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A student says that the product of a non-zero rational number and an irrational number is always irrational. Which is the best reason why the statement is correct?एक विद्यार्थी कहता है कि किसी शून्येतर परिमेय संख्या को अपरिमेय संख्या से गुणा करने पर गुणनफल सदैव अपरिमेय होता है। उसका कथन सही होने का सबसे उचित कारण कौन-सा है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: If the product were rational, dividing it by the non-zero rational number would make the irrational number rational.यदि गुणनफल परिमेय होता, तो उसे शून्येतर परिमेय संख्या से भाग देने पर अपरिमेय संख्या भी परिमेय हो जाती।
Let \(r\ne0\) be rational and \(x\) be irrational. If \(rx\) were rational, then \(x=(rx)/r\) would be rational, a contradiction. In exams, always check the condition \(r\ne0\).मान लें कि परिमेय संख्या \(r\ne0\) और अपरिमेय संख्या \(x\) है। यदि \(rx\) परिमेय हो, तो \(x=(rx)/r\) भी परिमेय होगा, जो विरोधाभास है। परीक्षा में \(r\ne0\) शर्त अवश्य जाँचें।
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