Identify The Non Irrational Number Among Cube Root 80, Pi, 2.87 Repeating, And Square Root 59

In the fascinating world of mathematics, numbers are broadly classified into two major categories: rational and irrational. Understanding the distinction between these two types of numbers is crucial for grasping various mathematical concepts and solving complex problems. Rational numbers are those that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. This means they can be written as terminating or repeating decimals. On the other hand, irrational numbers cannot be expressed in this form; their decimal representations are non-terminating and non-repeating. This article delves into the intricacies of rational and irrational numbers, focusing on identifying which values fall into each category. We will dissect the given options – $\sqrt[3]{80}$, π, $2.8\overline{7}$, and $\sqrt{59}$ – to determine which one is not an irrational number.

To truly understand which of the given values is not an irrational number, it's essential to first have a solid grasp of what irrational numbers are. Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. This is the core concept that sets them apart from rational numbers. The decimal representation of an irrational number is infinite and non-repeating, meaning the digits after the decimal point go on forever without any discernible pattern or repetition. Common examples of irrational numbers include the square root of any non-perfect square (such as $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$), and the mathematical constant pi (π), which represents the ratio of a circle's circumference to its diameter. Pi is famously known for its non-repeating decimal expansion, starting with 3.14159 and continuing infinitely without a repeating pattern. Another important characteristic of irrational numbers is that they arise frequently in geometry, trigonometry, and calculus. Their presence is fundamental to understanding concepts such as the lengths of diagonals in squares (involving $\sqrt{2}$) or the circumference and area of circles (involving π). Recognizing these key traits – non-fractional representation and non-repeating decimal expansion – is paramount in identifying irrational numbers amidst other numerical values. This detailed understanding will aid us in analyzing the given options and accurately determining which one deviates from the irrational category.

Now, let's meticulously examine each of the provided values to pinpoint the one that does not classify as an irrational number. This involves a careful assessment of their mathematical properties and decimal representations.

A. $\sqrt[3]{80}$ – The Cube Root Investigation

The first value we encounter is the cube root of 80, expressed as $\sqrt[3]{80}$. To ascertain whether this number is rational or irrational, we need to consider its prime factorization and whether it can be simplified into a form that reveals its nature. The prime factorization of 80 is $2^4 \times 5$. Thus, $\sqrt[3]{80}$ can be rewritten as $\sqrt[3]{2^4 \times 5}$, which further simplifies to $2 \sqrt[3]{2 \times 5}$ or $2 \sqrt[3]{10}$. Since 10 is not a perfect cube (i.e., it doesn't have an integer cube root), the cube root of 10, denoted as $\sqrt[3]{10}$, is an irrational number. Consequently, multiplying this irrational value by 2 still results in an irrational number. Therefore, $\sqrt[3]{80}$ is indeed an irrational number because it cannot be expressed as a simple fraction and its decimal representation is non-terminating and non-repeating. This detailed analysis confirms that option A, $\sqrt[3]{80}$, fits the definition of an irrational number.

B. π – The Transcendental Constant

Next, we have π (pi), one of the most well-known mathematical constants. Pi is defined as the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. However, the decimal representation of pi extends infinitely without any repeating pattern. This characteristic is the hallmark of an irrational number. In fact, pi is not just irrational but also transcendental, meaning it is not a root of any non-zero polynomial equation with rational coefficients. This places it firmly in the category of irrational numbers. Its significance spans various fields of mathematics, physics, and engineering, making it a fundamental constant. Therefore, π unequivocally qualifies as an irrational number due to its infinite and non-repeating decimal representation. The nature of pi as an irrational number is well-established and widely recognized in the mathematical community.

C. $2.8\overline{7}$ – The Repeating Decimal

The third value to consider is the repeating decimal $2.8\overline{7}$. The overline above the digit 7 indicates that this digit repeats infinitely, meaning the number is 2.877777.... To determine whether this number is rational or irrational, we need to check if it can be expressed as a fraction p/q, where p and q are integers. A repeating decimal can indeed be converted into a fraction, which is a key property of rational numbers. Let's demonstrate this conversion: Let x = 2.87777.... Multiply both sides by 10 to shift the decimal one place to the right: 10x = 28.7777.... Next, multiply both sides by 100 to shift the decimal two places to the right: 100x = 287.7777.... Now, subtract the first equation (10x = 28.7777...) from the second equation (100x = 287.7777...): 100x - 10x = 287.7777... - 28.7777... This simplifies to 90x = 259. Finally, divide both sides by 90 to solve for x: x = 259/90. Thus, the repeating decimal $2.8\overline{7}$ can be expressed as the fraction 259/90, where both 259 and 90 are integers. This definitively shows that $2.8\overline{7}$ is a rational number, as it meets the criterion of being expressible as a fraction. This outcome sets it apart from irrational numbers, which cannot be represented in such a way. Consequently, $2.8\overline{7}$ is our candidate for the value that is not an irrational number.

D. $\sqrt{59}$ – The Square Root Examination

Lastly, we need to analyze $\sqrt{59}$, the square root of 59. To determine its rationality, we need to assess whether 59 is a perfect square. A perfect square is an integer that is the square of another integer (e.g., 9 is a perfect square because $3^2$ = 9). The number 59 is not a perfect square; it falls between the perfect squares 49 ($7^2$) and 64 ($8^2$). Therefore, the square root of 59 cannot be expressed as an integer. When the square root of a non-perfect square is calculated, the result is a non-terminating and non-repeating decimal, which is the defining characteristic of an irrational number. Hence, $\sqrt{59}$ is an irrational number because its decimal representation extends infinitely without any repeating pattern, and it cannot be expressed as a simple fraction. This analysis confirms that $\sqrt{59}$ fits the criteria for being an irrational number.

In conclusion, after a thorough examination of the values $\sqrt[3]{80}$, π, $2.8\overline{7}$, and $\sqrt{59}$, we have identified that $2.8\overline{7}$ is the only value that is not an irrational number. This is because $2.8\overline{7}$ is a repeating decimal that can be expressed as a fraction (259/90), fitting the definition of a rational number. The other options, $\sqrt[3]{80}$, π, and $\sqrt{59}$, are all irrational numbers due to their non-terminating and non-repeating decimal representations. This exercise highlights the importance of understanding the fundamental differences between rational and irrational numbers in mathematical analysis. Recognizing these differences allows for accurate categorization and problem-solving in various mathematical contexts. Therefore, the definitive answer to the question, "Which of the following values is not an irrational number?" is C. $2.8\overline{7}$.