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 three digits b. \(103.351 \times 10^{2}\) to four digits c. 17.9915 to five digits d. \(3.365 \times 10^{5}\) to three digits

Short answer

a. \(3.42 \times 10^{-4}\) b. \(1.034 \times 10^{4}\) c. \(1.7992 \times 10^{1}\) d. \(3.37 \times 10^{5}\)

Step-by-step solution

Identify the first three significant digits

In the given number 0.00034159, the first three significant digits are 3, 4, and 1.

Round off the number

The next digit (5) is greater than or equal to 5, so we will round up our last significant digit (1) to 2. Thus, the rounded number is 0.000342.

Write in standard scientific notation

To write this number in standard scientific notation, we write it in the form \(a \times 10^n\), where \(1 \leq a < 10\) and n is an integer. So, the standard scientific notation will be \(3.42 \times 10^{-4}\). #b. \(103.351 \times 10^{2}\) to four digits#

Identify the first four significant digits

In the given number \(103.351 \times 10^2\), the first four significant digits are 1, 0, 3, and 3.

Round off the number

The next digit (5) is greater than or equal to 5, so we will round up our last significant digit (3) to 4. Thus, the rounded number is \(103.4 \times 10^2\).

Write in standard scientific notation

The resulting number is already in standard scientific notation, so there is no need to make any adjustments. Hence, the answer is \(1.034 \times 10^{4}\). #c. 17.9915 to five digits#

Identify the first five significant digits

In the given number 17.9915, the first five significant digits are 1, 7, 9, 9, and 1.

Round off the number

The next digit (5) is equal to 5, so we will round up our last significant digit (1) to 2. Thus, the rounded number will be 17.992.

Write in standard scientific notation

To write this number in standard scientific notation, we will have \(1.7992 \times 10^{1}\). #d. \(3.365 \times 10^{5}\) to three digits#

Identify the first three significant digits

In the given number \(3.365 \times 10^5\), the first three significant digits are 3, 3, and 6.

Round off the number

The next digit (5) is equal to 5, so we will round up our last significant digit (6) to 7. Thus, the rounded number will be \(3.37 \times 10^5\).

Write in standard scientific notation

The resulting number is already in standard scientific notation, so there is no need to make any adjustments. Hence, the answer is \(3.37 \times 10^{5}\).

Key concepts

Scientific Notation

Scientific notation is a way of writing very large or very small numbers in a compact form. This method employs powers of ten. It helps to make these numbers easier to work with and compare.
For example, instead of writing 0.000342, we can express this number in scientific notation as \(3.42 \times 10^{-4}\). This means that you move the decimal four places to the right to get the original number. On the other hand, a large number like 1,034,000 can be written as \(1.034 \times 10^{6}\).
When converting to scientific notation, you must follow the form \(a \times 10^{n}\), where \(1 \leq a < 10\) and \(n\) is an integer. This form ensures that the number \(a\) is always between 1 and 10, which makes it a proper scientific notation.

Rounding Numbers

Rounding numbers is the process of simplifying them to make them easier to work with. This is especially useful when exact values are not needed.
To round a number, you need to identify the digit that lies right after the one you're concerned with. If this digit is 5 or more, you increase the last digit you're keeping by one. If it's less than 5, you leave the number as is.
For example, to round 0.00034159 to three significant digits, you look at the first three digits, which are 3, 4, and 1. The next digit, 5, prompts you to round the number up to 0.000342.
Remember, rounding not only simplifies numbers—it can also help maintain the precision required for calculations.

Mathematical Notation

Mathematical notation involves various symbols and signs used to represent numbers and operations in mathematics clearly and concisely.
Some frequently used symbols include the multiplication sign (\(\times\)), the power of ten (\(10^{n}\)), and decimal points to signify fractions. Scientific notation uses these symbols to express numbers efficiently, especially with powers of ten and decimal movement.
For instance, in standard mathematical notation, the number \(103.351 \times 10^2\) is transformed into \(1.034 \times 10^{4}\). Incorporating these types of notations helps simplify complex problems and makes it easier to communicate ideas accurately in mathematics.
Being familiar with mathematical notation is crucial for making sense of scientific notation and engaging in many areas of math and science.

Significant Figures

Significant figures are the digits in a number that contribute to its accuracy. They are crucial for indicating the precision of a measurement or calculation.
To determine the significant figures, start counting from the first non-zero digit. For trailing zeroes in decimals and leading zeroes in whole numbers, their significance is dependent on context. For instance, in 0.00034159, the significant digits are 3, 4, and 1.
Significant figures are used in rounding as well. When instructed to write a number to a specific number of significant figures, such as three, ensure only those many digits are listed after any rounding.
Understanding significant figures helps you ensure that numerical results are not overly precise and align with the accuracy of the input measurements or data.