Mathematics (Specification A) | Degree of accuracy

Welcome to Your Study Notes: Degree of Accuracy

Hello Mathematicians! This chapter, Degree of Accuracy, is crucial because in the real world, things are rarely perfectly exact. Think about measuring ingredients, timing a race, or checking the distance to a star – every measurement has some level of uncertainty.

Understanding accuracy helps us handle these imperfect numbers correctly. It builds on your knowledge from the "Numbers and the number system" section by defining the boundaries of numbers we work with. Don't worry if this seems tricky at first; we'll break down rounding and bounds into simple, easy-to-follow steps!

Chapter Overview: Key Skills You Will Master

• Rounding numbers accurately (Decimal Places and Significant Figures).
• Determining the Upper Bound and Lower Bound of a rounded number.
• Understanding Error Intervals.

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Section 1: The Foundation – Rounding Numbers

Rounding is a way of simplifying a number while keeping it close to its original value. We mainly focus on two methods: rounding to Decimal Places and rounding to Significant Figures.

1.1 Rounding to Decimal Places (DP)

When you round to a specific decimal place, you are focusing only on the digits after the decimal point.

The Rule of Rounding (The 5-Rule)

To round a number to a specific DP (or any place value):

1. Identify the digit in the required place value. This is your Target Digit.
2. Look immediately to the digit on its right (the Deciding Digit).
3. If the Deciding Digit is 5 or more (5, 6, 7, 8, 9), you round the Target Digit up by 1.
4. If the Deciding Digit is 4 or less (0, 1, 2, 3, 4), you keep the Target Digit the same.
5. All digits after the Target Digit are removed (if they are after the decimal point).

Example: Round 4.7382 to 2 decimal places (2 d.p.).

4.7382
The Target Digit is 3 (in the second decimal place).
The Deciding Digit is 8. Since 8 is 5 or more, we round 3 up to 4.
Result: 4.74

Quick Tip: If rounding causes a chain reaction (e.g., rounding 9.99 to 1 d.p.), treat it like standard addition. 9.99 rounded to 1 d.p. becomes 10.0.

1.2 Rounding to Significant Figures (SF)

Rounding to Significant Figures (SF) means determining how precise a number is, starting the count from the very first non-zero digit. This method is often used in science and engineering.

Identifying Significant Figures: The Rules

1. Rule 1: Non-Zero Digits Count. All digits that are 1, 2, 3, 4, 5, 6, 7, 8, or 9 are always significant.
2. Rule 2: Middle Zeros Count. Zeros trapped between two significant figures are also significant (e.g., in 5003, the zeros are significant).
3. Rule 3: Leading Zeros DO NOT Count. Zeros that come before the first non-zero digit are just place holders and are not significant (e.g., in 0.0075, the three zeros are NOT significant).
4. Rule 4: Trailing Zeros (after the decimal) COUNT. Zeros at the end of a number after a decimal point ARE significant (e.g., 2.500 has four significant figures).
5. Rule 5: Trailing Zeros (in a large whole number) MAY NOT Count. In a number like 500, we don't know if the zeros were measured exactly or just placeholders. If asked to round 500 to 1 s.f., it stays 500. If we round 478 to 1 s.f., it becomes 500 (here, the zeros are just placeholders).

Memory Aid: Imagine driving a car. You only start counting the significant digits once you hit the first real gear (the first non-zero number).

Step-by-Step SF Rounding

Example: Round 0.005831 to 3 significant figures (3 s.f.).

1. Find the Start: The first significant figure is 5. (Rule 3: Ignore the leading zeros).
2. Count the Places: Count 3 places starting from 5: 5, 8, 3. The Target Digit is 3.
3. Look Right: The Deciding Digit is 1.
4. Round: Since 1 is 4 or less, we keep the Target Digit (3) the same.
5. Finish: We write the result, keeping placeholders before the 5 if needed, but dropping numbers after the 3.
   Result: 0.00583

Example 2: Round 34,752 to 2 significant figures (2 s.f.).

