Problem 12 - Technical Mathematics with Calculus

Question

Multiply each approximate number and retain the proper number of digits in your answer. $$(5.93 \times 7.28) \times(8.26 \times 1.38)$$

Step-by-Step Solution

Answer

The product of (5.93 \times 7.28) \times(8.26 \times 1.38), rounded to three significant digits, is 341.

Step 1: Multiply the First Pair of Numbers

Multiply the first pair of approximate numbers 5.93 and 7.28. Keep the intermediate result handy, but keep in mind that you may not need to round at this stage since you will perform further calculations.

Step 2: Multiply the Second Pair of Numbers

Now, multiply the second pair of approximate numbers 8.26 and 1.38. Just like in Step 1, keep the intermediate result and maintain all the digits for now.

Step 3: Multiply the Intermediate Results

Multiply the two intermediate results from Steps 1 and 2 to get the final product.

Step 4: Round to the Appropriate Number of Digits

Since the numbers provided are to three significant digits, round your final answer to three significant digits as well. This ensures you retain the proper number of significant figures in accordance with the rule for multiplication of significant figures.

Key Concepts

Approximate Numbers

When dealing with measurements or calculations, we often encounter approximate numbers. These are numbers that are not exact but represent a close estimation of the quantity being measured. In everyday life, approximate numbers are commonplace as it is often impractical or impossible to measure things precisely. For example, when you measure the length of a table, you may get a measurement like 1.53 meters. This number could be an approximation as the actual measurement could be slightly more or less.

In mathematics and science, we use approximate numbers to simplify calculations and to reflect the uncertainty in measurements. When multiplying approximate numbers, it's crucial to consider the precision of the numbers to avoid exaggerating the accuracy of the result. This is where the concept of significant digits comes into play, as they help identify which digits in an approximate number are meaningful and should be retained in calculations.

Significant Digits

The concept of significant digits, also known as significant figures, is fundamental in understanding how to represent the precision of an approximation. Significant digits include all the digits in a number that contribute to its accuracy, from the first non-zero digit to either the last digit if it’s non-zero or the last accurately known digit if it ends with a zero.

For instance, in the number 5.93, all three digits are significant because they provide specific information about the quantity measured. However, in a number like 5.930, the final zero is also significant if it is a measured value indicating precision up to the thousandths place. When multiplying numbers, the product should not have more significant figures than the least precise number used in the calculation. Therefore, if we multiply 5.93 (three significant digits) by 7.28 (three significant digits), we need to ensure that our final result also has three significant digits to accurately reflect the precision of our inputs. Correctly handling significant figures while performing operations is integral to maintaining the integrity of the data.

Scientific Notation

Handling very large or very small numbers can be cumbersome, particularly when dealing with significant figures. To address this, we use scientific notation to express numbers in a more compact form. Scientific notation follows the format where a number is represented as a product of a number between 1 and 10 and a power of 10. This makes it easier to work with and understand the scale of numbers, particularly in science and engineering where extreme values are common.

For example, the number 8.26, which is already in a suitable form for most calculations, can be written as \(8.26 \times 10^0\) in scientific notation, acknowledging it is of the order of unity. In calculations that involve multiplication, scientific notation becomes useful. Multiplying \(5.93 \times 10^0\) and \(7.28 \times 10^0\), and then \(8.26 \times 10^0\) by \(1.38 \times 10^0\), simplifies the process by allowing the manipulation of the coefficients and the exponents separately. Rounding our results accordingly, using scientific notation, ensures precision and ease of calculation, maintaining the significance of figures throughout the operation.