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Q: Can the sum of two irrational numbers be a rational number?

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The sum of the three can be rational or irrational.

The sum of a rational and irrational number must be an irrational number.

Can be rational or irrational.

Such a sum is always irrational.

Such a sum is always rational.

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Since the sum of two rational numbers is rational, the answer will be the same as for the sum of an irrational and a single rational number. It is always irrational.

It is always an irrational number.

It will be irrational. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.

The sum of two irrational numbers may be rational, or irrational.

It may be a rational or an irrational number.

Yes Yes, the sum of two irrational numbers can be rational. A simple example is adding sqrt{2} and -sqrt{2}, both of which are irrational and sum to give the rational number 0. In fact, any rational number can be written as the sum of two irrational numbers in an infinite number of ways. Another example would be the sum of the following irrational quantities [2 + sqrt(2)] and [2 - sqrt(2)]. Both quantities are positive and irrational and yield a rational sum. (Four in this case.) The statement that there are an infinite number of ways of writing any rational number as the sum of two irrational numbers is true. The reason is as follows: If two numbers sum to a rational number then either both numbers are rational or both numbers are irrational. (The proof of this by contradiction is trivial.) Thus, given a rational number, r, then for ANY irrational number, i, the irrational pair (i, r-i) sum to r. So, the statement can actually be strengthened to say that there are an infinite number of ways of writing a rational number as the sum of two irrational numbers.

No - the sum of any two rational numbers is still rational:

The sum of the three can be rational or irrational.

The sum of a rational and irrational number must be an irrational number.

Yes. The sum of two irrational numbers can be rational, or irrational.

No. In fact the sum of a rational and an irrational MUST be irrational.

It is always irrational.

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