Video Transcript
Consider the given sets of
numbers. Which sets does the number 2.5
belong to?
In this question, we are given five
sets of numbers and asked to determine which of these sets has 2.5 as a member. To do this, let’s start by
recalling what is meant by the five different given sets of numbers. First, we can recall that the set
of natural numbers is the set of counting numbers, that is, the set of nonnegative
integers. So it includes numbers like zero,
one, two, and so on. It is worth noting that some people
prefer not to include zero as a natural number, though this will not affect the
answer to this question since we are interested in the number 2.5.
Next, we can recall that the
integers are all of the positive and negative whole numbers. So they are numbers like zero, one,
two and negative one, negative two, and so on. Next, we can recall that the set of
rational numbers is the set of all quotients of integers such that the denominator
is nonzero. This set contains all numbers that
can be written as 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers and 𝑏 is nonzero.
The set of irrational numbers is
the complement of the set of rational numbers. We could write this in set
notation. However, it is usually easier to
think of this as just a complement of the set of rational numbers. It includes any number that cannot
be written as 𝑎 over 𝑏 for integers 𝑎 and 𝑏, where 𝑏 is nonzero. Finally, we can recall that the set
of real numbers is the union between the set of rational numbers and irrational
numbers. It contains all numbers that are
either rational or irrational.
We are now ready to answer this
question. We can start by rewriting 2.5 into
the fraction five over two. We can see that 2.5 is not a whole
number, so it is not a member of the natural numbers nor is it a member of the
integers. However, since we can write it as
the quotient of integers, it is a rational number. This allows us to conclude that it
is not irrational. Remember, there is no number that
is both rational and irrational. So showing that 2.5 is rational
means it cannot be irrational. Since 2.5 is a rational number, we
can conclude it must be a member of the union of the rational and irrational
numbers. So it is a real number.
Hence, we were able to show, of the
five given sets, 2.5 is only a member of the set of rational numbers and the set of
real numbers.
