Question

Round off each of the following numbers to the indicated number of significant digits, and write the answer in standard scientific notation. a. 0.00034159 to two digits b. \(103.351 \times 10^{2}\) to five digits c. 17.9915 to one digit d. \(3.365 \times 10^{5}\) to four digits

Short answer

The rounded numbers in standard scientific notation are: a. \(3.4 \times 10^{-4}\) b. \(1.0335 \times 10^{5}\) c. \(2.0 \times 10^{1}\) d. \(3.366 \times 10^{5}\)

Step-by-step solution

Identify the number of significant digits

For each number provided in the exercise, we need to identify the number of significant digits we need to round off. a. 0.00034159 to two digits b. \(103.351 \times 10^{2}\) to five digits c. 17.9915 to one digit d. \(3.365 \times 10^{5}\) to four digits

Round off the numbers to the desired significant digits

Round off each number to the specified number of significant digits: a. 0.00034159 becomes 0.00034 (rounded to two significant digits) b. \(103.351 \times 10^{2}\) becomes \(1.0335 \times 10^{5}\) (rounded to five significant digits) c. 17.9915 becomes 20 (rounded to one significant digit) d. \(3.365 \times 10^{5}\) becomes \(3.366 \times 10^{5}\) (rounded to four significant digits)

Express the rounded numbers in standard scientific notation

The numbers have already been rounded in step 2, and they are also expressed in standard scientific notation: a. 0.00034 is expressed as \(3.4 \times 10^{-4}\) in standard scientific notation. b. \(1.0335 \times 10^{5}\) is already in standard scientific notation. c. 20 is expressed as \(2.0 \times 10^{1}\) in standard scientific notation. d. \(3.366 \times 10^{5}\) is already in standard scientific notation.

Key concepts

Rounding

Rounding numbers is a crucial skill in mathematics and science, as it allows us to simplify figures while retaining their essential value. It involves reducing the number of digits in a number while keeping it as close as possible to the original value. This is often necessary to conform to the rules of significant figures, particularly in scientific calculations.

• To round a number, you first need to identify the digit at the position you are rounding to and look at the digit immediately to the right.
• If this digit is 5 or higher, you increase the rounding digit by one; if it is 4 or lower, you leave the rounding digit unchanged.
• All digits following the rounding digit are replaced by zeros if they are to the right of a decimal point, or simply omitted if they are to the left.

For example, in the number 0.00034159, when rounding to two significant digits, you start with "34." Since the next digit is a "1," the "34" remains unchanged, resulting in 0.00034.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a more concise and readable form. This notation involves representing numbers as a product of two parts: a coefficient and a power of ten. It is a powerful tool for dealing with significant figures and simplifying lengthy calculations.

• The coefficient typically ranges between 1 and 10 and is written with the desired number of significant figures.
• The power of ten indicates how many times the coefficient must be multiplied or divided by ten to obtain the original number.

For example, the number 0.00034 can be expressed as \(3.4 \times 10^{-4}\), where 3.4 is the coefficient and \(10^{-4}\) is the power of ten. This simplifies the number and focuses on its significant digits.

Chemistry Calculations

In chemistry, significant figures play a crucial role in calculations involving measurements and experimental data. Proper use of significant figures helps ensure that numerical results reflect the precision of the instruments used and the accuracy of the measurements.

• In chemical calculations, every measurement taken has an inherent degree of uncertainty, primarily due to the limitations of measuring devices.
• Recording and rounding measurements using the proper number of significant figures provide a true reflection of this uncertainty.
• During calculations such as mole conversions or reaction yields, the answers need to be expressed with the correct number of significant figures.

For instance, if you are calculating a concentration based on a measurement of 17.9915 grams and one significant digit is needed, the correct representation would be "20 grams" using significant figures. This ensures results are not over-precise, reflecting the true nature of the data's reliability.