Problem 2 - College Algebra with Modeling and Visualization

Question

Classify the number as one or more of the following: natural number, integer, rational number, or real number. \(20,082\) (Average cost in dollars of taition and fees at a private college in 2004 )

Step-by-Step Solution

Answer

20,082 is a natural number, integer, rational number, and real number.

Step 1: Understanding Natural Numbers

Natural numbers are the set of positive counting numbers starting from 1, 2, 3, and so on. Since 20,082 is a positive whole number, it qualifies as a natural number.

Step 2: Understanding Integers

Integers include all whole numbers, both positive and negative, as well as zero. The number 20,082 is a positive whole number, so it is also an integer.

Step 3: Understanding Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. The number 20,082 can be written as the fraction \( \frac{20082}{1} \), so it is also a rational number.

Step 4: Understanding Real Numbers

Real numbers include all the numbers on the number line, encompassing natural numbers, integers, rational numbers, and irrational numbers. Since 20,082 is both a natural, integer, and rational number, it is also a real number.

Key Concepts

Natural Numbers

Natural numbers are perhaps the simplest category in the number classification system. They are the basic set of positive counting numbers starting from 1, progressing to 2, 3, and so on. These numbers do not include zero, fractions, decimals, or negative numbers. They are the numbers we use in everyday counting like when listing items or determining quantity. In the context of the exercise, the number 20,082 fits perfectly into this category as it is a positive whole number, making it a natural number.

Integers

Integers broaden the family of numbers to include zero and negative whole numbers. This category consists of positive numbers (like natural numbers), their negative counterparts, and zero. To visualize, integers are evenly spaced on both sides of a number line with zero in the center. Numbers like -3, 0, 7, and indeed 20,082 are all integers. Hence, 20,082 is classified as an integer because it is a part of this important group of numbers that are essential in mathematics for operations involving subtraction, moving in opposite directions on the number line, etc.

Rational Numbers

Rational numbers encompass numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In simpler terms, if a number can be written as a fraction

• with both the numerator and the denominator as integers
• and the denominator not equalling zero,

then it is a rational number.
20,082 can be comfortably expressed as a fraction, for example, \[ \frac{20082}{1} \].
Thus, it is a rational number, as the definition suits it perfectly. Rational numbers are vital because they represent the complete spectrum of possible division results between integers, allowing more flexibility than plain integers.

Real Numbers

The realm of real numbers is extensive. It includes all the categories mentioned earlier:

• natural numbers
• integers
• rational numbers

It's a much graced family in mathematics as it also includes irrational numbers, which are numbers that cannot be precisely written as fractions of integers. Practically, every number you've encountered in basic mathematics comes under real numbers.
So, quite naturally, when a number is rational like 20,082, it is automatically a real number. Real numbers form the basis of most mathematical analysis, enabling us to tackle complex problems in both theoretical and practical scenarios.