Question

Give the full name of the following units and express the quantities in terms of the basic unit (e.g., \(1 \mathrm{~mL}=1\) milliliter \(=0.001 \mathrm{~L}\) ) (a) \(1 \mathrm{~cm}\) (b) \(1 \mathrm{dg}\) (c) \(1 \mathrm{~km}\) (d) \(1 \mu \mathrm{s}\) (e) \(1 \mathrm{ng}\)

Short answer

(a) 0.01 m (b) 0.1 g (c) 1000 m (d) 0.000001 s (e) 0.000000001 g

Step-by-step solution

Understanding the Conversion

Find out what the prefix stands for. Look up what 'centi', 'deci', 'kilo', 'micro', and 'nano' mean in terms of powers of ten.

Convert Centimeters to Meters

The prefix "centi" stands for \(10^{-2}\). Therefore, \(1 \, \text{cm} = 1 \, \text{centimeter} = 1 \, \times \, 10^{-2} \, \text{meters} = 0.01 \, \text{m}\).

Convert Decigrams to Grams

The prefix "deci" stands for \(10^{-1}\). Therefore, \(1 \, \text{dg} = 1 \, \text{decigram} = 1 \, \times \, 10^{-1} \, \text{grams} = 0.1 \, \text{g}\).

Convert Kilometers to Meters

The prefix "kilo" stands for \(10^{3}\). Therefore, \(1 \, \text{km} = 1 \, \text{kilometer} = 1 \, \times \, 10^{3} \, \text{meters} = 1000 \, \text{m}\).

Convert Microseconds to Seconds

The prefix "micro" stands for \(10^{-6}\). Therefore, \(1 \, \mu \text{s} = 1 \, \text{microsecond} = 1 \, \times \, 10^{-6} \, \text{seconds} = 0.000001 \, \text{s}\).

Convert Nanograms to Grams

The prefix "nano" stands for \(10^{-9}\). Therefore, \(1 \, \text{ng} = 1 \, \text{nanogram} = 1 \, \times \, 10^{-9} \, \text{grams} = 0.000000001 \, \text{g}\).

Key concepts

Unit Prefixes

Unit prefixes are very helpful as they allow us to express large and small numbers without writing a lot of zeros. These are short forms that you place before basic units such as meters (m), grams (g), or liters (L). For example, in "centimeter" ('centi-' is the prefix) we know it means something smaller than a meter.

• Centi- stands for 0.01 or \(10^{-2}\).
• Deci- means 0.1 or \(10^{-1}\).
• Kilo- translates to 1000 or \(10^{3}\).
• Micro- stands for 0.000001 or \(10^{-6}\).
• Nano- means 0.000000001 or \(10^{-9}\).

By knowing these prefixes, you can easily read and convert units without complex calculations. It makes dealing with measurements a whole lot easier.

Powers of Ten

The concept of powers of ten makes it easy to handle very large or very small numbers. Instead of writing down a long string of zeros, we can use exponents. This is highly efficient and easy to understand.
A positive power of ten, like \(10^3\), means you multiply 1 by 1000. Conversely, a negative power of ten, like \(10^{-2}\), means you multiply 1 by 0.01, or divide by 100.

• \(10^3 = 1000\)
• \(10^{-1} = 0.1\)
• \(10^{-6} = 0.000001\)

Recognizing these powers helps you perform conversions between units effortlessly. This knowledge aids you in solving a variety of problems smoothly.

Basic Units

Basic units are the fundamental units in any measurement system. In the metric system, these could be meters for distance, grams for mass, and seconds for time. They provide a standard for measuring quantities, which helps maintain consistency across scientific and everyday measurements.
For metric conversions, knowing the basic unit is vital as every larger or smaller unit is a multiple of the basic unit. For instance:

• 1 meter (m) is the basic unit of length.
• 1 gram (g) is the basic unit of mass.
• 1 second (s) is the basic unit of time.

By converting units to these basic forms, you ensure accuracy and comparability in any calculation or experimentation.

SI Units

SI units, or the International System of Units, are globally accepted as the standard units of measurement. This system is designed to make communication and calculation easier regardless of where you are in the world. It includes units like meters, kilograms, seconds, and more.
SI units are based on basic units and unit prefixes.

• Meter (m) for distance.
• Kilogram (kg) for mass.
• Second (s) for time.

Using SI units, scientists and engineers can accurately exchange information and collaborate across different languages and regions. It keeps everyone on the same page, preventing confusion and errors in measurement.