Problem 147 - College Algebra

Question

Putting Numbers into Perspective. A large number can be put into perspective by comparing it with another number. For example, we put the \(\$ 15.2\) trillion national debt (Example 12 ) and the \(\$ 2.17\) trillion the government collected in taxes (Exercise 115 ) by comparing these numbers to the number of U.S. citizens. For this project, each group member should consult an almanac, a newspaper, or the Internet to find a number greater than one million. Explain to other members of the group the context in which the large number is used. Express the number in scientific notation. Then put the number into perspective by comparing it with another number.

Step-by-Step Solution

Answer

The selected large number is 5 billion (chickens in the world), expressed in scientific notation as \(5.0 \times 10^{9}\). When compared with the human population of \(7.6 \times 10^{9}\), it's clear that there are fewer humans on Earth than chickens.

Step 1: Find a Large Number

Start by finding a large number. Let's choose 5 billion, which is the estimated number of chickens in the world.

Step 2: Express the Number in Scientific Notation

Express the number in scientific notation. Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. To convert 5 billion to scientific notation, we write it as \(5.0 \times 10^{9}\).

Step 3: Compare with Another Number

Now, put the number into perspective by comparing it with another number. Let's select the total global human population, which is estimated at 7.6 billion (or \(7.6 \times 10^{9}\) in scientific notation). By comparing these two numbers, it's clear that there are fewer humans on Earth than chickens.

Key Concepts

Understanding Large Numbers

When dealing with very large numbers, like billions and trillions, it can be hard to grasp exactly how big they are. These numbers often come up in contexts such as national debt figures or global population estimates. To make them easier to understand and work with, it's helpful to use scientific notation.

• Scientific notation allows you to express large numbers in a compact form by using powers of ten.
• For instance, 5 billion, which is 5,000,000,000 in decimal form, can be converted to scientific notation as \(5.0 \times 10^9\).
• This method not only simplifies calculations but also helps in comparing these numbers more effectively.

Breaking down these massively large numbers into scientific notation makes them easier to read and comprehend, especially in scientific and academic contexts.

Comparing Numbers Accurately

Comparing numbers in scientific notation is straightforward and can provide insight into relative magnitudes. When numbers are in the form of powers of ten, you can immediately see if one number is larger or smaller. Let's look at how to compare two large quantities effectively:

• Consider two numbers: the world's chicken population at \(5.0 \times 10^9\) and the global human population at \(7.6 \times 10^9\).
• Even without doing complex calculations, you can observe that \(7.6 \times 10^9\) is greater than \(5.0 \times 10^9\).
• This comparison shows that there are more humans than any single large group like chickens.

Using scientific notation helps highlight these differences quickly, making such comparisons straightforward and effective in quantitative discussions.

Applying College Algebra Concepts

College algebra often involves manipulating and comparing large numbers, so understanding scientific notation is crucial. This skill is particularly useful for students who deal with mathematical models or large data sets in their coursework. Here's why college algebra students should be familiar with these concepts:

• Converting numbers between different forms (such as decimal to scientific notation) is a key algebraic skill.
• When working with equations that involve large numbers, accurate manipulation and comparison are necessary for correctness.
• Algebraic problems often require estimation, and scientific notation aids in making rough calculations more manageable.

Having a handle on large numbers and their comparisons using scientific notation empowers students to tackle a range of mathematical challenges, from finance to data science, within their studies.