimate \( = \frac{40}{10 \times 0.8} \)
Estimate \( = \frac{40}{8} \)
Estimate \( = 5 \)
The estimated value is 5.
Key Takeaway: When asked to 'estimate', round *first* (usually to 1 s.f.) and *then* calculate.
Section 3: Limits of Accuracy (Upper and Lower Bounds)
When a value has been rounded, we need to know the range of values it could have been before rounding. This range is defined by the Lower Bound (LB) and the Upper Bound (UB).
Imagine you measure a pencil as 15 cm to the nearest centimetre. It wasn't exactly 15 cm. It could have been 14.8 cm or 15.3 cm, but if it was 15.5 cm, you would have rounded up to 16 cm.
3.1 The Magic Rule of Bounds
To find the Upper and Lower Bounds, you need to know the degree of accuracy (the unit it was measured or rounded to). Let this unit be \(U\).
The maximum possible error is always half of the unit of accuracy: \( \frac{U}{2} \)
- Lower Bound (LB): Rounded Value \( - \frac{U}{2} \)
- Upper Bound (UB): Rounded Value \( + \frac{U}{2} \)
Important Note on the Upper Bound: The Upper Bound is the value that the number goes up to, but does not include. If the number reached the UB exactly, it would usually be rounded up to the next value. However, in IGCSE calculations, we often use the UB itself to ensure we cover the full range of possible values.
Example: Finding Bounds
A weight is measured as 3.5 kg, correct to one decimal place (1 d.p.).
1. Find the unit of accuracy (\(U\)):
The measurement is to 1 d.p. The unit is 0.1 kg.
2. Find half the unit (\( \frac{U}{2} \)):
\( \frac{0.1}{2} = 0.05 \) kg
3. Calculate the Bounds:
- Lower Bound (LB): \( 3.5 - 0.05 = \mathbf{3.45} \) kg
- Upper Bound (UB): \( 3.5 + 0.05 = \mathbf{3.55} \) kg
This means the actual weight \(W\) is in the range: \( 3.45 \le W < 3.55 \)
Did You Know? (Bounds for Significant Figures)
The rules for finding bounds work exactly the same way if the rounding was done using significant figures. You just need to identify the place value of the last significant figure.
Example: A distance is 600 m, correct to 1 significant figure.
- The 1st significant figure is 6. This digit is in the Hundreds place.
- The unit of accuracy (\(U\)) is the place value of the last significant digit, which is 100 m.
- Half the unit is \( \frac{100}{2} = 50 \) m.
- LB: \( 600 - 50 = \mathbf{550} \) m
- UB: \( 600 + 50 = \mathbf{650} \) m
Key Takeaway: Bounds create the range: \( LB \le \text{Actual Value} < UB \). Always use half the unit of measurement.
Section 4: Calculations with Upper and Lower Bounds
When you use measurements in a formula (like calculating area or speed), the error from the individual measurements combines. To find the maximum or minimum possible result, you must choose the appropriate bounds for *each* variable.
Let A and B be two measured quantities. We use LB(A), UB(A), LB(B), and UB(B).
4.1 Rules for Addition and Subtraction
These rules are straightforward: to get the maximum sum, you add the maximum possible parts. To get the minimum difference, you subtract the largest possible amount from the smallest possible amount.
Addition (\( A + B \))
- Maximum Result: UB(A) + UB(B)
- Minimum Result: LB(A) + LB(B)
Subtraction (\( A - B \))
- Maximum Result: UB(A) - LB(B) (Largest possible number minus the smallest possible number)
- Minimum Result: LB(A) - UB(B) (Smallest possible number minus the largest possible number)
Think of it this way: To make the result of subtraction as small as possible, you want the first number (A) to be tiny (LB) and the number you take away (B) to be huge (UB).
4.2 Rules for Multiplication and Division
The same logic applies: to maximise the result, you use the 'biggest' bounds in the numerator and the 'smallest' bounds in the denominator (and vice versa for the minimum result).
Multiplication (\( A \times B \))
- Maximum Result: UB(A) \(\times\) UB(B)
- Minimum Result: LB(A) \(\times\) LB(B)
Division (\( \frac{A}{B} \))
To remember this, use the trick LODI: Lower on top, Divide by Inverse (UB on bottom).
- Maximum Result: \(\frac{\text{UB(A)}}{\text{LB(B)}}\) (Largest numerator divided by smallest denominator)
- Minimum Result: \(\frac{\text{LB(A)}}{\text{UB(B)}}\) (Smallest numerator divided by largest denominator)
Worked Example: Speed
A car travels a Distance \(D = 150\) km (nearest 10 km) in a Time \(T = 2.0\) hours (nearest 0.1 hour).
Find the maximum possible average speed.
Recall: Speed \( = \frac{\text{Distance}}{\text{Time}} \).
Step 1: Find the bounds for D and T.
- Distance (D): Rounded to nearest 10. Half unit is 5.
LB(D) = 145 km, UB(D) = 155 km. - Time (T): Rounded to nearest 0.1. Half unit is 0.05.
LB(T) = 1.95 h, UB(T) = 2.05 h.
Step 2: Apply the maximum division rule.
Maximum Speed \( = \frac{\text{Maximum Distance}}{\text{Minimum Time}} \)
Maximum Speed \( = \frac{\text{UB(D)}}{\text{LB(T)}} \)
Maximum Speed \( = \frac{155}{1.95} \)
Maximum Speed \( \approx 79.487... \text{ km/h} \)
Step 3: Round the final answer appropriately (3 s.f. standard).
Maximum Speed \( = \mathbf{79.5} \) km/h (3 s.f.)
⚠️ CRITICAL ADVICE: DO NOT ROUND INTERMEDIATE STEPS!
Always use the exact bounds (3.45, 1.95, etc.) in your fraction/calculation. Only round your final answer to the required degree of accuracy (usually 3 s.f. or as specified in the question).
Quick Review: Bounds Calculations
- A + B (Max): UB + UB
- A - B (Max): UB - LB
- A \(\times\) B (Min): LB \(\times\) LB
- A \(\div\) B (Min): \(\frac{\text{LB}}{\text{UB}}\)
Section 5: Applying Accuracy in Context (C1.9.3)
The final part of accuracy often requires you to think about the real-world problem and choose a sensible level of accuracy for your final answer.
- If you calculate the number of people attending an event, you must round to the nearest whole number, as you cannot have 0.5 of a person.
- If you calculate the price of an item in dollars, you might round to 2 decimal places for cents (e.g., \$15.48).
If the question does not specify the required accuracy, the standard rules apply:
Standard Accuracy Rules (for non-exact answers):
Unless specified otherwise:
- Give non-exact numerical answers correct to 3 significant figures.
- Give angles in degrees correct to 1 decimal place.
- If your answer is exact (e.g., 5, 1/3, or \(\sqrt{4}\)), leave it exact!
Remember: Always show your full working before rounding the final result, especially when dealing with bounds problems!