Hi. This week, we have seen infinity appear in
many different ways. We have seen it in decimals,
in infinite sums, and even hidden inside geometry
itself. But today, we ask a new question. Are
all infinities the same size? Well, at first,
this feels like an impossible question. But intuitively,
we might guess that some infinity sets are larger
than others. For example, maybe the positive
integers are twice as large as the even numbers,
since the even numbers are only half of the integers.
That feels reasonable. But mathematics surprises
us by concluding that the even numbers and the
positive integers are exactly the same size of
infinity. That already feels strange. After all,
the even numbers appear to be only half of the
positive integers. But in infinity, appearances
can be misleading. And it even gets stranger.
The rational numbers, all fractions, also turn
out to have the same size of infinity as the
integers. Wow! the rational numbers seem vastly
larger than the integers. After all, between
any two integers, there are infinitely many fractions.
So at this point, it may begin to feel like perhaps
all infinities are the same size. But they're
not. Let's climb one step higher. The real numbers
consist of both rational and irrational numbers.
we've already encountered two famous irrational
numbers, square root of 2 and pi. Those are numbers
whose decimals continue forever without repeating.
There are also an infinitely many irrational
numbers. So here's a question maybe my grandkids
would ask. If we compare the infinity of irrational
numbers to the infinity of rational numbers,
which infinity is bigger? Following our previous
pattern, we might expect that they are the same
size. But they are not. The irrational numbers
form a larger infinity. And what really astonishes
me is that if you could somehow choose a number
at random from the number line, the probability
of selecting a rational number is zero. And the
probability of selecting an irrational number
is 100%. Now, that does not mean rational numbers
do not exist. They are still infinite. But irrational
numbers are so overwhelmingly abundant that rational
numbers almost disappear in comparison. The rationals
include all integers and all fractions. and yet
they form only a tiny part of the number line.
Almost every number is irrational. This is where
infinity becomes something fundamentally different
from simply very large. Infinity now has hierarchy,
depth, structure. Some infinities are vastly
larger than others. This was one of the great
turning points in mathematics. Infinity stopped
being viewed as merely an endless quantity and
became something mathematicians could study,
compare, and explore. And if all of this feels
difficult to absorb, you are not alone. Infinity
stretches the mind. So what part of this story
feels most difficult for you to believe? Be sure
to join us tomorrow for our final reflection
on infinity. We will push our growth mindset
even further as we consider eternal ideas.