Problem 88 - Introductory Algebra for College Students

Question

Determine whether natural numbers, whole numbers, integers, rational numbers, or all real numbers are appropriate for each situation. Recorded heights of students on campus

Step-by-Step Solution

Answer

The suitable set to represent the recorded heights of students on campus is \'rational numbers\' as they include all positive fractions and decimal numbers

Step 1: Look at Types of Numbers

Begin by assessing which types of numbers make sense for the situation. The heights of students can't be negative, so it rules out sets that include negative numbers like integers. Also, because height is often measured with precision, it is reasonable to believe the measurements could include decimals.

Step 2: Eliminate Inappropriate Number Sets

Since natural numbers and whole numbers only consist of non-negative numbers without decimal places, it would not fit the need for precise measurements. Real numbers, on the other hand, contain all numbers, including irrational numbers which means numbers that can't be expressed as fractions. This doesn't make sense when talking about the heights.

Step 3: Chose Appropriate Number Set

The only set remaining, that best fits the context of the problem and includes fractional or decimal values is the set of rational numbers.

Key Concepts

Natural Numbers

Natural numbers are the simplest set of numbers that we typically use when counting basic objects. They consist of all the positive numbers such as 1, 2, 3, and so on. These numbers do not include fractions, decimals, or negative numbers.
In everyday life, natural numbers are often used in situations where counting is involved. For example, counting apples in a basket or counting steps taken in a day.
It's important to note that natural numbers start from 1 and do not include zero as some people might incorrectly assume. They form the foundation of more complex number sets.

Integers

Integers expand upon the concept of natural numbers to include zero and negative numbers. This means that integers are whole numbers that can be positive, negative, or zero.
For instance, -3, 0, and 7 are all integers. When dealing with situations like temperature measurement in a city, integers become important because temperatures can be below zero.
Though integers expand the counting process beyond natural numbers, they still do not accommodate fractions or decimals. This makes them unsuitable for situations where precision in measurement is essential.

Rational Numbers

Rational numbers include any number that can be expressed as a fraction or ratio. This means they comprise numbers that can be written as the quotient of two integers, where the denominator is not zero.

• Examples include 1/2, 4, and 0.75
• They can be whole numbers or have fractional parts

Rational numbers cover all situations where precision with decimal or fractional values is needed.
When recording heights of students, this set captures any number that includes fractions, making them ideal for such precise measurements. This set effectively meets our needs when both whole numbers and fractions are to be accounted for.

Real Numbers

The set of real numbers is vast, containing both rational and irrational numbers. Every number that can exist on a continuous number line is a real number.
Rational numbers are part of this set, along with irrational numbers like π (pi) and √2, which cannot be exactly expressed as simple fractions.
Real numbers are crucial for comprehensive measurements and calculations but may include unnecessary complexity where irrational numbers are not needed, such as in recording student heights. However, they remain important in higher mathematics and physics for covering more abstract concepts and measurements.