Problem 40 - Technical Mathematics with Calculus

Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$\sqrt{\frac{653}{601}}$$

Step-by-Step Solution

Answer

The result of \(\sqrt{\frac{653}{601}}\) to three significant figures is approximately 1.04.

Step 1: Understand the Precision

When working with approximate numbers and square roots, we must pay attention to significant figures. The number 653 has three significant figures, and 601 also has three significant figures. The result of the operation should also be reported with three significant figures.

Step 2: Perform the Division

First, divide 653 by 601. The more precise the calculation at this stage, the fewer rounding errors will propagate to the final result. Using a calculator, we find that 653 divided by 601 is approximately equal to 1.086523. However, we should keep more digits through intermediate calculations to minimize rounding errors.

Step 3: Take the Square Root

Take the square root of the result from Step 2. The square root of 1.086523, calculated to several decimal places for accuracy, is approximately 1.04238. This is not yet rounded to the proper number of significant figures.

Step 4: Final Rounding

Round the result from Step 3 to three significant figures, as this was the least number of significant figures in the given numbers. The value 1.04238 rounded to three significant figures is 1.04. Therefore, the final answer should be reported with this precision.

Key Concepts

Understanding Significant Figures

Grasping the concept of significant figures is crucial in mathematics, especially when dealing with approximate numbers. Significant figures are the digits in a number that are known with certainty, plus one digit that is an estimate or uncertain. It's a way to express the precision of a number.

For instance, in the number 653, all three digits are significant because they are all known with certainty. The same goes for the number 601. When performing operations with these numbers, it's essential to maintain the level of precision given by the smallest number of significant figures. This is why, after the calculation of \( \sqrt{\frac{653}{601}} \) is performed, the final result should retain that level of precision. It's a balancing act between maintaining the integrity of the data and providing a practical and usable figure.

Square Root Calculations and Precision

The calculation of square roots is a common operation that requires careful consideration of precision. When working with square roots, every step from start to finish can introduce rounding errors, which is why precision is key.

Continuing with our example, once we divide 653 by 601, we proceed to determine the square root of the quotient. Using a calculator, we strive for accuracy by calculating to several decimal places which in this case yields approximately 1.04238.

However, it's important to note that this number is not the final answer since we yet have to round it to maintain the correct number of significant figures. This intermediate step, although not the end result, plays a vital role in achieving a precise and accurate final value.

Mitigating Rounding Errors

Rounding errors are discrepancies that result from altering a number to have fewer digits. They're practically inevitable when dealing with numbers that have a long string of digits after the decimal point, but we can minimize their impact with careful calculations.

In the process of performing mathematical operations with approximate numbers, it’s best practice to keep more digits than needed through the intermediate steps. This way, when we finally round the number to its appropriate significant figures, the compounding effect of rounding errors is reduced.

For the problem of \( \sqrt{\frac{653}{601}} \) we calculated the square root to several decimal places before rounding down to three significant figures. By doing so, the final answer of 1.04 is as close to the true value as possible, given the limitations of significant figures.