In the proof that ab+cd, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd is a rational number.

Answer 1

To prove that ad + bcbd is a rational number, we need to demonstrate that it can be expressed as the ratio of two integers.

We know that a, b, c, and d are all integers, and b and d are non-zero. Since integers are closed under addition and multiplication, we can conclude that ab, cd, and bc are all integers.

Since ad + bcbd is a sum of integers, it is also an integer.

Now, let's consider the denominator of the fraction. Since b and d are both non-zero integers, we know that bcd is also a non-zero integer. Hence, the denominator is non-zero.

By definition, a rational number is any number that can be expressed as the ratio of two integers, with the denominator not being zero.

Therefore, ad + bcbd satisfies the definition of a rational number and can be expressed as the ratio of two integers, making it a rational number.