Question

Prove the following statement directly from the definitions. The difference of any two rational numbers is a rational number. Proof: Suppose r and s are any two rational numbers. By definition of rational, r

Answer 1

To prove that the difference between two rational numbers is rational, express r and s as a/b and c/d. Find their difference with a common denominator (bd) to get (ad - bc)/(bd). Since the numerator and denominator are integers and the denominator is not zero, the result is a rational number.

Step-by-step explanation:

The statement to prove is: 'The difference between any two rational numbers is a rational number'. To establish this, let's consider two rational numbers r and s, such that r = a/b and s = c/d, where a, b, c, and d are integers, and b and d are not zero. A rational number by definition can be expressed as the quotient of two integers, with the denominator being non-zero.

The difference of r and s is r - s = (a/b) - (c/d). To combine these two fractions, we find a common denominator which is the product of the two denominators, bd. Rewriting the fractions with the common denominator gives us (ad)/(bd) - (bc)/(bd). Combining these two fractions gives (ad - bc)/(bd).

Now, since both the numerator (ad - bc) and the denominator (bd) are integers, and the denominator bd is not zero (as b and d are both not zero), the number (ad - bc)/(bd) fits the definition of a rational number. Therefore, we have shown that the difference between two rational numbers is itself a rational number.

Answer 2

- Integers, b ≠ 0 and d ≠ 0.

- r - s = (ad - bc)/bd.

- products and differences of integers are integers.

- zero product property

- quotient.

Step-by-step explanation:

Rational numbers are integers that can take the form of numerator divided by a denominator, that is to say in the form of 'a/b;. So, we are given from the question or problem to prove and fill in the gap the correct information or data:

For the first part of the question:

The missing data in the gap = Integers; b ≠ 0 and d ≠ 0. That is to say the denominators can not be zero or say that the denominators are non zero. This is one of the properties or let us say one of the ways of identifying a Rational number.

For the second part:

The missing data in the gap = a/b - c/d = (ad - bc)/bd. This is the representation in a simplified form.

For the third part:

The missing data or information = products and differences of integers are integers.

For the fourth part:

The missing data or information: zero product property, that is to say the product of the numerators is not equal to zero, or it is non zero.

For the fifth part:

The missing data or information: quotient. r - s is a quotient of two integers.