The sum of a rational number and an irrational number is

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Study for the Algebra 1 Honors End-of-Course Test. Study with flashcards and multiple-choice questions, each question has hints and explanations. Get ready for your exam!

Multiple Choice

The sum of a rational number and an irrational number is

Explanation:
Adding a rational number and an irrational number always gives an irrational result. The reason is simple: if you suppose their sum were rational, then subtracting the rational part would leave the irrational number as a difference of two rational numbers, which would be rational. That contradicts the irrationality of the second number. So the sum cannot be rational. For example, 1/2 plus sqrt(2) cannot be written as a fraction or a terminating/repeating decimal, so it is irrational. The irrational part can’t be canceled out by a rational amount, so the whole sum stays irrational.

Adding a rational number and an irrational number always gives an irrational result. The reason is simple: if you suppose their sum were rational, then subtracting the rational part would leave the irrational number as a difference of two rational numbers, which would be rational. That contradicts the irrationality of the second number. So the sum cannot be rational. For example, 1/2 plus sqrt(2) cannot be written as a fraction or a terminating/repeating decimal, so it is irrational. The irrational part can’t be canceled out by a rational amount, so the whole sum stays irrational.

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