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Q: The sum of a rational number and an irrational number and an irrational number is what?

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

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

The value of the sum depends on the values of the rational number and the irrational number.

It is always irrational.

The sum is irrational.

An irrational number.

It is an irrational number.

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.

Any, and every, irrational number will do.

Such a sum is always irrational.

irrational

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

Let R1 = rational number Let X = irrational number Assume R1 + X = (some rational number) We add -R1 to both sides, and we get: -R1 + x = (some irrational number) + (-R1), thus X = (SIR) + (-R1), which implies that X, an irrational number, is the sum of two rational numbers, which is a contradiction. Thus, the sum of a rational number and an irrational number is always irrational. (Proof by contradiction)

Yes

Yes.

It is always an irrational number.

Wrong. It is always an irrational number.

It is always 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.

Yes.

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

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

Yes, always.

It is true.

Let your sum be a + b = c, where "a" is irrational, "b" is rational, and "c" may be either (that's what we want to find out). In this case, c - b = a. If we assume that c is rational, you would have: a rational number minus a rational number is an irrational number, which can't be true (both addition and subtraction are closed in the set of rational numbers). Therefore, we have a contradiction with the assumption that "c" (the sum in the original equation) is rational.