Question

Round off each of the following numbers to the indicated number of significant digits and write the answer in standard scientific notation. a. \(4341 \times 10^{2}\) to three significant digits b. \(93.441 \times 10^{3}\) to three significant digits c. \(0.99155 \times 10^{2}\) to four significant digits d. 9.3265 to four significant digits

Short answer

The rounded numbers in standard scientific notation are: a. \(4.34 \times 10^{4}\) b. \(9.34 \times 10^{4}\) c. \(9.916 \times 10^{1}\) d. \(9.327 \times 10^{0}\)

Step-by-step solution

Rounding off to the required significant digits

We will round off each number to the given significant digits. a. Round off \(4341 \times 10^{2}\) to three significant digits \(4340 \times 10^{2}\) (After rounding) b. Round off \(93.441 \times 10^{3}\) to three significant digits \(93.4 \times 10^{3}\) (After rounding) c. Round off \(0.99155 \times 10^{2}\) to four significant digits \(0.9916 \times 10^{2}\) (After rounding) d. Round off 9.3265 to four significant digits 9.327 (After rounding)

Write the numbers in standard scientific notation

We will now write the rounded numbers in step 1 in standard scientific notation. a. \(4340 \times 10^{2}\) in standard scientific notation \(4.34 \times 10^{4}\) b. \(93.4 \times 10^{3}\) in standard scientific notation \(9.34 \times 10^{4}\) c. \(0.9916 \times 10^{2}\) in standard scientific notation \(9.916 \times 10^{1}\) d. 9.327 in standard scientific notation \(9.327 \times 10^{0}\) The final answers are: a. \(4.34 \times 10^{4}\) b. \(9.34 \times 10^{4}\) c. \(9.916 \times 10^{1}\) d. \(9.327 \times 10^{0}\)

Key concepts

Scientific Notation

Scientific notation is a method used to write very large or very small numbers in a compact form. It plays a significant role in fields such as chemistry, physics, and engineering where precise measurements are crucial.

At its core, scientific notation expresses numbers as a product of two factors: a decimal and a power of ten. The decimal—also known as the significand or mantissa—typically has one nonzero digit to the left of the decimal point, ensuring the number of significant figures is clear. Here’s the general form:
\[a \times 10^n\]where \(a\) is the decimal part and \(n\) is the exponent indicating the power of ten.

Let's consider an example from the exercise we are looking at: \(4341 \times 10^2\). If we want to express this number with three significant digits in scientific notation, we first round it to 4340 (as we'll explain in the next section) and then rewrite it as \(4.34 \times 10^4\). This makes the number easier to read and understand, particularly in scientific calculations.

Rounding Numbers

Rounding numbers is a fundamental arithmetic technique essential for handling significant figures and reporting measurements accurately in chemistry and other scientific disciplines.

The process typically involves truncating a number to a specific number of significant digits based on certain rules. When the digit immediately right of the last significant figure is greater than or equal to 5, we round up; otherwise, we round down.

For example, to round the number \(93.441 \times 10^3\) to three significant digits, we focus on the first three digits—934—since the fourth digit, 4, is less than 5, we leave the third digit as-is. This gives us \(93.4 \times 10^3\). Precision is vital in chemistry education, especially when dealing with measurements and calculations where the correct application of rounding rules has significant impacts on the outcome.

Chemistry Education

In chemistry education, understanding how to manipulate and interpret numbers, like using scientific notation and rounding numbers, is crucial. Chemistry often deals with quantities that span a huge range from incredibly small like atomic masses to large like amounts of substances in moles.

It is in the laboratory and classroom where these skills become practical. Students learn the importance of significant figures when making measurements or when calculating yields in reactions. For instance, when we round the number \(0.99155 \times 10^2\) to four significant digits to get \(0.9916 \times 10^2\), we are ensuring that the final result reflects the precision of the measurement tools used during the experiment.

The attention to detail in scientific notation and rounding is not just mathematical rigor—it reflects the reality of experimental uncertainties and is pivotal for students to grasp, for later scientific research or even just informed decision-making in our technological world.