Question

To 4 d.p. a number is \(9.9190 .\) State (a) the maximum possible value, (b) the minimum possible value of the original number.

Short answer

Answer: The maximum possible value of the original number is 9.91905, and the minimum possible value is 9.91895.

Step-by-step solution

Determine the rounding error

Since the number is given to 4 decimal places, the maximum possible error due to rounding is half of the value at the 5th decimal place. Therefore, the maximum error due to rounding is \(\frac{0.0001}{2} = 0.00005.\)

Find the maximum possible value

To find the maximum possible value of the original number, we need to add the rounding error to the given number. Maximum value = 9.9190 + 0.00005 = 9.91905

Find the minimum possible value

To find the minimum possible value of the original number, we need to subtract the rounding error from the given number. Minimum value = 9.9190 - 0.00005 = 9.91895

Answer

(a) The maximum possible value of the original number is 9.91905. (b) The minimum possible value of the original number is 9.91895.

Key concepts

Understanding Decimal Places

When we talk about decimal places, we're referring to the number of digits that appear after the decimal point in a number. For example, in the number 9.9190, there are four decimal places. This level of detail indicates how precisely a number is expressed. Decimal places are crucial when rounding numbers because they determine the accuracy and precision of the value being represented.

Rounding to a certain number of decimal places involves approximating a number to reduce its complexity while maintaining a level of precision. For instance, if you're asked to round the number 9.919045 to four decimal places, it results in 9.9190. The number was shortened, but it still closely represents the original value.

Understanding and effectively using decimal places helps in accurately communicating numerical information, especially in calculations and measurements.

Calculating Maximum Value with Rounding Errors

Determining the maximum value involves considering the rounding error which may affect a number. Whenever a number is rounded, there may be slight adjustments to the number's true value.

Imagine you have a number rounded to 4 decimal places as 9.9190. To calculate the maximum possible value of the original number, you add the maximum rounding error to the rounded value. This is because the true number could be slightly higher than the rounded version to the nearest decimal place.

Here's how it works:

• Determine the rounding error: find half the value of the next decimal place (the 5th decimal in this case), which is 0.00005.
• Add this rounding error to the rounded number: 9.9190 + 0.00005 = 9.91905.

This potential "bump" up due to the rounding error provides the upper boundary of what the original value could have been before rounding. This process ensures you understand the flexibility around the possibly rounded number.

Deciphering Minimum Value in Rounding Contexts

Just like the maximum value, the minimum value is impacted by rounding. To find the minimum possible value, we consider that the original number could have been slightly lower than what is visible after rounding.

Minimum value is found by subtracting the rounding error from the given rounded number. The true original value could potentially be lower by this error margin, indicating the bottom range of where the number could sit before rounding.

For instance, with the rounded number 9.9190:

• Calculate the rounding error: again, it's 0.00005 for the 5th decimal place.
• Subtract this error from the rounded number: 9.9190 - 0.00005 = 9.91895.

This calculation shows that the original number might have been this low, forming the lower boundary of the possible range of true values. Understanding the minimum value ensures holistic comprehension of the rounding implications.

Exploring Mathematical Precision

Mathematical precision refers to how exact a value is represented in mathematics. Precision is critical in many areas, like sciences, engineering, and finance, where accuracy can greatly impact results and decisions.

Precision in mathematics can be influenced by where and how numbers are rounded. Rounding may be inevitable in calculations involving long decimals or requiring succinct representation, but it's essential to keep track of potential errors.

• Precision ensures every digit in a number is as meaningful as necessary, providing clarity and accuracy.
• Greater precision often involves using more decimal places, giving a closer approximation to the true value of a number.

In contrast, reducing the number of decimal places can introduce rounding errors, as seen in the exercise where a number rounded to four decimal places had potential maximum and minimum values due to such errors.

Therefore, mathematical precision not only affects how numbers are presented but also how results can vary based on these slight adjustments. Understanding this concept helps you accurately interpret and work with numbers in practical applications.