Problem 4 - Astronomy: At Play in the Cosmos

Question

One number is said to be an "order of magnitude" larger than another number if a. it is 2 times larger. d. it is 10 times larger. b. it is 3 times larger. e. it is 100 times larger. c. it is 5 times larger.

Step-by-Step Solution

Answer

It is 10 times larger (option d).

Step 1: Understanding the Concept of Order of Magnitude

An 'order of magnitude' refers to the class of scale or magnitude of any quantity, where each class is a factor of 10. This means, when we talk about one number being an order of magnitude larger, we mean it is 10 times larger.

Step 2: Review the Answer Options

Look at the options provided: (a) 2 times larger, (b) 3 times larger, (c) 5 times larger, (d) 10 times larger, and (e) 100 times larger. Knowing that 'order of magnitude' means 10 times larger, we identify the correct answer among these choices.

Step 3: Select the Correct Answer

The correct answer is (d), as 10 times larger is the definition of being one order of magnitude larger than another number.

Key Concepts

Scale of Magnitude

The term 'scale of magnitude' is often used to describe the relative size or extent of something, measured in a consistent way. When we talk about a scale of magnitude in mathematics or science, we refer to how much larger one quantity is compared to another. A fundamental unit for measuring these scales is the factor of 10. This makes things easier because instead of dealing with bulky numbers, we use a base-10 system, which allows for simple comparisons.
For example, a comparison where one number is 10 times larger than another is said to differ by one order of magnitude. This kind of scale is useful in many fields, as it helps compare vastly different quantities easily and quickly. It offers a way to grasp huge ranges of numbers without getting overwhelmed. Whether you're dealing with distances in astronomy or sizes of populations in biology, understanding the scale of magnitude is crucial.

Factor of 10

The 'factor of 10' is a simple but powerful concept. In essence, it represents multiplication by 10. Each time you move an 'order of magnitude' up, you multiply by 10. Likewise, moving down by an 'order of magnitude' means dividing by 10.
The beauty of this system lies in its simplicity. By using factors of 10, you can easily perform calculations and make rough estimates without needing a calculator. This method is not only used in scientific notation but is also helpful in everyday life. For instance, understanding that something is a 'factor of 10' stronger or weaker can provide immediate insight into its actual importance or risk.

Understanding Numerical Scale

Understanding numerical scale means appreciating how numbers relate to each other in terms of size and importance. By practicing how to think in orders of magnitude, it becomes easier to evaluate the real-world significance of numbers quickly.
Numerical scale considers factors like base measurements and units that help bridge the gap between small and large numbers. For instance, kilometers vs. meters or seconds vs. milliseconds. Having a grasp of these scales helps in converting numbers from one unit to another, maintaining their context and feasibility.

• Consider a city population of 1,000,000 people compared to a small town of 10,000 people. Here, the city is two orders of magnitude larger than the town.
• A bacterium might be measured in micrometers, whereas a human is measured in meters, showcasing the concept of orders of magnitude in biological scales.

These insights into numerical scales enable better communication and understanding across various fields and situations.