Rational Vs. Irrational Numbers: Sums & Products Explained
Let's dive into the fascinating world of numbers! We're going to explore whether certain sums and products result in rational or irrational numbers. Don't worry, it's not as daunting as it sounds. We'll break it down step by step with clear explanations. So, grab your thinking caps, and let's get started!
(a) $\frac{2}{11}+\frac{17}{19}$
Understanding Rational Numbers. To kick things off, let's remind ourselves what rational numbers are. A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where p and q are integers, and q is not zero. Basically, if you can write a number as a simple fraction, it's rational. Examples include 1/2, -3/4, 5 (which can be written as 5/1), and even terminating or repeating decimals like 0.75 or 0.333...
Analyzing the Sum of Fractions. Now, let's tackle the problem at hand: $\frac{2}{11}+\frac{17}{19}$. We're adding two fractions together. Both $\frac{2}{11}$ and $\frac{17}{19}$ are rational numbers because they are already expressed in the form of a fraction where both the numerator and denominator are integers.
Performing the Addition. To add these fractions, we need a common denominator. The common denominator for 11 and 19 is their product, which is 209. So, we rewrite the fractions as follows:
Now we can add them:
Determining Rationality. The result of the addition is $\frac{225}{209}$. Since 225 and 209 are both integers, and 209 is not zero, the resulting fraction is a rational number. This illustrates a fundamental property: The sum of two rational numbers is always a rational number. So, the final answer for part (a) is that the result is a rational number because it is the sum of two rational numbers, which can be expressed as a fraction of two integers.
(b) $16+\sqrt{13}$
Understanding Irrational Numbers. Before we proceed, let's define irrational numbers. An irrational number is a number that cannot be expressed as a fraction $\frac{p}{q}$, where p and q are integers. These numbers have decimal representations that are non-terminating and non-repeating. Classic examples include $\sqrt{2}$, $\pi$, and e.
Analyzing the Sum of an Integer and a Square Root. In this case, we have $16+\sqrt{13}$. The number 16 is an integer, and integers are rational numbers (since they can be written as a fraction with a denominator of 1, e.g., 16 = 16/1). However, $\sqrt{13}$ is a different beast. The square root of 13 is an irrational number because 13 is not a perfect square. When you take the square root of a non-perfect square, you get a decimal that goes on forever without repeating.
Why the Sum is Irrational. When you add a rational number (like 16) to an irrational number (like $\sqrt13}$), the result is always an irrational number. Think about it$, you'll still end up with a never-ending, non-repeating decimal. There's no way to turn that into a simple fraction.
To illustrate this, consider that if $16 + \sqrt{13}$ were rational, then we could express it as a fraction $\frac{a}{b}$. Then, we could write $\sqrt{13} = \frac{a}{b} - 16 = \frac{a - 16b}{b}$. This would imply that $\sqrt{13}$ is rational, which we know is false. Therefore, the sum must be irrational.
Therefore, the result of $16+\sqrt{13}$ is an irrational number because it is the sum of a rational number (16) and an irrational number ($\sqrt{13}$). Adding a rational number to an irrational number will always yield an irrational number.
(c) $20 \times \sqrt{5}$
Reviewing the Properties. We're looking at the product of two numbers: 20 and $\sqrt{5}$. We know from our previous discussions that 20 is a rational number (an integer, which is a subset of rational numbers) and $\sqrt{5}$ is an irrational number (the square root of a non-perfect square).
The Product of Rational and Irrational Numbers. So, what happens when you multiply a rational number by an irrational number? The answer is almost always an irrational number. The only exception is when the rational number is zero. If you multiply any irrational number by zero, you get zero, which is a rational number. But in our case, we're multiplying by 20, not zero.
Explanation. Multiplying an irrational number by a non-zero rational number scales the irrational number but doesn't make it rational. The non-repeating, non-terminating decimal part of the irrational number remains, preventing the product from being expressed as a simple fraction. Essentially, you're just making the "irrationality" bigger, not eliminating it.
To prove this, suppose that $20 \times \sqrt{5}$ were rational. Then we could write it as a fraction $\frac{p}{q}$. This would mean that $\sqrt{5} = \frac{p}{20q}$, implying that $\sqrt{5}$ is rational since p and 20q are integers. But we know that $\sqrt{5}$ is irrational, leading to a contradiction. Thus, the product must be irrational.
Thus, the result of $20 \times \sqrt{5}$ is an irrational number because it is the product of a non-zero rational number (20) and an irrational number ($\sqrt{5}$). The product of a non-zero rational number and an irrational number is always irrational.
In summary:
- (a) $\frac{2}{11}+\frac{17}{19}$ results in a rational number because the sum of two rational numbers is always rational.
- (b) $16+\sqrt{13}$ results in an irrational number because the sum of a rational number and an irrational number is always irrational.
- (c) $20 \times \sqrt{5}$ results in an irrational number because the product of a non-zero rational number and an irrational number is always irrational.