Approximation & Errors: Your Friendly Guide to "Close Enough!"
Hey everyone! Ever told a friend you'll meet them in "about 10 minutes"? Or heard that a city has "around 2 million" people? That's approximation! In this chapter, we're going to learn the proper way to make numbers simpler and easier to work with. We'll also explore something called "errors". Don't worry, it doesn't mean you made a mistake! In math, an "error" is just the small difference between a measurement and the real, exact value.
This is a super useful skill in everyday life, from cooking and shopping to science experiments. So, let's get started!
Part 1: The Art of Approximation
So, What Are Approximate Values?
An approximate value is a number that's close to the exact value, but is simpler or more convenient to use. We often can't know the exact value, or we just don't need to.
For example, the exact population of a town might be 48,782 people. But it's much easier to say it's approximately 49,000.
The most common way to find an approximate value is by rounding.
Estimation Strategy 1: Rounding Off
This is the one you've probably seen before! It follows one simple rule.
The Golden Rule of Rounding: Look at the digit to the right of the place you are rounding to.
- If it's 5 or more (5, 6, 7, 8, 9), you round up (add 1 to the rounding digit).
- If it's 4 or less (4, 3, 2, 1, 0), it stays put (the rounding digit doesn't change).
After rounding, all the digits to the right of the rounding digit are dropped (if they are after a decimal point) or changed to zeros (if they are before a decimal point). Let's see it in action!
Rounding to a Certain Number of Decimal Places (d.p.)
This is used to make long decimal numbers shorter.
Example: Round 8.736 to 1 decimal place.
Find the digit in the first decimal place. It's 7. This is our rounding digit.
Look at the digit to its right. It's 3.
Is it 5 or more? No, it's 4 or less. So, the 7 stays put.
Drop all the digits after the 7.
Answer: 8.7 (correct to 1 d.p.)
Rounding to a Certain Place Value
This is used for whole numbers.
Example: Round 4,815 to the nearest hundred.
Find the digit in the hundreds place. It's 8.
Look at the digit to its right. It's 1.
Is it 5 or more? No. So, the 8 stays put.
Change all digits to the right of the 8 into zeros.
Answer: 4,800 (correct to the nearest hundred)
Rounding to a Certain Number of Significant Figures (s.f.)
Don't worry if this seems tricky at first! Significant figures are just the "important" digits in a number that give it meaning.
How to spot Significant Figures:
Rule 1: All non-zero digits (1-9) are ALWAYS significant. (e.g., in 128, there are 3 s.f.)
Rule 2: Zeros between non-zero digits are ALWAYS significant. (e.g., in 506, there are 3 s.f.)
Rule 3: Zeros at the beginning of a number are NEVER significant. (e.g., in 0.025, there are only 2 s.f., the 2 and the 5)
Rule 4: Zeros at the end of a number are significant ONLY if there is a decimal point. (e.g., 2.30 has 3 s.f., but 230 only has 2 s.f.)
Example: Round 0.07852 to 2 significant figures.
Find the first significant figure. It's 7 (Rule 3). The second one is 8. So, 8 is our rounding digit.
Look at the digit to the right of 8. It's 5.
Is it 5 or more? Yes! So we round up the 8 to a 9.
Drop the digits after our new 9.
Answer: 0.079 (correct to 2 s.f.)
Other Estimation Strategies: Rounding Up & Rounding Down
Sometimes, the "5 or more" rule doesn't make sense for a real-life situation. We have to use common sense!
Rounding Up
You round up when you need to be sure you have enough of something.
Example: 45 students are going on a trip. Each mini-bus can hold 10 students. How many buses do you need?
Calculation: 45 / 10 = 4.5 buses.
You can't hire half a bus! If you round down to 4 buses, some students get left behind. You must round up to 5 buses to fit everyone.
Rounding Down
You round down when you are limited by what you have, like money or materials.
Example: You have $30. A cinema ticket costs $12. How many tickets can you buy?
Calculation: 30 / 12 = 2.5 tickets.
