Question

Express each of the following in scientific notation: a. \(0.00000769 \mathrm{~g}\) b. \(36,457,549\) (population of California) c. \(299,790,000 \mathrm{~m} / \mathrm{s}\) (speed of light) d. \(299,398,484\) (U.S. population)

Short answer

\(0.00000769\mathrm{~g}\) is \(7.69 \times 10^{-7}\mathrm{~g}\), 36,457,549 is \(3.646 \times 10^7\), 299,790,000 m/s for the speed of light is \(2.998 \times 10^8 \text{m/s}\), and 299,398,484 is \(2.994 \times 10^8\).

Step-by-step solution

Expressing 0.00000769 g in scientific notation

To convert the given number into scientific notation, find the non-zero digits' place after the decimal and count the number of decimal places we move until we have a number between 1 and 10. In this case, we move the decimal point 7 places to the right. So, 0.00000769 g becomes 7.69 times 10 to the power of -7. In scientific notation, this is written as: \(7.69 \times 10^{-7} \text{g}\).

Expressing 36,457,549 in scientific notation

Move the decimal point 7 places to the left to make the number fall between 1 and 10. This gives: 3.6457549. We then round it to a reasonable number of significant figures. If we choose to round to four significant figures, the number becomes 3.646. Since we moved the decimal 7 places, the number in scientific notation is \(3.646 \times 10^7\).

Expressing 299,790,000 m/s in scientific notation

Move the decimal point 8 places to the left to get a number between 1 and 10, which gives us 2.9979. Rounding to four significant figures, the speed of light is written as \(2.998 \times 10^8 \text{m/s}\).

Expressing 299,398,484 in scientific notation

Moving the decimal point 8 places to the left gives us 2.99398484. Rounding to four significant figures gives us 2.994, so in scientific notation, the U.S. population is expressed as \(2.994 \times 10^8\).

Key concepts

Significant Figures

Understanding significant figures is important in scientific measurements, as they convey the precision of those measurements. Significance in this context refers to the trustworthy digits in a number that contribute to its accuracy. To identify significant figures, we follow these rules:

• All non-zero digits are significant.
• Any zeros between non-zero digits are also significant.
• Leading zeros before the first non-zero digit are not significant, as they merely indicate the position of the decimal point.
• Trailing zeros in a number with a decimal point are significant.

For example, in the number 0.00000769, the digits 7, 6, and 9 are significant, but the leading zeros are not. Similarly, the number 299,398,484 has eight significant figures.

When performing calculations, the number of significant figures can be adjusted. For instance, if we round 299,398,484 to four significant figures, we get 2.994. This rounding process aims to retain the meaningfulness of the number while making it more manageable. The concept of significant figures is closely tied to scientific notation, as it helps determine how many digits to present after converting to that notation.

Decimal Places

Decimal places refer to the number of digits to the right of the decimal point in a numerical expression. They can significantly impact the precision and accuracy of calculations. For instance, the number 0.00000769 has five decimal places. Converting a number into scientific notation often involves shifting the decimal point to create a new number between 1 and 10. This is then followed by an exponent that reflects the number of places the decimal has shifted.

It's critical to differentiate between significant figures and decimal places. While significant figures focus on the meaningful digits in a number, regardless of the position of the decimal point, decimal places specifically count the digits after the decimal. In our example from the original exercise, starting with the value 299,790,000 m/s, the decimal place counting would begin after moving the decimal to immediately follow the first non-zero digit, thus affecting the determination of significant figures. When rounding to a certain number of significant figures, the number of decimal places will be adjusted accordingly.

Exponent Notation

Exponent notation, also known as scientific notation, is a convenient way to express very large or very small numbers. In this form of notation, numbers are written as a product of two factors: a decimal greater than or equal to 1 but less than 10, and a power of 10. This method simplifies numbers by eliminating trailing zeros or lengthy strings of leading zeros.

For example, converting the number 299,790,000 m/s to scientific notation, we shift the decimal 8 places to the left, obtaining 2.9979. We then round this to four significant figures, resulting in 2.998. Finally, we append the exponent, signifying that the original decimal point was moved 8 places, to get the scientific notation:

\(2.998 \times 10^8 \text{m/s}\)

Exponent notation not only makes numbers easier to read and work with but also ensures consistency in the representation of quantities in scientific and mathematical documents. The exponent indicates how many places to move the decimal point to return to the original number. If the original number is large, the exponent is positive, signifying a shift to the left. Conversely, for small numbers, the exponent is negative, corresponding to a shift to the right of the decimal point.