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.
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.
To prove that ad+bcbd is a rational number, we need to show that it can be expressed as a fraction of two integers. Since a, b, c, and d are all integers, we can proceed with the following steps:
1. Recall that a rational number can be expressed as the ratio of two integers. So, our goal is to manipulate the expression ad+bcbd into a fraction.
2. Notice that ad is already an integer since a and d are both integers.
3. Now, let's consider bcbd. Since b and d are non-zero integers, their product bcbd is also an integer.
4. Therefore, ad + bcbd is the sum of two integers, which can still be expressed as an integer.
5. Since any integer can be expressed as a fraction by dividing it by 1, we can write ad + bcbd as the fraction (ad + bcbd) / 1.
6. Simplifying this, we get (ad + bcbd) / 1 = (ad + bcbd).
7. Thus, we have expressed ad + bcbd as a fraction (ad + bcbd) / 1, where the numerator and denominator are both integers.
8. Since it can be expressed as a fraction of two integers, ad + bcbd is a rational number.
Therefore, we have proven that ad+bcbd is a rational number.