Calculus was one of the most important
theories ever developed in mathematics.
But why do we need it?
Subscribe now to the series, the history
of maths, on the YouTube channel, What
makes it tick? Mathematicians sometimes
love to push things to the limit and to
see what happens. For example, one of the
fields of their research is series. If
you take the series 1+1 on 2 +
1 on 3 + 1 4 + +
+, you go to the infinite. The series
doesn't converge. Strangely enough, if you
take the series 1 + 1 on 2 +
1 on 4 + 1 on 8, you don't
reach the infinite, you reach a limit.
Again it's this beautiful opportunity to
show it graphically. You take a square,
a beautiful square and the side is length 1.
If you cut it into two equal pieces
this is a half, 1 on 2. Now let's take
this piece, and again you cut it into 2
pieces. This square here is 1 on 4.
The upper square, the same style. Again,
you divide into 2 pieces. Each is equal to
1 on 8. And you can go up to the
infinite and you will get the whole
square. And the square is 1, the surface of
1. So it shows, it's proof there is a
limit to this theory
1 on 2, 1 on 4, 1 on 8 etc. If you try to
reach the limit, you sometimes find
fascinating things. And one of the most
famous examples is this one.
Let's take 1 + 1 on X to the power of X.
How far does it go?
Is there a limit? Yes there is a limit. Let's
take this 1, it makes 1 + 1 on 1 to
the power of 1, this equals 2. Now let's take
X equal to 2. It's 1 + 1 on 2 to the power
of 2, which is 2.25. And if you go to the
infinite, you finally get a number 2.71
etc. which is E, another magic number
exactly like P like the golden number. So
if you love mathematics you will love to
push things to the limit and you may be,
will be surprised by extraordinary
results once more. Join us next time to
find out how nothing happens by chance.
Subscribe now to watch our full series
on the history of maths on your YouTube
channel, What makes it tick?
