Understanding Accuracy in Maths
Accuracy in mathematics is crucial when dealing with measurements, approximations, and rounded values. It's about understanding how much a number has been rounded, and how to express this rounding in terms of limits and bounds. Accuracy helps to indicate the range of possible values that a rounded number could represent.
Here, we’ll explore important concepts such as limits of accuracy, upper and lower bounds, and how to apply these in calculations.
What Are Limits of Accuracy?
Limits of accuracy describe the range of values that a rounded number could have. When we round a number, it’s important to recognise the degree of accuracy, such as rounding to a certain number of decimal places or significant figures. The limits of accuracy are given using inequality symbols, and we express these using lower and upper bounds.
How to Work Out Limits of Accuracy
To find the limits of accuracy for a rounded number:
- Identify the degree of accuracy to which the number has been rounded. This could be based on decimal places, significant figures, or place value (such as the nearest ten, nearest hundred, or nearest tenth).
- Halve the degree of accuracy to determine how much the number can vary by:
- Upper limit (maximum): Add half of the degree of accuracy to the rounded number.
- Lower limit (minimum): Subtract half of the degree of accuracy from the rounded number.
Example: Rounded to the nearest whole number
If a number is rounded to the nearest whole number (say, 4.2), the degree of accuracy is 0.5 (half of 1). Therefore:
- Upper limit: 4.2 + 0.5 = 4.7
- Lower limit: 4.2 - 0.5 = 3.7
Thus, the number 4.2 could represent any value in the range 3.7 ≤ number < 4.7.
Truncation and Limits of Accuracy
When a number is truncated (i.e., part of it is discarded without rounding), the calculation of limits works slightly differently:
- Upper limit (maximum): Increase the last digit of the number by 1 (for decimal numbers, or the last non-zero digit for whole numbers).
- Lower limit (minimum): The truncated number itself.
For example, if 3.56 is truncated to 3.5, the limits would be:
- Upper limit: Increase the last digit (3.5 becomes 3.6).
- Lower limit: The original truncated number (3.5).
Thus, the error interval for truncation would be 3.5 ≤ number < 3.6.
What Are Upper and Lower Bounds?
In mathematics, the lower bound is the smallest possible value that a rounded or truncated number could represent, while the upper bound is the largest possible value. When a number is rounded, it could have values ranging between the lower and upper bounds.
How to Find Upper and Lower Bounds
To determine the upper and lower bounds of a number:
- Identify the degree of accuracy (place value) to which the number has been rounded (e.g., nearest tenth, nearest hundredth).
- Halve the degree of accuracy:
- Add this value to the original number to find the upper bound (maximum).
- Subtract this value from the original number to find the lower bound (minimum).
Example: Rounded to 1 decimal place
If 5.67 is rounded to 5.7 (rounded to 1 decimal place), the degree of accuracy is 0.05. Therefore:
- Upper bound: 5.7 + 0.05 = 5.75
- Lower bound: 5.7 - 0.05 = 5.65
So, the error interval for the number 5.7 is 5.65 ≤ number < 5.75.
Upper and Lower Bounds for Truncated Numbers
For a truncated number:
- Upper bound: Increase the last digit of the number (if it's a decimal) or the last non-zero digit (if it's a whole number).
- Lower bound: Use the truncated number itself.
For example, if 8.15 is truncated to 8.1:
- Upper bound: Increase the last digit (8.1 becomes 8.2).
- Lower bound: The truncated number itself (8.1).
The error interval is 8.1 ≤ number < 8.2.
How to Apply and Interpret Upper and Lower Bounds
When working with upper and lower bounds, it’s important to understand how they relate to calculations:
- The lower bound is the smallest possible value a number could take.
- The upper bound is the largest possible value the number could represent.
To apply upper and lower bounds in calculations involving two numbers:
- When adding or subtracting, the error intervals are combined by adding or subtracting the corresponding upper and lower bounds.
- When multiplying or dividing, you must use the upper bound of one number and the lower bound of the other to ensure the result is accurate.
Key Symbols
- ≤ (less than or equal to): Used to show the lower bound.
- < (less than): Used to show the upper bound.
To write the error interval for a rounded or truncated number, we use the notation:
Lower bound ≤ number < Upper bound
This means that the number can take any value from the lower bound, up to but not including the upper bound.
By understanding these concepts, you will improve your accuracy in maths and how to handle rounded or truncated numbers in calculations. This knowledge is essential for achieving correct results in both exams and real-life applications.
