Question

part c
now that youve created your hypotheses, its time to prove them. first, look at the sum of two rational numbers. lets say they are two rational numbers, x and y. because theyre rational, they can be written as a ratio of integers. let x = \frac{a}{b} and y = \frac{c}{d}, where a, b, c, and d are integers and b and d do not equal 0. the process for finding the sum x + y in terms of a, b, c, and d is shown.

statement	reason
= (\frac{d}{d})(\frac{a}{b}) + (\frac{b}{b})(\frac{c}{d})	create common denominators.
= \frac{ad}{bd} + \frac{bc}{bd}
= \frac{ad + bc}{bd}	simplify.

based on this sum and using the closure property of integers, what conclusion can you make about the sum of two rational numbers? explain your answer.

Explanation:

1. Recall the form of rational numbers: A rational number can be expressed as \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q

eq0\). So we let \(x = \frac{a}{b}\) and \(y=\frac{c}{d}\) with \(a,b,c,d\in\mathbb{Z}\) and \(b
eq0\), \(d
eq0\).

1. When we find the sum \(x + y=\frac{a}{b}+\frac{c}{d}\), we first find a common denominator (which is \(bd\) since \(b\) and \(d\) are non - zero integers). Using the rule of adding fractions \(\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}\) (by multiplying the numerator and denominator of \(\frac{a}{b}\) by \(d\) and \(\frac{c}{d}\) by \(b\) respectively).
2. Then, by the rule of adding fractions with the same denominator, \(\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad + bc}{bd}\).
3. Now, consider the closure property of integers: The sum and product of two integers are also integers. Since \(a,b,c,d\) are integers, \(ad\) and \(bc\) are integers (by closure of integers under multiplication), so \(ad+bc\) is an integer (by closure of integers under addition). Also, \(bd\) is an integer (by closure of integers under multiplication) and \(bd

eq0\) (because \(b
eq0\) and \(d
eq0\), and the product of two non - zero integers is non - zero).

1. Since \(\frac{ad + bc}{bd}\) is in the form of a ratio of two integers (where the denominator \(bd

eq0\)), \(\frac{ad + bc}{bd}\) is a rational number. And since \(x=\frac{a}{b}\) and \(y = \frac{c}{d}\) are rational numbers, this shows that the sum of two rational numbers is also a rational number.

Answer:

The sum of two rational numbers is a rational number. This is because if we let the two rational numbers be \(x=\frac{a}{b}\) and \(y = \frac{c}{d}\) (where \(a,b,c,d\) are integers and \(b
eq0,d
eq0\)), their sum \(x + y=\frac{ad+bc}{bd}\). By the closure property of integers, \(ad\), \(bc\) are integers (so \(ad + bc\) is an integer) and \(bd\) is a non - zero integer. Since \(\frac{ad + bc}{bd}\) is a ratio of two integers with a non - zero denominator, it is a rational number.