Section 1-5: Critical Thinking and Number Sense

Welcome back to Professor Baker's Math Class! In this section, we'll be tackling Section 1-5, focusing on critical thinking and number sense. This is all about understanding the numbers we encounter every day and using them effectively.

Learning Objectives

  • Coping with Measurements: Understand and interpret the myriad measurements the average consumer encounters daily.
  • Magnitudes: Grasp the relative sizes of numbers, particularly very large and very small numbers.
  • Powers of 10: Effectively use powers of 10 to represent and manipulate numbers.
  • Estimation: Develop and apply estimation techniques to simplify calculations and make informed decisions.

Understanding Powers of 10

Powers of 10 are your friends when dealing with very large or very small numbers. Let's review some examples:

  • Positive Powers of 10:
    • $10^3 = 1,000$ (one thousand)
    • $10^6 = 1,000,000$ (one million)
    • $10^9 = 1,000,000,000$ (one billion)
    • $10^{12} = 1,000,000,000,000$ (one trillion)
  • Negative Powers of 10:
    • $10^{-2} = 0.01$ (one hundredth)
    • $10^{-3} = 0.001$ (one thousandth)
    • $10^{-6} = 0.000001$ (one millionth)
    • $10^{-9} = 0.000000001$ (one billionth)

Quick Review: Exponents

Let's quickly review some exponent rules that will be useful:

  • Negative Exponents: $a^{-n} = \frac{1}{a^n}$. For example, $10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001$
  • Zero Exponent: If $a \neq 0$, then $a^0 = 1$.

Basic properties of exponents:

  • $a^p a^q = a^{p+q}$. For example, $10^2 \times 10^3 = 10^{2+3} = 10^5 = 100,000$
  • $\frac{a^p}{a^q} = a^{p-q}$. For example, $\frac{10^6}{10^4} = 10^{6-4} = 10^2 = 100$
  • $(a^p)^q = a^{pq}$. For example, $(10^3)^2 = 10^{3 \times 2} = 10^6 = 1,000,000$

Real-World Examples

Let's look at some examples of how these concepts apply to real-world situations:

Example 1: Comparing Computer Memory

Imagine a computer from the 1980s with 64 kilobytes of memory, compared to a modern computer with 4 gigabytes. How much larger is the memory of today's computer?

Solution:

New memory size / Old memory size = $\frac{4 \times 10^9 \text{ bytes}}{64 \times 10^3 \text{ bytes}} = \frac{4}{64} \times 10^{9-3} = 62,500$.

The new computer has over 60,000 times as much memory as the old one!

Example 2: Understanding National Debt

If the national debt is $10.6 trillion and there are 305 million people in the U.S., how much does each person owe?

Solution:

Debt per person = $\frac{10.6 \times 10^{12}}{305 \times 10^6} = \frac{10.6}{305} \times 10^6 = 0.0348 \times 10^6 = $34,800$.

Example 3: Estimating Costs

You're traveling to Canada. Gas is $3.77/gallon in the U.S. and $1.10/liter in Canada. Is gas cheaper in Canada?

Solution: 1 Canadian dollar is about 1 U.S. dollar and 1 liter costs 1.10 Canadian dollars, gas costs about 1 U.S. dollar per liter. A quart is about a liter, there are about 4 liters in a gallon, gas in Canada costs about 4 U.S. dollars per gallon. Thus, gas is cheaper at the station in the United States.

Practice Problems

Check out the Section 1-5 assignment for practice problems and use the provided answers to check your work. Remember, practice makes perfect! Keep exploring and questioning the numbers around you. You've got this!