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If we define "opposite" as the additive inverse, the sum is zero.

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When x is not zero, it can be (x + 1/x).

Q: What is the sum of any rational number and its opposite?

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The sum of any finite set of rational numbers is a rational number.

an irrational number PROOF : Let x be any rational number and y be any irrational number. let us assume that their sum is rational which is ( z ) x + y = z if x is a rational number then ( -x ) will also be a rational number. Therefore, x + y + (-x) = a rational number this implies that y is also rational BUT HERE IS THE CONTRADICTION as we assumed y an irrational number. Hence, our assumption is wrong. This states that x + y is not rational. HENCE PROVEDit will always be irrational.

No. The sum of an irrational number and any other [real] number is irrational.

Yes

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)

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If the opposite is meant to be the additive opposite and not the multiplicative opposite, then their sum is zero. The reason is that is what defines an additive opposite!

The sum of any finite set of rational numbers is a rational number.

It is a rational number.

True.

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

Because that is how its additive inverse is defined!

Any, and every, irrational number will do.

Yes.

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. In fact the sum of a rational and an irrational MUST be irrational.

Because the opposite of 8 for example is -8 and 8+(-8) = 0

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