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Below is a proof showing that the sum of a rational number and an irrational number is an irrational number.
Let a be a rational number and b be an irrational number.
Assume that a + b = x and that x is rational.
Then b = x – a = x + (–a).
x + (–a) is rational because _______________________.
However, it was stated that b is an irrational number. This is a contradiction.
Therefore, the assumption that x is rational in the equation a + b = x must be incorrect, and x should be an irrational number.
In conclusion, the sum of a rational number and an irrational number is irrational.
Which of the following best completes the proof?
A. it is the sum of two rational numbers.
B. it is the sum of two irrational numbers.
C. it represents a non-terminating, non-repeating decimal.
D. its terms cannot be combined.
The sum of two rational numbers must be rational, based on this, the sum of x and -a, which are both said to be rational, must have a rational answer. However, the answer which is stated to be B is an irrational number which is a contradiction.
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