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. \(1.0340 \times 10^{4}\) c. \(1.7992 \times 10^{1}\) d. \(3.37 \times 10^{5}\)

Step-by-step solution

(a) Rounding \(0.00034159\) to three significant digits#

To round \(0.00034159\) to three significant digits, we need to identify the first three significant digits, which are \(3\), \(4\), and \(1\). Now round the number to these three digits. The next digit after the third significant digit is \(5\), so we round up, giving us \(0.000342\). Now, let's convert this into standard scientific notation: \[ 0.000342 = 3.42 \times 10^{-4} \]

(b) Rounding \(103.351 \times 10^{2}\) to four significant digits#

First, let's express the number \(103.351 \times 10^{2}\) as a decimal number: \[ 103.351 \times 10^{2} = 10335.1 \] Now, we need to round it to four significant digits. The first four significant digits are \(1\), \(0\), \(3\), and \(3\). Observe that the next digit is \(5\), so we round up, giving us \(10340\). Now, let's express the rounded number in standard scientific notation: \[ 10340 = 1.0340 \times 10^{4} \]

(c) Rounding \(17.9915\) to five significant digits#

To round \(17.9915\) to five significant digits, we need to identify the first five significant digits, which are \(1\), \(7\), \(9\), \(9\), and \(1\). Now round the number to these five digits. The next digit after the fifth significant digit is \(5\), so we round up, giving us \(17.992\). Now, let's convert this into standard scientific notation: \[ 17.992 = 1.7992 \times 10^{1} \]

(d) Rounding \(3.365 \times 10^{5}\) to three significant digits#

To round \(3.365 \times 10^{5}\) to three significant digits, we need to identify the first three significant digits, which are \(3\), \(3\), and \(6\). Now round the number to these three digits. The next digit after the third significant digit is \(5\), so we round up, giving us \(3.37 \times 10^{5}\). Already in standard scientific notation, the rounded number is: \[ 3.37 \times 10^{5} \]

Key concepts

Significant Digits

Significant digits are important because they indicate the precision of a measured or calculated number. A common rule when counting significant digits is to look at the digits starting from the first non-zero number. In the example of rounding the number 0.00034159, the first three significant digits are 3, 4, and 1. These digits tell us that the measurement is precise up to these three numbers. Leading zeros are not counted as significant. They merely tell us the position of the decimal point. By being mindful of significant digits, we ensure that our calculations are precise and aligned with the data's accuracy.

Rounding Numbers

Rounding numbers is a way to simplify them while keeping them close to the original value. This is especially important when we need to express numbers to a specific number of significant digits. We apply the rule: if the digit right after our desired place is 5 or more, we round up. For example, when rounding 103.351 to four significant digits, the fifth digit was a 5, prompting us to round up to 10340. This rounding process helps us handle figures easily without losing too much accuracy, especially in scientific calculations where too many digits might not contribute to the precision we need.

Standard Form

The standard form in mathematics, particularly in scientific notation, allows us to write very large or very small numbers conveniently. It expresses numbers as a product of a number between 1 and 10 and a power of ten. For instance, after rounding 0.000342, we can write it as 3.42 x 10^{-4}. This method is especially useful in scientific fields where extreme values frequently occur. Scientific notation ensures that these numbers remain manageable and easily comparable while keeping them precise and correctly scaled.

Exponential Representation

Exponential representation makes handling repeated multiplication of the same base much simpler. By representing numbers as a base raised to an exponent, we can significantly simplify calculations. When converting numbers to scientific notation, we find the number between 1 and 10 and multiply it by 10 raised to an appropriate power. For example, 17.992 can be expressed as 1.7992 x 10^{1}. Here, 10^{1} tells us that the decimal point is moved one place. Exponential representation is crucial for simplifying mathematical expressions in both, theory and application, by providing a clear format for complex numbers.