Problem 6 - 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.) $$ -\sqrt{5} $$

Step-by-Step Solution

Answer

\(-\sqrt{5}\) is classified as an irrational and real number.

Step 1: Determine Rational or Irrational

A number is considered rational if it can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\). To check if \(-\sqrt{5}\) is rational, try to express it in this form. However, since we cannot find integers \(a\) and \(b\) such that \(-\sqrt{5} = \frac{a}{b}\), we can conclude that \(-\sqrt{5}\) is an irrational number.

Step 2: Determine Real or Imaginary

A number is considered real if it can be located on the number line, which includes both rational and irrational numbers. On the other hand, a number is imaginary if it is the product of a real number and the imaginary unit \(i\). Since \(-\sqrt{5}\) is irrational without any imaginary unit multiplied, we can conclude that it is a real number.

Step 3: Conclusion

Based on the steps above, we can classify \(-\sqrt{5}\) as an irrational and real number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers, and the denominator is not zero. This means any number that fits the form \( \frac{a}{b} \), with \(a\) and \(b\) both integers, is considered rational. Let's look at some examples:

• The number \( \frac{1}{2} \) is rational because it’s a fraction with an integer numerator (1) and an integer denominator (2).
• Another example is 6, which can be expressed as \( \frac{6}{1} \), showing it's rational.

When dealing with decimals, rational numbers can either terminate or repeat. For example, 0.75 (which is \( \frac{3}{4} \)) or 0.333... (which is \( \frac{1}{3} \)) are also rational numbers.

In the exercise, \(-\sqrt{5}\) cannot be expressed as \(\frac{a}{b}\) with integers \(a\) and \(b\), which is why it isn't a rational number.

Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction of two integers. This means you cannot write them as \( \frac{a}{b} \), where both \(a\) and \(b\) are integers. Instead, their decimal form is non-repeating and non-terminating.

• Common examples of irrational numbers include \( \pi \) and \( e \).
• Also, square roots of non-perfect squares, such as \( \sqrt{2} \) and \( \sqrt{3} \), are irrational.

The number \(\sqrt{5}\) is irrational because its decimal representation goes on forever without repeating any pattern. Thus, \(-\sqrt{5}\) is also irrational. It is simply \( \sqrt{5} \) with a negative sign, maintaining the non-repeating decimal quality of an irrational number.

Real Numbers

Real numbers encompass all the numbers that we use in everyday life, including both rational and irrational numbers. They are the numbers that can be found on the number line. Real numbers do not include imaginary numbers, which involve the imaginary unit \(i\).

Let’s clarify what real numbers cover:

• Rational numbers such as 7, \( \frac{2}{3} \), and 0.5.
• Irrational numbers like \( \sqrt{2} \) and \( \pi \).

Real numbers do not include imaginary numbers, such as \( i \) or 3\(i\), where they can't be placed on a typical number line. Because \(-\sqrt{5}\) can be plotted as a point on the number line, it is indeed a real number. This combination of being real yet not expressible as a fraction places it firmly in the category of irrational and real numbers.