Problem 7 - A Graphical Approach to College Algebra

Question

Classify each number as one or more of the following: natural number, integer, rational number, or real number. \(-25\) (The percent change in the number of Yahoo searches from 2011 to 2012 )

Step-by-Step Solution

Answer

-25 is an integer, rational, and real number.

Step 1: Identify Natural Numbers

Natural numbers are positive integers starting from 1, such as 1, 2, 3, and so on. Since \( -25 \) is a negative number, it does not fit the classification of a natural number.

Step 2: Determine if It's an Integer

Integers include all whole numbers, both positive and negative, as well as zero. Since \( -25 \) is a negative whole number, it is classified as an integer.

Step 3: Check for Rational Number Classification

A rational number can be expressed as a fraction of two integers where the denominator is not zero. \( -25 \) can be written as \( \frac{-25}{1} \). Therefore, \( -25 \) is a rational number.

Step 4: Verify Classification as a Real Number

Real numbers include all rational and irrational numbers. Since \( -25 \) is a rational number, it is automatically classified as a real number.

Key Concepts

Natural Numbers

Natural numbers are the numbers we naturally count with. They start from 1 and go on as 1, 2, 3, etc. Natural numbers are always positive and do not include zero or negatives. This makes them the simplest number category and they are often used in everyday counting, like counting apples or books. For instance,

• 1: First place
• 5: Five fingers
• 10: Ten toes

Natural numbers are limited to whole, positive numbers only. As a result, any negative number, like -25, cannot be classified as a natural number.

Integers

Integers expand upon the set of natural numbers, including zero, positive whole numbers, and their negatives. So, while natural numbers are part of integers, integers also contain negative numbers and zero. Here’s a simple breakdown:

• Positive integers: 1, 2, 3, 4, ...
• Negative integers: -1, -2, -3, -4, ...
• Zero: 0

For example, imagine measuring temperatures. If it's 5 degrees above or below zero, both are integers. Therefore, a number like -25 fits this category easily because it is a negative whole number.

Rational Numbers

Rational numbers include any number that can be expressed as the fraction of two integers, where the denominator isn’t zero. In essence, if you can write a number in the form \( \frac{a}{b} \), with both \( a \) and \( b \) as integers and \( b eq 0 \), it is rational.

• Examples of rational numbers: \( \frac{1}{2}, \frac{-3}{4}, 5 \text{ (as } \frac{5}{1}), 0 \text{ (as } \frac{0}{1}) \)
• -25 fits because \( \frac{-25}{1} \) shows it can be expressed as a ratio of two integers.

Rational numbers cover a wide realm, not just integers but also fractions and decimals that end or repeat.

Real Numbers

Real numbers are the broadest type of numbers you often encounter. They include all numbers that can lie on the number line, encompassing both rational and irrational numbers. If it exists on the line drawn in real life, it’s a real number, such as:

• Rational numbers: 3, -5.2, \( \frac{14}{3} \)
• Irrational numbers: \( \pi \), \( \sqrt{2} \)

Everything from counting numbers to infinite decimals lies within the realm of real numbers. Because every rational number is also a real number, a number like -25 isn't just an integer or rational, but also a real number, illustrating the expansive nature of this category.