Real numbers
Grade Class 10
Question:
<p><span class="math-tex">\((2+\sqrt{2})\)</span> is</p>
<p style="display:inline">an integer</p>
<p style="display:inline">a rational number</p>
<p style="display:inline">A real number</p>
<p style="display:inline">an irrational number</p>
Step-by-Step Solution
Key Concept: To prove irrationality, use the method of contradiction by assuming the expression is rational and applying the closure property of rational numbers under subtraction to isolate a known irrational component.
<p><span class="math-tex">\((2+\sqrt{2})\)</span> is an irrational number.<br />
If it is rational, then the difference of two rational is rational.<br />
<span class="math-tex">\(\therefore(2+\sqrt{2})-2=\sqrt{2}\)</span> = irrational, which is a contradiction.<br />
Hence, <span class="math-tex">\((2+\sqrt{2})\)</span>, is an irrational number.</p>
Correct Answer: D