Video Transcript
What property of addition states
that the sum of any two rational numbers is always a rational number?
In this question, we are asked to
recall which of the properties of addition tells us that the sum of any two rational
numbers must itself be rational. We can begin by writing the
property given in the question out in full using set notation. The property states that if we have
any two rational numbers, let’s say 𝑎 over 𝑏 and 𝑐 over 𝑑 for integers 𝑎, 𝑏,
𝑐, and 𝑑 with 𝑏 and 𝑑 nonzero, then the sum of 𝑎 over 𝑏 and 𝑐 over 𝑑 must be
a rational number.
We can recall that this is called
the closure property of the addition of rational numbers. The reason this is called the
closure property is that we cannot use the addition of rational numbers to generate
a nonrational number. It is closed in the sense that we
will only ever deal with rational numbers using this operation on rational
numbers. It is also worth noting that this
property actually comes from our definition of how we add rational numbers and the
properties of multiplying and adding integers.
We know that we can add rational
numbers by rewriting them to have the same denominator. So 𝑎 over 𝑏 plus 𝑐 over 𝑑 is
equal to 𝑎𝑑 plus 𝑏𝑐 over 𝑏𝑑. We can show that the denominator of
this fraction is never zero, since 𝑏 and 𝑑 are nonzero. And we know that for a product of
integers to be zero, one of the factors must be zero. Similarly, we can show that the
numerator and denominator must be integers since they are the sum and product of
integers and we know that the sum and product of integers are both closed
operations.
Of course, this is not necessary to
answer the question. However, it is useful to understand
where these properties come from. We can conclude that the closure
property for the addition of rational numbers tells us that the sum of two rational
numbers is always rational.