1. Find the Start: 3 is the 1st s.f., 4 is the 2nd s.f. Target Digit is 4.
2. Look Right: The Deciding Digit is 7.
3. Round: 7 is 5 or more, so we round 4 up to 5.
4. Finish: Crucially, because this is a whole number, we must fill the remaining spaces with zeros to keep the place value correct.
   Result: 35,000 (The number must still be around thirty-four thousand!)

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Section 2: Error Intervals – Upper and Lower Bounds

Now we tackle the concept of Error Intervals. When someone tells you a distance is 10 metres to the nearest metre, they are giving you a rounded number. The actual distance could be slightly less or slightly more than 10.

The Upper Bound (UB) and Lower Bound (LB) define the range within which the actual, unrounded number must lie.

2.1 Finding the Maximum Possible Error (MPE)

The key to bounds is identifying the degree of accuracy the number has been rounded to (e.g., nearest 10, nearest 0.1, nearest whole number).

Step 1: Determine the Error Gap

The maximum possible difference between the rounded value and the true value is called the Maximum Possible Error (MPE).

MPE = \(\frac{\text{Degree of Accuracy}}{2}\)

Example: If a measurement is 15 cm to the nearest cm (degree of accuracy = 1 cm):

• MPE = \(1 \text{ cm} / 2 = 0.5 \text{ cm}\).

Example: If a measurement is 2.4 kg to the nearest 0.1 kg (degree of accuracy = 0.1 kg):

• MPE = \(0.1 \text{ kg} / 2 = 0.05 \text{ kg}\).

2.2 Calculating the Upper and Lower Bounds

Once you have the MPE, finding the bounds is straightforward:

Lower Bound (LB) = Rounded Value - MPE
Upper Bound (UB) = Rounded Value + MPE

Example: A length L is measured as 4.0 metres, correct to 1 decimal place. Find the bounds.

1. Identify Accuracy: Nearest 0.1 metre.
2. Calculate MPE: \(0.1 / 2 = 0.05\) metres.
3. Calculate Bounds:
   • LB = \(4.0 - 0.05 = 3.95\)
   • UB = \(4.0 + 0.05 = 4.05\)

The Analogy of the Fence: The Upper Bound (4.05) is the first number that would actually round up to the next rounded value (4.1). Everything up to, but not including, 4.05, rounds down to 4.0. Imagine a fence at 4.05. If you stand exactly on the fence line, you belong to the next field!

2.3 Defining the Error Interval

We use inequalities to write the range of possible values (the Error Interval). If \(x\) is the true value:

LB \(\le x < \) UB

Crucial Detail: The Lower Bound includes the equality sign (\(\le\)) because 3.95 would round up to 4.0. The Upper Bound uses only the less than sign (\(<\)) because 4.05 itself would round up to 4.1, meaning 4.05 is the starting point of the next interval, not included in the 4.0 interval.

For the example above (\(L = 4.0\)):
The Error Interval is:

\(3.95 \le L < 4.05\)

2.4 Bounds with Significant Figures

The process for finding bounds when rounding to Significant Figures is exactly the same, but determining the degree of accuracy can be harder.

Example: A mass \(M\) is 7000 g, correct to 1 significant figure. Find the error interval.

1. Find the significant figure place: The 1st s.f. is 7, which is in the thousands place.
2. Determine Accuracy: The rounding must have been to the nearest 1000. (Degree of Accuracy = 1000).
3. Calculate MPE: \(1000 / 2 = 500\).
4. Calculate Bounds:
   • LB = \(7000 - 500 = 6500\)
   • UB = \(7000 + 500 = 7500\)
5. Error Interval: \(6500 \le M < 7500\)

Did you know? Measurements are often only written with the number of significant figures justified by the accuracy of the measuring instrument. Writing 10.00 cm suggests a far more accurate measurement than just 10 cm!

Keep practicing these steps, and you will master accuracy in no time! Good luck!