Question

Challenge Express each quantity in regular notation along with its appropriate unit. \(\begin{array}{llllll}{\text { a. } 3.60 \times 10^{5} \mathrm{s}} & {\text { b. } 5.4 \times 10^{-5} \mathrm{g} / \mathrm{cm}^{3}} & {\text { c. } 5.060 \times 10^{3} \mathrm{km}} & {\text { d. } 8.9 \times 10^{10} \mathrm{Hz}}\end{array}\)

Short answer

The short answer is: a. \(360,000 \mathrm{s}\) b. \(0.000054 \, \mathrm{g}/\mathrm{cm}^{3}\) c. \(5,060 \mathrm{km}\) d. \(89,000,000,000 \mathrm{Hz}\)

Step-by-step solution

Quantity a

For quantity a: \(3.60 \times 10^{5} \mathrm{s}\), we will move the decimal point five places to the right since the power is +5. \(3.60 \times 10^{5} \mathrm{s} = 360,000 \mathrm{s}\)

Quantity b:

For quantity b: \(5.4 \times 10^{-5} \mathrm{g} / \mathrm{cm}^{3}\), we will move the decimal point five places to the left since we have a power of -5. \(5.4 \times 10^{-5} \mathrm{g} / \mathrm{cm}^{3} = 0.000054 \, \mathrm{g} / \mathrm{cm}^{3}\)

Quantity c:

For quantity c: \(5.060 \times 10^{3} \mathrm{km}\), we will move the decimal point three places to the right since it's a positive power of +3. \(5.060 \times 10^{3} \mathrm{km} = 5,060 \mathrm{km}\)

Quantity d:

For quantity d: \(8.9 \times 10^{10} \mathrm{Hz}\), we will move the decimal point ten places to the right since the power is +10. \(8.9 \times 10^{10} \mathrm{Hz} = 89,000,000,000 \mathrm{Hz}\) So the quantities in regular notation with their appropriate units are: a. \(360,000 \mathrm{s}\) b. \(0.000054 \, \mathrm{g}/\mathrm{cm}^{3}\) c. \(5,060 \mathrm{km}\) d. \(89,000,000,000 \mathrm{Hz}\)

Key concepts

Converting Scientific Notation

Scientific notation is a way to express very large or very small numbers by combining a decimal with a power of ten. You’ll often encounter it in science and engineering because it simplifies calculations and ensures clarity. To convert from scientific notation to regular or standard notation, you follow a simple process determined by the exponent on the ten.
If the exponent is positive, like in the case of the time measurement, which was expressed as \(3.60 \times 10^{5} \text{s}\), you move the decimal point to the right, equal to the value of the exponent. This moved the decimal five places, rendering \(360,000 \text{s}\). Conversely, if the exponent is negative, such as with the density \(5.4 \times 10^{-5} \text{g/cm}^{3}\), the decimal moves to the left. Here, the decimal moved five places to the left, giving us the value \(0.000054 \text{g/cm}^{3}\).
Remember that the number of spaces you move the decimal point should exactly match the exponent of ten. This straightforward method allows you to express values in a more relatable way, fitting into everyday contexts and making comparison more intuitive.

Measurement Units

Measurement units are standardized quantities used to express and compare the physical properties of objects, like length, mass, speed, and time. A correct unit is critical for the meaningful expression of a measurement. For instance, distances are often measured in meters or kilometers, and mass can be in grams or kilograms. Examples from our exercise are seconds \(\text{s}\) for time, grams per cubic centimeter \(\text{g/cm}^{3}\) for density, kilometers \(\text{km}\) for distance, and hertz \(\text{Hz}\) for frequency.
In science, the International System of Units (SI) provides a standard that's used globally, which includes units like meters, kilograms, and seconds. It's vital that when you convert scientific notation or perform any calculation, the units are also taken into account, since they provide the scale and context for the numerical value. For instance, expressing a length in meters rather than kilometers can drastically change the perceived distance.

Significant Figures

Significant figures are a concept in measurement that deals with precision. They include all the reliable digits in a number, indicating how precisely the value is measured or known. When we converted the distance to \(5,060 \text{km}\), we maintained the significant figures from the scientific notation \(5.060 \times 10^{3} \text{km}\).
It's important to include the right amount of significant figures when converting from scientific to standard notation. Too few can lead to a loss of precision, while too many can falsely imply a higher precision than what the measurement supports. Keep in mind, zeros can be significant or merely placeholders. For example, in \(360,000 \text{s}\), the zeros are not significant unless specified. Failing to consider significant figures can result in inaccuracies in science and engineering, where precision is often critical. When reporting measurements, always strive to use the same level of precision as the initial data provided, ensuring consistency and reliability in your results.