Problem 87 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

Based on the situation, the most appropriate number set for representing shoe sizes of students on campus is rational numbers.

Step 1: Understanding Number Sets

Natural numbers are numbers used for counting and ordering (1, 2, 3, ...). Whole numbers are similar to natural numbers, but also include zero (0, 1, 2, ...). Integers include all natural numbers, zero, and negative whole numbers (..., -3, -2, -1, 0, 1, 2, 3, ...). Rational numbers are those that can be expressed as a fraction, which means they include integers and fractions but do not include numbers with an infinite non-repeating decimal, like \(\pi\). Real numbers, on the other hand, include all rational numbers and irrational numbers, which can be represented on the number line.

Step 2: Analyzing the Situation

The context talks about shoe sizes. Shoe sizes typically start from size 1 and can be whole and some half numbers (like size 8.5) for more precise fitting. They do not include negative numbers which exclude integers. They also don't use numbers with non-terminating and non-repeating decimals, which exclude irrational, thereby, real numbers.

Step 3: Assign the Number Set

Given the information in step 2, the most appropriate number sets to represent shoe sizes would be whole numbers and rational numbers. However, the most specific and suitable set would be rational numbers since it allows for half-size representation.

Key Concepts

Natural Numbers

Natural numbers are often referred to as counting numbers. They begin from 1 and go on infinitely (1, 2, 3, 4, ...). If you're making a list of items, such as counting apples or students in a classroom, you'd be using natural numbers. It's important to remember that natural numbers do not include zero or negative numbers. They are simply for counting distinct, whole items.

Whole Numbers

Whole numbers build upon natural numbers by including zero. So, they are comprised of 0, 1, 2, 3, and so on. This makes them especially useful for scenarios where you might have none of an item, like stocking a cupboard till it’s empty. When talking about shoe sizes, considering only whole numbers would restrict sizes to full numbers like 7, 8, or 9, missing out on precise fits like 8.5.

Integers

Integers expand upon whole numbers by including negative numbers as well. The full set of integers can look something like ..., -3, -2, -1, 0, 1, 2, 3, .... They are particularly useful when you need to express losses or negative values, such as changes in bank balances. However, for shoe sizes, integers aren’t a fit because you can't have a negative shoe size. Only positive and sometimes fractional numbers are relevant to this context.

Rational Numbers

Rational numbers are incredibly flexible when it comes to representing different values. They include all integers, whole numbers, and fractions. Basically, any number that can be expressed as a ratio between two integers qualifies as a rational number. For example, 1/2 and 8.5 are both considered rational numbers. This makes them ideal for shoe sizes, as these can sometimes be precise, involving fractions or decimals (like 10.5). Rational numbers cover these needs without including infinitely non-repeating decimals, which are reserved for irrational numbers.