Express each mass in grams, both in standard notation and in scientific notation: (a) \(536 \mathrm{mg}\) (b) \(8.26 \mathrm{dg}\) (c) \(0.0057 \mu \mathrm{g}\) (d) \(139 \mathrm{~kg}\) (e) \(836 \mathrm{ng}\) (f) \(0.073 \mathrm{Mg}\)

When dealing with very large or very small numbers in science and mathematics, scientific notation is a method used for simplifying their expression and making calculations easier. It is represented as the product of a number between 1 and 10 and a power of 10, which is written as \( a \times 10^b \). This format aids in reducing errors and simplifying operations like multiplication and division.

Take for example the mass conversion exercise for 536 milligrams to grams. The expression \( 536 \times 10^{-3} g \) showcases how scientific notation streamlines representing very small values. For masses like \(0.0057 \mu g\), scientific notation becomes crucial because it effectively communicates the scale of the measurement, resulting in \(5.7 \times 10^{-9} g\) and evading the confusion of many leading zeros in standard notation.

The metric system is an internationally adopted decimal system of measurement. Its units are based on factors of ten, making conversions a process of moving the decimal point rather than complex multiplication. Commonly used metric units for mass include grams (g), kilograms (kg), and milligrams (mg), among others. Each unit is a factor of 10 apart from the others, which simplifies conversions significantly.

In our textbook exercise, we see units like milligrams (mg), decigrams (dg), micrograms (\(\mu g\)), nanograms (ng), and megagrams (Mg). These are all metric system units representing different scales of mass, where the standard reference for mass is the gram (g). The conversion demonstrates how these varied units relate to each other through factors of ten, such as 1 mg equaling \(1 \times 10^{-3}\) grams.

Unit conversion is a fundamental skill in science that allows us to compare and utilize measurements in different units. By employing conversion factors that are essentially ratios equating different quantities of the same measure, we can transition from one unit to another with ease.

For instance, consider converting 836 nanograms to grams as shown in the exercise solutions. Using the conversion factor \(1 \times 10^{-9}\), we can calculate the weight in grams as \(836 \times 10^{-9} g\). It is vital to keep track of units during conversion to ensure accuracy, as it prevents mixing up of scales, which can lead to errors in scientific work.

Standard notation is the regular way of writing numbers without any exponents or scientific formatting. It's the familiar form we use in everyday life and it's especially useful for representing measurements that are close to our everyday experience.

In mass conversion, standard notation allows us to relate more directly to the quantities involved. For a kilograms to grams conversion, like the \(139 \mathrm{kg}\) example in our exercise, the converted mass is clearly and simply shown as \(139,000 \mathrm{g}\) in standard notation. This helps in visualizing the actual quantity, rather than the condensed scientific notation which might be less intuitive, especially for larger or common masses.