Problem 88 - Introductory and Intermediate Algebra for College Students 4th

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

Rational numbers (\(Q\)) are likely the most appropriate to describe the recorded heights of students on campus.

Step 1: Analyze the Physical Situation

In measuring the height of a person, it can be a fraction of a unit (e.g. meters, centimeters, inches), and more likely to be non-negative. Usually, students' heights cannot be exactly measured to the last possible decimal due to limits of the measuring device, so although theoretically they could be any real number, in practical measurements, they are usually confined to rational numbers.

Step 2: Examine the Types of Number Sets

Here are the number sets: Natural numbers (\(N\)) are all positive integers (e.g. 1, 2, 3...). Whole numbers (\(W\)) include all natural numbers plus zero (0). Integers (\(Z\)) include all whole numbers and their negatives (...-3, -2, -1, 0, 1, 2, 3...). Rational numbers (\(Q\)) are any numbers that can be written as a fraction \(\frac{a}{b}\) where \(a, b\) are integers and \(b \neq 0\). Real numbers (\(R\)) include all numbers that can be plotted on a number line.

Step 3: Match the Practical Situation with the Suitable Number Set

In context, since heights are always non-negative, negative numbers and zero can be ruled out, eliminating integers from consideration. Height can be fractional, so natural numbers and whole numbers don't fit. Although heights could be any real number, in practical measurements, they are usually rational due to limits of the measuring device.

Key Concepts

Understanding Real Numbers

In mathematics, real numbers are a vital concept that encompasses almost the entire set of numbers we commonly use. Simply put, they include all numbers that can be represented on a number line. This means real numbers consist of:

• Whole numbers (e.g., 0, 1, 2, 3,...)
• Integers, which are whole numbers and their negatives (...-3, -2, -1, 0, 1, 2, 3...)
• Fractions (numbers like 1/2, 2/3)
• Decimals (numbers such as 3.14, 2.71)
• Irrational numbers, numbers that cannot be expressed as simple fractions (like the square root of 2 or π)

Every real number can be pinpointed exactly on a number line, which makes them foundational in both pure and applied mathematics. However, not all problems in mathematics require all of these numbers. Real numbers are used in situations where the exact point on a continuum is necessary, like measuring certain scientific phenomena or distances. Understanding how real numbers work forms the basis of many advanced mathematical concepts.

Measuring Height and Choosing Number Sets

Measuring height, whether in centimeters, inches, or meters, often involves numbers that are not whole. For example, a person might be 5 feet and 7.5 inches tall or 168.3 centimeters. When measuring something like height, we often deal with:

• Decimal numbers
• Fractions

This is because measuring tools often have finite precision, meaning they limit how exactly the measurement can be made. Though theoretically, height could take any real number value, practically, it is mostly expressed as rational numbers.

Rational numbers sufficiently capture the measurement because they allow for precise readings which are fractions or decimals. In accurate height measurements, continuous real number representation is not typically necessary; rational approximations serve well.

Rational Number Sets

Rational numbers fit into a specific number set known as set \(Q\). Each number in this set can be expressed as a fraction, written as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). This means:

• Every integer is a rational number since it can be written as itself over one, like \(-3 = \frac{-3}{1}\)
• Decimals that terminate or repeat are rational numbers, like 0.75 (since \(0.75 = \frac{3}{4}\)

Rational numbers are crucial in situations requiring precision but still confined to the range of fractional or decimal values. They include almost all calculation outputs found in simple computations like dividing measurements or monetary amounts. In practical contexts, such as recording heights, using rational numbers provides a close enough approximation without the complication of non-repeating decimals or needing infinite precision.

Exploring Number Sets in Mathematics

Mathematics categorizes numbers into various sets which helps in analyzing different problems.Here's a quick look at some primary sets:

• Natural Numbers (N): Begins from 1 and goes up (e.g., 1, 2, 3...); useful in counting.
• Whole Numbers (W): All natural numbers plus zero (e.g., 0, 1, 2, 3...); used in measuring quantities.
• Integers (Z): Extends whole numbers to include negatives (e.g., ...-2, -1, 0, 1, 2...); covers more comprehensive calculations.
• Rational Numbers (Q): All numbers that can exist as \(\frac{a}{b}\); encompass fractions and repeating/terminating decimals.
• Real Numbers (R): An extended set including all rational numbers and irrational numbers.

Understanding these sets helps in selecting the right category for different operations and situations. Each reading, like measurements of height or calculating interest, maps efficiently indoors different number sets depending on complexity and accuracy required.