Video Transcript
Use “is a member of” or “is not a
member of” to complete the following: 𝜋 what the set of real numbers.
In this question, we are asked to
fill in a blank in an expression. We can recall that the two symbols
we are given mean is or is not a member of. So we need to determine if 𝜋 is a
member of the real numbers.
To do this, let’s start by
recalling what we mean by the set of real numbers. By definition, it is the union of
the set of rational numbers and the set of irrational numbers. We can also recall that the set of
rational numbers is the set of all quotients of integers, where the denominator is
nonzero. Equivalently, it is the set of all
numbers whose decimal expansion terminates or is repeating.
On the other hand, the set of
irrational numbers is the complement of the set of rational numbers. They are the numbers which cannot
be written as the quotient of two integers. or equivalently the numbers with a
nonrepeating infinite decimal expansion. It is far beyond the scope of this
video and lesson to prove that 𝜋 is irrational.
However, it is a well-known fact
that 𝜋 has a nonrepeating infinite decimal expansion. So we can state that 𝜋 is an
irrational number. We can write this in set notation
as 𝜋 is a member of the set of irrational numbers. We can then note that since 𝜋 is
an irrational number, we can say that it is either rational or irrational. So it is a member of the union of
these sets. This is equivalent to saying that
it is a real number.
And another way of thinking about
this is to say that all irrational numbers are real numbers, since the set of
irrational numbers is a subset of the real numbers. Hence, the answer is that 𝜋 is a
member of the real numbers.
