Question

What number should replace the question mark in each of the following? (a) \(1 \mathrm{~cm}=? \mathrm{~m}\) (d) \(1 \mathrm{dm}=? \mathrm{~m}\) (b) \(1 \mathrm{~km}=? \mathrm{~m}\) (e) \(1 \mathrm{~g}=? \mathrm{~kg}\) (c) \(1 \mathrm{~m}=? \mathrm{pm}\) (f) \(1 \mathrm{cg}=? \mathrm{~g}\)

Short answer

\(1 \text{cm} = 0.01 \text{m}, 1 \text{dm} = 0.1 \text{m}, 1 \text{km} = 1000 \text{m}, 1 \text{g} = 0.001 \text{kg}, 1 \text{m} = 10^{12} \text{pm}, 1 \text{cg} = 0.01 \text{g}\).

Step-by-step solution

- Converting Centimeters to Meters

To convert centimeters to meters, divide the number of centimeters by 100 because 1 meter is equal to 100 centimeters. So, \(1 \text{cm} = \frac{1}{100} \text{m} = 0.01 \text{m}\).

- Converting Decimeters to Meters

To convert decimeters to meters, divide the number of decimeters by 10 because 1 meter consists of 10 decimeters. Thus, \(1 \text{dm} = \frac{1}{10} \text{m} = 0.1 \text{m}\).

- Converting Kilometers to Meters

To convert kilometers to meters, multiply the number of kilometers by 1000 since 1 kilometer equals 1000 meters. Consequently, \(1 \text{km} = 1 \times 1000 \text{m} = 1000 \text{m}\).

- Converting Grams to Kilograms

To convert grams to kilograms, divide the number of grams by 1000 because 1 kilogram is composed of 1000 grams. Hence, \(1 \text{g} = \frac{1}{1000} \text{kg} = 0.001 \text{kg}\).

- Converting Meters to Picometers

To convert meters to picometers, multiply the number of meters by \(10^{12}\) since 1 meter equals \(10^{12}\) picometers. As a result, \(1 \text{m} = 1 \times 10^{12} \text{pm} = 10^{12} \text{pm}\).

- Converting Centigrams to Grams

For converting centigrams to grams, divide the number of centigrams by 100 because 1 gram equals 100 centigrams. Therefore, \(1 \text{cg} = \frac{1}{100} \text{g} = 0.01 \text{g}\).

Key concepts

Unit Conversion

Unit conversion is a fundamental process in science, mathematics, and daily life, enabling us to compare and compute with different measurement systems. It involves changing a quantity expressed in one type of unit to another type, while keeping its value unchanged. For example, if you know the length of an object in centimeters and need to find it in meters, you'd divide the number by 100, because there are 100 centimeters in a meter.

Similarly, to convert decimeters to meters you divide by 10, and to convert kilometers to meters you multiply by 1000. For mass, converting grams to kilograms means dividing by 1000, and centigrams to grams involves dividing by 100 as well. Remember, to ensure accuracy in unit conversion, use the correct conversion factor for each unit category.

When doing unit conversions, it's essential to:

• Understand the relationship between the units involved.
• Know the correct conversion factors.
• Perform the necessary multiplication or division operation.

Following this structured approach empowers students to handle any conversion confidently.

Measurement Units

Measurement units serve as a standard for quantifying physical properties like length, mass, and volume. In our exercise, we're dealing with units such as centimeters (cm), meters (m), kilometers (km), grams (g), and kilograms (kg). These units belong to the metric system, which is based on powers of ten, making it straightforward to convert between them.

The metric system is universally used in science and most countries worldwide for its simplicity in conversions and consistency. Understanding how these units relate to each other is key to resolving problems involving them. For example, the prefix 'kilo' means one thousand units, so a kilometer is one thousand meters, and a kilogram is one thousand grams. Knowing these prefixes, and the underlying structure of the metric system aids in quick mental conversions and accurate computations.

The importance of understanding measurement units is not just for academic purposes but also for practical applications like cooking, construction, medicine, and almost any trade or profession.

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a concise form. It uses powers of ten to simplify numbers, so they become easier to work with, especially in calculations. In the expression \( a \times 10^n \), \( a \) is a number greater than or equal to 1 but less than 10, and \( n \) is an integer.

In our exercise, converting 1 meter to picometers involves scientific notation: \(1 \text{ m} = 1 \times 10^{12} \text{ pm}\), because a picometer is one trillionth of a meter. Scientific notation is not only indispensable in dealing with microscopic and astronomical distances but also in various fields of science where extreme precision and large scale differences are common.

Learning to use scientific notation can simplify complex calculations and help students understand and navigate the vast scales present in the universe, from the infinitesimal to the immense.