Problem 12 - College Algebra with Modeling and Visualization

Question

For the measured quantity, state the set of numbers that most appropriately describes it. Choose from the natural numbers, integers, and rational numbers. Explain your answer Populations of states

Step-by-Step Solution

Answer

Populations are best described by the set of natural numbers.

Step 1: Understand Population Count

When considering population, it's important to recognize that we are counting the number of individuals in a state. This type of number will always be a positive whole number or zero, as you can't have a negative count or a fraction of a person in the population.

Step 2: Identify Suitable Number Set

Given that population is a count of people, the natural numbers most appropriately describe populations. Natural numbers are the set of all positive integers starting from 1, but since populations can also be zero, we need to think about whether there's a more suitable set.

Step 3: Compare with Other Sets

The integers include all whole numbers, both positive and negative, but negative numbers do not make sense in the context of populations. Rational numbers include fractions and decimals, which are also inappropriate because population counts cannot be fractional.

Step 4: Choose the Appropriate Set

The most suitable set to describe populations is the set of natural numbers, considering that these numbers can describe all possible positive counts and zero, depending on the definition being used (including zero or starting from one). If zero needs to be considered, some definitions of natural numbers may include it.

Key Concepts

Natural Numbers

Natural numbers are the simplest and most familiar set of numbers used in counting. They begin from 1 and increase by 1 each time, extending indefinitely without any upper limit. In certain contexts, especially in set theory or modern mathematics, natural numbers may also include 0.
This is important when we discuss countable things, such as people in a population, because populations cannot be negative or fractional.

• Natural numbers: 1, 2, 3, 4, 5, ...
• Sometimes include 0: 0, 1, 2, 3, 4, ...

Counting populations requires whole numbers. Since you cannot have half a person or a negative number of people, natural numbers fit perfectly.
They allow for the flexibility needed to define the starting point of a population count without confusion, supporting both the exclusion or inclusion of zero.

Integers

Integers are a broader category than natural numbers. They include all whole numbers along with their negative counterparts. This means integers encompass positive numbers, zero, and negative numbers, forming a series like this: ...-3, -2, -1, 0, 1, 2, 3,....
While integers are crucial in various mathematical operations, their application to describing population counts is limited.

• Integers include negative numbers, which do not make sense for populations.
• Population never involves negative whole numbers, rendering this set inappropriate for this context.

Despite their importance in many mathematical fields, such as algebra and arithmetic, integers are not necessary when dealing with populations. This is because population figures only reflect non-negative counts of people, aligning more closely with natural numbers.

Rational Numbers

Rational numbers comprise a large set of numbers that includes all the fractions and decimals, alongside integers themselves. In other words, a rational number can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
Rational numbers allow for precise measurements and divisions, which are essential in areas such as engineering, science, and statistics.

• Numbers like \( \frac{1}{2} \), \( 2.75 \), and 0.333... are rational numbers.
• They express quantities that fall between integers, offering a continuous range of values.

However, when it comes to population, rational numbers are unnecessary. Populations cannot be fractional, as individuals are counted in whole numbers בלבד. In this context, rational numbers introduce complexity without benefit, because no population will be accurately depicted by a fraction.