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.
