Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Classify the number as to type. (For example, \(\frac{1}{2}\) is rational and real, whereas \(\sqrt{5}\) is irrational and real.) $$ \frac{3}{8} $$

Step-by-Step Solution

Answer

The number \(\frac{3}{8}\) is rational and real.

Step 1: Determine if the number is rational or irrational

The given number is \(\frac{3}{8}\). A number is considered rational if it can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b\) is non-zero. Since \(\frac{3}{8}\) can be expressed in this form (with \(a\) being \(3\) and \(b\) being \(8\)), it is a rational number.

Step 2: Determine if the number is real or imaginary

A number is considered real if it can be represented on the number line. As \(\frac{3}{8}\) is a rational number, it can be represented on the number line. Thus, it is a real number. In conclusion, the given number \(\frac{3}{8}\) can be classified as a rational and real number.

Key Concepts

Rational Numbers

Rational numbers are those numbers that can be written as a fraction, or a ratio, of two integers. These fractions are in the form \(\frac{a}{b}\), where both \(a\) and \(b\) are integers and \(b\) is not zero. This means every whole number, negative or positive integers, and decimals that repeat or terminate can be classified as rational numbers. For example:

• The number \( \frac{3}{8} \) is rational because it can be expressed as a fraction with integers.
• The number \(4\) is rational because it can be written as \(\frac{4}{1}\).
• Repeating decimals such as \(0.333...\), which can be represented as \(\frac{1}{3}\), and terminating decimals like \(2.75\) that can be written as \(\frac{11}{4}\), are also rational.

Rational numbers cover a wide range of values that we use often in everyday math when counting, performing basic arithmetic, and scaling measurements. Since they can be expressed as fractions, they easily represent parts of a whole.

Irrational Numbers

Irrational numbers are quite the opposite of rational numbers; they can't be easily expressed as a fraction of two integers. Instead, they are represented by non-repeating and non-terminating decimals. Examples include numbers like \(\sqrt{2}\) or \(\pi\).

• The square root of a non-perfect square - like \(\sqrt{5}\) - is irrational because it cannot be written exactly as a fraction.
• Mathematical constants such as \(\pi\), which continue infinitely without repetition, are irrational because there isn't a precise fraction representing it.

These numbers often arise in geometry, algebra, and trigonometry, challenging us because they can't be simplified into neat, tidy fractions. Despite being less intuitive, they are just as real as rational numbers!

Real Numbers

Real numbers encompass both rational and irrational numbers. This means any number that can be laid down on the number line is a real number. Simply put, they include every number that you might typically encounter in daily life, as well as those you'd find in more advanced mathematics.

• Rational numbers like \(\frac{3}{8}\) and \(2\) are real because they can be plotted on the number line.
• Irrational numbers, such as \(\sqrt{5}\) or \(\pi\), though more complex, also exist on the number line alongside rational numbers.

A handy way to think about real numbers is to visualize the entire stretch from negative infinity to positive infinity on the number line, filling the space where both rational and irrational numbers live. This broad category does not include imaginary numbers, which are a more abstract concept used in advanced fields like engineering and physics.'