Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
When we add a rational number [tex]\(a\)[/tex] and an irrational number [tex]\(b\)[/tex], the sum [tex]\(a + b\)[/tex] will always be an irrational number. Here’s why:
1. Definition of Rational and Irrational Numbers:
- A rational number is any number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
- An irrational number is any number that cannot be expressed as a simple fraction; this includes numbers like [tex]\(\sqrt{2}\)[/tex], [tex]\(\pi\)[/tex], and [tex]\(e\)[/tex].
2. Sum of Rational and Irrational Numbers:
- If we add a rational number to an irrational number, the irrationality of the irrational number will dominate the sum.
3. Illustrative Example:
- Let [tex]\(a = \frac{1}{2}\)[/tex], which is a rational number.
- Let [tex]\(b = \sqrt{2}\)[/tex], which is an irrational number.
Adding these together:
[tex]\[ a + b = \frac{1}{2} + \sqrt{2} \][/tex]
4. Justification:
- Despite adding [tex]\(\frac{1}{2}\)[/tex], the result [tex]\(\frac{1}{2} + \sqrt{2}\)[/tex] cannot be expressed as a fraction of two integers. This means the square root component's irrationality remains, making the entire expression irrational.
5. Conclusion:
- Therefore, [tex]\(a + b = \frac{1}{2} + \sqrt{2}\)[/tex] is approximately [tex]\(1.9142135623730951\)[/tex], an irrational number. The sum of a rational number and an irrational number is always an irrational number.
Thus, in general, if [tex]\(a\)[/tex] is a rational number and [tex]\(b\)[/tex] is an irrational number, then [tex]\(a + b\)[/tex] is always an irrational number.
1. Definition of Rational and Irrational Numbers:
- A rational number is any number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
- An irrational number is any number that cannot be expressed as a simple fraction; this includes numbers like [tex]\(\sqrt{2}\)[/tex], [tex]\(\pi\)[/tex], and [tex]\(e\)[/tex].
2. Sum of Rational and Irrational Numbers:
- If we add a rational number to an irrational number, the irrationality of the irrational number will dominate the sum.
3. Illustrative Example:
- Let [tex]\(a = \frac{1}{2}\)[/tex], which is a rational number.
- Let [tex]\(b = \sqrt{2}\)[/tex], which is an irrational number.
Adding these together:
[tex]\[ a + b = \frac{1}{2} + \sqrt{2} \][/tex]
4. Justification:
- Despite adding [tex]\(\frac{1}{2}\)[/tex], the result [tex]\(\frac{1}{2} + \sqrt{2}\)[/tex] cannot be expressed as a fraction of two integers. This means the square root component's irrationality remains, making the entire expression irrational.
5. Conclusion:
- Therefore, [tex]\(a + b = \frac{1}{2} + \sqrt{2}\)[/tex] is approximately [tex]\(1.9142135623730951\)[/tex], an irrational number. The sum of a rational number and an irrational number is always an irrational number.
Thus, in general, if [tex]\(a\)[/tex] is a rational number and [tex]\(b\)[/tex] is an irrational number, then [tex]\(a + b\)[/tex] is always an irrational number.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.