Now that you've created your hypotheses, it's time to prove them. First, look at the sum of two rational numbers. Let's say they are two rational numbers, [tex]$x$[/tex] and [tex]$y$[/tex]. Because they're rational, they can be written as a ratio of integers. Let [tex]$x=\frac{a}{b}$[/tex] and [tex]$y=\frac{c}{d}$[/tex], where [tex]$a, b, c$[/tex], and [tex]$d$[/tex] are integers and [tex]$b$[/tex] and [tex]$d$[/tex] do not equal 0. The process for finding the sum [tex]$x+y$[/tex] in terms of [tex]$a, b, c$[/tex], and [tex]$d$[/tex] is shown.
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.
Answer (1)
Rational numbers x and y can be expressed as x = b a and y = d c , where a , b , c , d are integers and b , d = 0 .
The sum of two rational numbers is x + y = b a + d c = b d a d + b c .
Since integers are closed under multiplication and addition, a d + b c and b d are integers, and b d = 0 .
Therefore, the sum b d a d + b c is a rational number: The sum of two rational numbers is a rational number.
Explanation
Problem Analysis and Setup We are given that x and y are rational numbers, which means they can be expressed as fractions b a and d c respectively, where a , b , c , d are integers and b , d = 0 . We are also given the sum x + y = b a + d c = b d a d + b c . We need to determine what conclusion can be made about the sum of two rational numbers based on the given sum and the closure property of integers.
Analyzing the Numerator The numerator of the sum is a d + b c . Since a , b , c , and d are integers, and integers are closed under multiplication, a d and b c are integers. Furthermore, since integers are closed under addition, a d + b c is an integer.
Analyzing the Denominator The denominator of the sum is b d . Since b and d are integers, and integers are closed under multiplication, b d is an integer. Also, since b = 0 and d = 0 , it follows that b d = 0 .
Conclusion Therefore, the sum x + y = b d a d + b c is a ratio of two integers, where the denominator is not zero. By definition, this means that x + y is a rational number.
Final Answer Thus, we can conclude that the sum of two rational numbers is a rational number.
Examples
Imagine you're baking a cake and a recipe calls for 2 1 cup of flour and 4 1 cup of sugar. Both 2 1 and 4 1 are rational numbers. When you combine them, you have 2 1 + 4 1 = 4 2 + 4 1 = 4 3 cup of dry ingredients. The total amount, 4 3 cup, is also a rational number, demonstrating that the sum of rational quantities in a recipe remains rational.
