Problem 18 - A Graphical Approach to College Algebra

Question

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers. Golf scores relative to par

Step-by-Step Solution

Answer

The integer set is most appropriate for golf scores relative to par.

Step 1: Understand Golf Scores Relative to Par

Golf scores relative to par can be negative, zero, or positive numbers. The score indicates how many strokes above or below par a golfer has taken for the course.

Step 2: Identify the Type of Numbers Needed

To express golf scores, you need a set of numbers that includes negative numbers, zero, and positive numbers. This means that the natural numbers are not sufficient since they start from 1 and only include positive numbers.

Step 3: Choose the Integer Set

The appropriate set of numbers for describing golf scores relative to par is the integers. This set includes all whole numbers, both positive and negative, as well as zero, which allows representation of scores like -3, 0, and +5.

Key Concepts

natural numbers

Natural numbers are the most basic set of numbers that we often learn about first in mathematics. These are the numbers used when counting objects in a straightforward, step-by-step manner. Typically, natural numbers begin at 1 and go on indefinitely: 1, 2, 3, 4, and so on.
Natural numbers do not include 0 or any negative numbers. They are used when we want to count the number of items, such as the number of apples in a basket, the number of students in a classroom, or any other scenario where whole, positive numbers are appropriate.

• They start from 1: 1, 2, 3, 4,...
• Used for counting and ordering.
• Do not include zero or negative numbers.

A perfect example of their use is in situations where no concept of zero or below zero needs to be represented, such as counting population or items that cannot inherently be negative.

rational numbers

Rational numbers are a bit different from natural numbers as they represent a broader set of possibilities. A rational number is any number that can be expressed as the fraction of two integers where the denominator is not zero. This means any number that can be written as \(rac{a}{b}\) where both \(a\) and \(b\) are integers, and \(b eq 0\).
Rational numbers include more than just whole numbers, they also account for fractions and decimals, both terminating and repeating, like 0.5, -3, \(rac{1}{3}\), and 2.75.

• Expressed as fractions: \(rac{p}{q}\) where both \(p\) and \(q\) are integers.
• Include fractions, integers, and finite or repeating decimals.
• Can be positive, negative, or zero.

This makes rational numbers very versatile, allowing them to describe a wide array of real-world scenarios where precision through fractions is necessary, such as dividing a pizza into equal parts or measuring distances with partial units.

whole numbers

Whole numbers are a familiar and friendly set of numbers, which you can think of as natural numbers with the addition of zero. They form the sequence: 0, 1, 2, 3, 4, and so on, extending infinitely.
This set is particularly useful when you need to include the concept of zero. Zero is often used to indicate the absence of something, which is crucial in many practical situations.

• Start at zero: 0, 1, 2, 3,...
• Includes all positive integers and zero.
• No negative numbers.

Whole numbers are ideal in situations like calculating savings, zero balance cases, or counting where zero is a meaningful number. They are used when you need a non-negative result, such as in calculating age or floor numbers in buildings.