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== == The set of natural numbers is {1, 2, 3, ...} The set of integers is {..., -3, -2, -1, 0, 1, 2, 3, ...} All natural numbers are integers.

A rational number is an integer 'A' divided by a natural number 'B'; i.e. A / B. Suppose we add two rational numbers: A / B + C / D This is algebraically equal to (AD + BC) / BD Since A and C are integers and B and D are natural numbers, then AD and BC are integers because two integers multiplied yields an integer. If you add these together, you get an integer. So we have an integer (AD + BC) on the top.

B and D are natural numbers. Multiply them and you get a natural number. So we have a natural number BD on the bottom.

Since (AD + BC) / BD is a rational number, A / B + C / D is a rational number.

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Since a rational number is, by definition, one that can be written a a ratio of 2 integers, adding 2 rationals is tantamount to adding 2 fractions, which always produces a fraction (ratio of 2 integers) for the answer, so the answer is, by definition, rational. llllaaaaaaaaaaaaaalllllllllaaaaaaaaaalllllllllllaaaaaaaaaaaalaaaaaaaa

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Q: Why is the sum of any two rational numbers a rational number?

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

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

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, it is.

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

Every time. The sum of two rational numbers MUST be a rational number.

It is a rational number.

Yes.

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 a rational number.

Either way, you'll end up with a rational number, but you won't get a sum if you multiply.

True.

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