Is The Square Root Of 90 A Rational Number

Is the Square Root of 90 a Rational Number? A Deep Dive into Irrationality

The question of whether the square root of 90 is a rational number is a fundamental concept in mathematics, touching upon the core principles of number systems. Understanding this requires a grasp of rational and irrational numbers, prime factorization, and the properties of square roots. Let's explore this question thoroughly.

Understanding Rational and Irrational Numbers

Before tackling the square root of 90, let's define our terms. A rational number is any number that can be expressed as a fraction p/q, where p and q are integers, and q is not zero. Examples include 1/2, 3, -5/7, and even 0 (which can be expressed as 0/1). These numbers can be represented as terminating or repeating decimals.

An irrational number, on the other hand, cannot be expressed as a simple fraction. Their decimal representation is non-terminating and non-repeating. Famous examples include π (pi), e (Euler's number), and the square root of most integers.

Prime Factorization: The Key to Understanding Square Roots

Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...). Prime factorization is crucial for simplifying square roots.

Let's find the prime factorization of 90:

90 = 2 x 45 = 2 x 3 x 15 = 2 x 3 x 3 x 5 = 2 x 3² x 5

Now we can express the square root of 90 using this factorization:

√90 = √(2 x 3² x 5)

Simplifying the Square Root of 90

We can simplify this square root by taking out any perfect squares from under the radical symbol. Since 3² is a perfect square, we can simplify as follows:

√90 = √(3² x 2 x 5) = 3√(2 x 5) = 3√10

This simplified form shows that the square root of 90 is equal to 3 times the square root of 10.

Is √10 Rational? The Crucial Question

The question now becomes: is √10 a rational number? To determine this, we need to consider if 10 can be expressed as a fraction of two perfect squares. Let's assume, for the sake of contradiction, that √10 is rational. This means we can write it as:

√10 = p/q (where p and q are integers, q ≠ 0, and p/q is in its simplest form)

Squaring both sides, we get:

10 = p²/q²

10q² = p²

This equation reveals that p² is a multiple of 10. Since 10 = 2 x 5, p² must contain both 2 and 5 as factors. This means p itself must contain both 2 and 5 as factors (because if a prime number divides a square, it must divide the number itself). Therefore, we can write p as:

p = 2 x 5 x k = 10k (where k is an integer)

Substituting this back into our equation:

10q² = (10k)² = 100k²

Dividing both sides by 10:

q² = 10k²

This shows that q² is also a multiple of 10, and consequently, q must also contain both 2 and 5 as factors.

The Contradiction: We initially assumed that p/q was in its simplest form (meaning p and q share no common factors). However, we've shown that both p and q must contain 2 and 5 as factors. This is a contradiction, proving our initial assumption false.

Therefore, √10 cannot be expressed as a fraction of two integers and is, consequently, irrational.

Conclusion: √90 is Irrational

Since √90 simplifies to 3√10, and √10 is irrational, it follows that √90 is also an irrational number. It cannot be expressed as a fraction p/q where p and q are integers, and its decimal representation is non-terminating and non-repeating.

Further Exploration: Properties of Irrational Numbers

The irrationality of √90 highlights some important properties of irrational numbers:

• Non-terminating and non-repeating decimals: The decimal representation of √90 goes on forever without any repeating pattern. This is a defining characteristic of irrational numbers.
• Closure under addition and multiplication: While the sum or product of two rational numbers is always rational, the sum or product of a rational and irrational number is usually irrational. Similarly, the product of two irrational numbers can be rational (e.g., √2 x √2 = 2), irrational (e.g., √2 x √3 = √6), or even a complex number.
• Density: Irrational numbers are densely packed on the number line. Between any two rational numbers, no matter how close, you can always find an irrational number.

Practical Applications and Significance

While seemingly abstract, understanding rational and irrational numbers is crucial in various fields:

• Engineering and Physics: Many physical constants, like the speed of light or gravitational constant, are irrational numbers. Accurate calculations often require approximations.
• Computer Science: Representing irrational numbers in computer systems necessitates approximations using floating-point arithmetic, leading to potential errors in calculations.
• Geometry: Irrational numbers appear frequently in geometric calculations involving circles, triangles, and other shapes. For instance, the diagonal of a unit square is √2, an irrational number.

The seemingly simple question of whether the square root of 90 is rational leads to a deeper understanding of number systems and their properties. By employing prime factorization and proof by contradiction, we've established the irrationality of √90, reinforcing our understanding of fundamental mathematical concepts with practical implications across various scientific and computational domains. This exploration serves as a reminder that even seemingly straightforward mathematical questions can unveil intricate and fascinating aspects of the number system.