Question

Round each of the following numbers to four significant figures, and express the result in standard exponential notation: (a) 102.53070, (b) 656,980, (c) \(0.008543210\), (d) \(0.000257870,(\mathrm{e})-0.0357202 .\)

Short answer

The short answers are: a) \(1.025 \times 10^2\) b) \(6.570 \times 10^5\) c) \(8.540 \times 10^{-3}\) d) \(2.590 \times 10^{-4}\) e) \(-3.572 \times 10^{-2}\)

Step-by-step solution

Identifying Significant Figures

The first step is to identify the significant figures in the given number (102.53070). Starting from the left, the first non-zero digit (1) is the first significant figure. Continue counting significant figures until reaching the desired number (4). In this case, the significant figures are 1, 0, 2, and 5.

Rounding

The next step is to round the number. The fourth significant figure is the last one to keep, so anything after that needs to be rounded off. Look at the next digit after the fourth significant figure, which is 3. Since it's less than 5, the fourth significant figure (5) stays the same, and everything after it is removed.

Expressing in Standard Exponential Notation

Now, express the rounded number (102.5) in standard exponential notation. Move the decimal point until there is only one non-zero digit to the left of it. In this case, move the decimal point one position to the left. The result is 1.025. Multiply this by 10 raised to the power of the number of positions moved (1). The final result in standard exponential notation is \(1.025 \times 10^2\). #b) Rounding 656,980 to four significant figures#

Identifying Significant Figures

The significant figures in the given number (656,980) are 6, 5, 6, and 9.

Rounding

The digit after the fourth significant figure is 8, which is 5 or greater; therefore, add 1 to the fourth significant figure (9), making it 10. Since we cannot have 10 as a single digit, drop the zero and add 1 to the next digit on the left (6), making the rounded number 657,000.

Expressing in Standard Exponential Notation

Express the rounded number (657,000) in standard exponential notation. Move the decimal point 5 positions to the left, giving 6.570. Multiply this by 10 raised to the power of 5. The final result in standard exponential notation is \(6.570 \times 10^5\). #c) Rounding 0.008543210 to four significant figures#

Identifying Significant Figures

The significant figures in the given number (0.008543210) are 8, 5, 4, and 3.

Rounding

The digit after the fourth significant figure is 2, which is less than 5. The fourth significant figure (3) remains unchanged, and everything after it is removed.

Expressing in Standard Exponential Notation

Express the rounded number (0.00854) in standard exponential notation. Move the decimal point 3 positions to the right, giving 8.540. Multiply this by 10 raised to the power of -3. The final result in standard exponential notation is \(8.540 \times 10^{-3}\). #d) Rounding 0.000257870 to four significant figures#

Identifying Significant Figures

The significant figures in the given number (0.000257870) are 2, 5, 7, and 8.

Rounding

The digit after the fourth significant figure is 7, which is 5 or greater. Add 1 to the fourth significant figure (8), giving 9.

Expressing in Standard Exponential Notation

Express the rounded number (0.000259) in standard exponential notation. Move the decimal point 4 positions to the right, giving 2.590. Multiply this by 10 raised to the power of -4. The final result in standard exponential notation is \(2.590 \times 10^{-4}\). #e) Rounding -0.0357202 to four significant figures#

Identifying Significant Figures

The significant figures in the given number (-0.0357202) are 3, 5, 7, and 2.

Rounding

The digit after the fourth significant figure is 0, which is less than 5. The fourth significant figure (2) remains unchanged, and everything after it is removed.

Expressing in Standard Exponential Notation

Express the rounded number (-0.03572) in standard exponential notation. Move the decimal point 2 positions to the right, giving -3.572. Multiply this by 10 raised to the power of -2. The final result in standard exponential notation is \(-3.572 \times 10^{-2}\).

Key concepts

Rounding Numbers

Rounding numbers is a fundamental technique in mathematics, especially useful in simplifying numbers while preserving their important values. For example, when a number is rounded to four significant figures, we focus only on the first four digits starting from the first non-zero digit. If the next digit is 5 or more, we round up. If it's less than 5, we round down.
This process eliminates the insignificant digits, which helps in simplifying calculations and presenting clearer data. For instance, rounding 102.53070 gives us 102.5 when focusing on four significant figures, because the digit after the fourth significant figure is 3, which is less than 5.

Exponential Notation

Exponential notation represents numbers as a product of a number and a power of ten, allowing for a compact representation of both very large and very small numbers. It helps make multiplication and division easier, especially with complex numbers.
To transform a number into exponential notation, move the decimal point so that only one non-zero digit remains to the left. Count the number of places moved, which becomes the exponent on the power of ten. For example, 657,000 becomes 6.570 multiplied by 10 to the power of 5 because the decimal is moved five places left. This transforms a cumbersome number into a more manageable form.

Scientific Notation

Scientific notation is a specific form of exponential notation typically used in science to handle very large or very small numbers succinctly. It is written as the product of a number (greater than or equal to 1 but less than 10) and 10 raised to an integer power.
To express a number in scientific notation, convert it into exponential notation with one significant digit on the left of the decimal and adjust the exponent accordingly. For example, the number 0.008543210, when rounded to four significant figures, is written as 8.540 x 10^-3. The negative exponent indicates that the original number was less than one.