You don't have enough money for 3 tickets, so you must round down. You can buy 2 tickets.
Key Takeaway for Part 1
Approximation helps us simplify numbers. Rounding off is the most common method. For real-life problems, sometimes we need to use logic to decide whether to round up (to have enough) or round down (based on a limit).
Part 2: Oops! The World of Measurement Errors
Why Do Errors Happen?
Imagine trying to measure the length of your desk with a ruler. Is it exactly 60 cm? Or is it 60.1 cm? Or maybe 60.08 cm? The truth is, no measurement is ever 100% perfect. This small uncertainty is what we call an error in measurement. It's not a mistake; it's just a natural part of measuring.
The BIGGEST Possible Error: Maximum Absolute Error
The size of the possible error depends on your measuring tool. If your ruler only shows centimetres, you can only measure "to the nearest centimetre".
The Maximum Absolute Error is the largest possible difference between the measured value and the actual value. It's easy to find!
$$ \text{Maximum Absolute Error} = \frac{\text{Smallest unit of measurement}}{2} $$
Example 1: A book's width is measured as 15 cm, correct to the nearest cm.
The smallest unit is 1 cm.
Maximum Absolute Error = 1 cm / 2 = 0.5 cm.
This means the actual width of the book is somewhere between (15 - 0.5) cm and (15 + 0.5) cm. So, it's between 14.5 cm and 15.5 cm.
Example 2: A bag of flour weighs 2.5 kg, correct to the nearest 0.1 kg.
The smallest unit is 0.1 kg.
Maximum Absolute Error = 0.1 kg / 2 = 0.05 kg.
Is the Error a Big Deal? Relative & Percentage Error
An error of 1 cm is a huge deal if you're measuring an ant, but it's tiny if you're measuring a football pitch! To understand how "big" an error is compared to the measurement, we use Relative and Percentage Error.
Relative Error
This compares the maximum error to the measured value.
$$ \text{Relative Error} = \frac{\text{Maximum Absolute Error}}{\text{Measured Value}} $$Percentage Error
This is just the relative error shown as a percentage, which is easier to understand.
$$ \text{Percentage Error} = \text{Relative Error} \times 100\% $$Or all in one step:
$$ \text{Percentage Error} = \frac{\text{Maximum Absolute Error}}{\text{Measured Value}} \times 100\% $$Quick Review: Error Formulas
1. Maximum Absolute Error = (Smallest Unit) / 2
2. Relative Error = (Max. Absolute Error) / (Measured Value)
3. Percentage Error = Relative Error x 100%
Part 3: Putting It All Together
Let's Solve a Problem!
You've learned all the concepts, now let's use them. It's easier than it looks!
Problem: The height of a door is measured as 200 cm, correct to the nearest cm. Find the percentage error of this measurement.
Step-by-step Solution:
Step 1: Find the Maximum Absolute Error.
The measurement is to the nearest cm, so the smallest unit is 1 cm.
Maximum Absolute Error = 1 cm / 2 = 0.5 cm.
Step 2: Find the Percentage Error.
We use the formula: $$ \text{Percentage Error} = \frac{\text{Maximum Absolute Error}}{\text{Measured Value}} \times 100\% $$
The Measured Value is 200 cm.
Percentage Error = $$ \frac{0.5}{200} \times 100\% $$
Percentage Error = $$ 0.0025 \times 100\% $$
Percentage Error = 0.25%
Answer: The percentage error in the measurement is 0.25%. See? You've got this!
Challenge Yourself!
Now it's your turn to think like a mathematician. This is about judging what makes sense in the real world.
Question: You are buying fence panels to put around a garden. You calculate that you need exactly 22.3 metres of fencing. The shop only sells panels that are 1 metre long.
Should you round down and buy 22 panels, or round up and buy 23 panels? Why?
Thinking it through: If you round down, you won't have enough fencing, and there will be a gap! You must round up to make sure the entire garden is fenced in. This is a great example of choosing the right estimation strategy for the situation.