Question

Indicate the number of significant digits in the answer when each of the following expressions is evaluated (you do not have to evaluate the expression). a. \((6.25) /(74.1143)\) b. (1.45)(0.08431)\(\left(6.022 \times 10^{23}\right)\) c. \((4.75512)(9.74441) /(3.14)\)

Short answer

Number of significant digits in the answers for each expression: a. \(3\) significant digits b. \(3\) significant digits c. \(3\) significant digits

Step-by-step solution

1. Identifying Operations

First, let's identify the operations in each expression: a. Division: \((6.25) /(74.1143)\) b. Multiplication: (1.45)(0.08431)\(\left(6.022 \times 10^{23}\right)\) c. Multiplication and Division: \((4.75512)(9.74441) /(3.14)\)

2. Expression A: Division

For division, the answer should have the same number of significant digits as the least precise number involved. In this case, we have: Number 1: 6.25 (3 significant digits) Number 2: 74.1143 (6 significant digits) Now, choose the number with the least number of significant digits (3) as our answer's significant digits.

3. Expression B: Multiplication

For multiplication, the answer should have the same number of significant digits as the number with the least number of significant digits: Number 1: 1.45 (3 significant digits) Number 2: 0.08431 (4 significant digits) Number 3: \(6.022 \times 10^{23}\) (4 significant digits, scientific notation) In this case, the number with the least number of significant digits is 1.45 with 3 significant digits. The answer will also have 3 significant digits.

4. Expression C: Multiplication and Division

For this expression, we should first multiply the two numbers, then divide by the third number: Multiplication: Number 1: 4.75512 (6 significant digits) Number 2: 9.74441 (6 significant digits) Answer: same as the least number of significant digits, so 6 significant digits. Division: Number 1: answer from the multiplication step (6 significant digits) Number 2: 3.14 (3 significant digits) Answer: same as the least accurate number, so 3 significant digits.

5. Results

Number of significant digits in the answers for each expression: a. 3 significant digits b. 3 significant digits c. 3 significant digits

Key concepts

Division

When you perform division in mathematics, especially with significant digits in mind, it's important to keep a few key rules handy. The rule of thumb for division is that the result should have the same number of significant digits as the number with the fewest significant digits in the original numbers. For example, if you were dividing two numbers like 6.25 by 74.1143,

• 6.25 has 3 significant digits.
• 74.1143 has 6 significant digits.

You would limit your result to 3 significant digits, which is the smallest of the two. This ensures that the precision or "accuracy" is consistent with the least detailed measurement you began with. To put it simply, you're only as precise as your least precise measurement allows.

Multiplication

Multiplication with significant digits works similarly to division in terms of rules. When multiplying two or more numbers, the number of significant digits in the result is determined by the factor with the least significant digits. Let's break it down with an example akin to one from our exercise: multiplying 1.45, 0.08431, and a number in scientific notation like \(6.022 \times 10^{23}\). Here:

• 1.45 has 3 significant digits.
• 0.08431 has 4 significant digits.
• \(6.022 \times 10^{23}\) also has 4 significant digits.

In this multiplication, your final answer would be rounded to 3 significant digits, aligning it with the factor that has the least precision. These rules ensure that any limitations in precision from your inputs are carried through to your output, maintaining a consistent standard of accuracy.

Scientific Notation

Scientific notation is a handy tool for dealing with very large or small numbers, removing the bulk of writing out many zeros. It represents numbers using powers of ten. For instance, \(6.022 \times 10^{23}\) is a lot more concise than writing the whole number out. When working with scientific notation in multiplication or division, the number of significant digits is based on the coefficient. In our example, 6.022 is the coefficient and has 4 significant digits. Keep these 4 significant digits in the final result unless another number in your calculation dictates fewer significant digits through the rules of multiplication or division.By focusing on the coefficient's significant digits, you maintain both the readability of large numbers and the accuracy needed for scientific calculations, making it a powerful method in mathematics and science alike.