Hi. Yesterday, we opened the door to infinity
through prime numbers. Today, infinity appears
in a familiar yet unexpected place, inside a
simple fraction. In math, we often take for granted
that numbers can be expressed in different ways.
For example, one half equals 0 .5. No surprises
and nothing mysterious. A fraction and a decimal.
Two different ways to describe the same number.
Now consider the simple fraction one -third.
We see this number in life simply by dividing
a pizza into three equal slices, then one slice
is one -third of the pizza. When we convert one
-third into a decimal, we get 0 .333... The threes
never end. Hmm. Suddenly, hidden beneath this
simple fraction, we discover an infinite process.
Infinity was there the whole time. At first,
this infinity may not seem too mysterious. We
still have a pattern to follow. Just keep adding
more threes. But now let's combine the three
pieces of pizza and put them back together. In
other words, one -third plus one -third plus
one -third. From the fraction perspective, the
answer is obviously one. Three -thirds. make
one whole. Writing the same problem using decimal
produces 0 .333 dot dot dot plus 0 .333 dot dot
dot plus 0 .333 dot dot dot which equals 0 .99
dot dot dot. And now we seem to have a paradox.
The fraction representation says the answer is
1, but the decimal representation appears to
say the answer is something slightly less. So,
which answer is correct? This is where infinity
forces us to think more carefully about what
numbers actually are. What happens when two correct
ways of describing the same number appear to
disagree? We normally think both descriptions
are completely compatible. But how do we reconcile
when they appear to give different answers? Historically,
questions like this cause enormous confusion
in mathematics. The question at play is how can
something go on forever and still equal something
exact? The answer eventually came through an
important idea called a limit. A limit describes
the value of a process approaches as it continues
forever. And in this case, we continue adding
more nines. We go from 0 .9 to 0 .99 to 0 .999.
And as we add nines, the limit of the number
approaches 1 perfectly. There is no gap left.
So mathematicians define 0 .99... to equal 1
exactly. Not approximately and not close enough.
Exactly the same number. And this is where infinity
can change the way we think. Infinity is no longer
just counting forever. Now it begins reshaping
what numbers themselves can mean. A simple fraction.
A familiar decimal. And suddenly we discover
there was an infinity iceberg. hidden beneath
them all along. So, what do you think about mathematicians
declaring that 0 .99 dot dot dot exactly equals
1? Does it feel reasonable? Does it feel like
cheating? Is it surprising that mathematicians
agree on something that initially feels so strange?
Maybe this feels a bit unsettling. And don't
ignore that feeling. Give it some space. And
welcome to the world of infinity, where logic
sometimes stretches beyond intuition. And we
will see that intuition stretch further this
week as we uncover stranger stories of infinity.