Question

When the calculation \((2.31)\left(4.9795 \times 10^{3}\right) /\left(1.9971 \times 10^{4}\right)\) is performed, how many significant digits should be reported for the answer? You should \(n o t\) need to perform the calculation.

Short answer

The answer should be reported with three significant digits.

Step-by-step solution

Identify the number of significant digits in each number

We need to identify how many significant digits are included in each number: 1. 2.31 has three significant digits because all non-zero digits are significant. 2. 4.9795 x 10^3 has five significant digits (including the four in the decimal part 4.9795). 3. 1.9971 x 10^4 has five significant digits (including the four in the decimal part 1.9971).

Using significant digits rules for multiplication and division

For multiplications and divisions, the final answer should have as many significant digits as the LEAST number of significant digits provided in the numbers used in the operation. In this calculation, we are performing both multiplication and division. Let's divide it into two parts, and for each part, use the rules above.

Multiplying the two numbers with significant digits

We have: (2.31) (4.9795 x 10^3) Here, 2.31 has three significant digits and 4.9795 x 10^3 has five significant digits. The smallest number of significant digits is 3, so the result of this multiplication should have only three significant digits. Let's say the product is: A x 10^B (where A has three significant digits)

Dividing the result with the remaining number having significant digits

Next, we need to divide the result A x 10^B by (1.9971 x 10^4). Since A has three significant digits and 1.9971 x 10^4 has five significant digits, we should use the smallest number of significant digits, which is 3. Dividing A x 10^B by 1.9971 x 10^4, the final answer should have three significant digits. So, you should report the answer with three significant digits.

Key concepts

Multiplication and Division Rules

In math, when you're doing calculations with numbers, you need to know how many significant digits to keep in your final answer. There are some rules for this that help keep your work precise and accurate.

When you're multiplying or dividing numbers, the rule is simple: you should use the fewest number of significant digits from the numbers you are working with. This ensures your answer is as precise as the least precise measurement involved.

Imagine you have numbers like these:

• 3.14 (3 significant digits)
• 5.764 (4 significant digits)

After performing a multiplication or division with these numbers, your result should only have 3 significant digits, which is the lower number of the two. This helps keep results consistent in terms of their precision.

Scientific Notation

Scientific notation is a way of writing very big or very small numbers easily. It uses powers of ten to simplify these numbers into a more manageable form. This is super handy when dealing with lots of zeros in calculations.

For example, instead of writing out 5,000,000, you can write it as \(5 \times 10^{6}\). The \(10^{6}\) part means you move the decimal point six places to the right. It makes writing and reading these numbers quicker.

In scientific notation, each number part is made of a coefficient (a number like 5 in \(5 \times 10^{6}\)) and a power of 10. All digits in the coefficient are significant, making it a useful format for working with significant figures.

Digits Counting

When you're working with numbers and need to know how many significant figures they have, you go through a simple process called 'digits counting'. You look at each part of the number to see which digits are important to the value.

Here’s how it breaks down:

• All non-zero numbers are significant. So, in 234, all three digits are significant.
• Zeros between numbers are significant, like in 107, where all 3 digits count.
• Leading zeros (those before any non-zero digit) are not significant, as seen in 0.0056, which has 2 significant digits.
• Trailing zeros in a number with a decimal point are significant. For instance, 45.00 has four significant digits.

Using these rules helps you know exactly how many digits to keep in your calculations.

Significant Digits

Significant digits (or significant figures) are crucial in measurement and calculation. They indicate the precision of a number. Each significant figure in a measurement adds to the accuracy and reliability of the data.

Significant digits come into play when you perform any mathematical operation, especially multiplication or division. The number of significant digits tells you the degree of precision of the result.

For example, when you're dividing a number like 1234 (which has four significant figures) by a number like 2.4 (which has two significant figures), your answer should also have two significant figures. This maintains a balance of precision across your calculations.

Keeping an eye on significant digits helps ensure that your results are both consistent and accurate, reflecting the true nature of the numbers you're working with.