Question

Evaluate the expression in two ways. (a) Calculate entirely on your calculator using appropriate parentheses, and then round the answer to two decimal places. (b) Round both the numerator and the denominator to two decimal places before dividing, and then round the final answer to two decimal places. Does the second method introduce an additional roundoff error? $$ \frac{333+\frac{1.98}{0.74}}{4+\frac{6.25}{3.15}} $$

Short answer

The short answer to this question will depend on the calculator used for calculation. After performing the steps described, one can compare the results from both methods to check if an additional roundoff error is introduced by the second method.

Step-by-step solution

Direct Calculator Input and Calculation

Input the expression directly into a calculator as follows: \( \frac{333+\frac{1.98}{0.74}}{4+\frac{6.25}{3.15}} \). Make sure to use parentheses to denote correct order of operations. Then calculate the result and round to two decimal places.

Round Values and Perform Division Operations

Round the values 1.98 and 0.74 to two decimal places before dividing, and do the same for 6.25 and 3.15. You should get the following results: \( \frac{1.98}{0.74} \) rounded to two decimal places is 2.68 and \( \frac{6.25}{3.15} \) rounded to two decimal places is 1.98.

Calculate Expression with Rounded Values

Using the rounded results from Step 2 in place of the original division operations, calculate the new expression: \( \frac{333+2.68}{4+1.98} \). Compute this result and round to two decimal places.

Compare Results

Compare the results from Steps 1 and 3. If the results are different, this indicates that the second method introduces an additional roundoff error.

Key concepts

Numerical Approximation

Numerical approximation is the process of finding a number close to the exact mathematical value of an expression, which may not be possible (or practical) to calculate with complete precision. This is particularly significant in computations involving irrational numbers or complex expressions where we seek a value to a certain degree of accuracy.

One common method of numerical approximation involves rounding numbers to a specified number of decimal places during intermediate steps of a calculation. This is evident in the example provided, where intermediate values like the result of \( \frac{1.98}{0.74} \) after calculating, would be approximated before further computations. It's essential to recognize that every act of rounding off can introduce errors in the final answer. As the exercise shows, the degree and impact of these errors depend on when and how the approximation is applied during the sequence of calculations.

Order of Operations

The order of operations is a fundamental convention in mathematics that dictates the sequence in which multiple operations should be carried out to achieve the correct result. Commonly remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it ensures that calculations are standardized and unambiguous.

In the illustrative example, the importance of order of operations becomes clear when inputting the expression into a calculator. The use of parentheses ensures that the division within numerator and denominator is performed before their addition to 333 and 4, respectively. If you neglect this order, the results can drastically differ. Additionally, when approximating values, respecting the order of operations remains critical. Errors incurred by rounding should be minimized by following the correct sequence, as failing to do so might magnify these inaccuracies.

Calculator Computations

Calculator computations refer to the numerical operations performed using a calculator, which may include functions like addition, subtraction, multiplication, division, and more advanced calculations. Calculators follow the order of operations rigorously but they are also subject to internal rounding based on their built-in algorithms and display settings.

Inputting an expression directly into a calculator and using appropriate parentheses will generally give a more accurate result, as the calculator computes to its maximum internal precision before rounding for the display. In contrast, rounding components of the expression manually before using a calculator, as seen in Step 2 of the solution, can lead to compounded errors—one from the manual rounding and another from the calculator's display rounding.

Rounding Decimals

Rounding decimals is the process of adjusting the number of digits after the decimal point to limit the precision to a predetermined level, often for simplicity or practical applications. The method usually involves looking at the first digit discarded, and if it is 5 or above, the last kept digit is incremented by one. Conversely, if it is below 5, the last kept digit remains the same.

Applying this in calculations, as shown in the second part of the exercise, significantly influences the outcome, especially when rounding is performed at intermediate stages. This intervention can introduce a 'roundoff error', where the approximation deviates from what would have been calculated if all the digits were retained throughout the calculation. As the solution comparison suggests, additional roundoff error can indeed alter the results, which is an essential lesson in accuracy versus approximation in numerical analysis.