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Prove that the rational numbers are closed under multiplication. That is, prove that if a and b are rational numbers, then a - b is a rational number.

A) Use the distributive property
B) Assume a and b are irrational
C) Proof by contradiction
D) Prove by induction

User AshMan
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Final answer:

To prove that the rational numbers are closed under multiplication, we can use the distributive property and the definition of rational numbers. Assuming that a and b are rational numbers, we can show that their product, a * b, is also a rational number.

Step-by-step explanation:

To prove that the rational numbers are closed under multiplication, we can use the distributive property. Let's assume that a and b are rational numbers, and we want to show that their product, a * b, is also a rational number.

1. By the definition of rational numbers, a can be written as a = p/q and b can be written as b = r/s, where p, q, r, and s are integers and q and s are non-zero.

2. Using the distributive property, we have a * b = (p/q) * (r/s) = (p * r) / (q * s).

3. Since p, q, r, and s are integers and integers are closed under multiplication, p * r and q * s are also integers.

4. Therefore, (p * r) / (q * s) is a quotient of two integers, which is the definition of a rational number.

5. Thus, we have shown that if a and b are rational numbers, then their product, a * b, is also a rational number. Therefore, the rational numbers are closed under multiplication.

User Ido Schacham
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