Ladies and Gentlemen, in the left corner, with an unknown imaginary weight, hailing
from the complex domain, we have i to the i to the i. His opponent, also with an imaginary
weight and also hailing from the complex domain is...uh...also i to the i to the i. Ok, well,
uh, neverthless, an epic battle is about to take place to determine what these competitors
are equal to and who is bigger. Let’s get ready to rumble! Alright, now that the dramatic
introduction is out of the way, let’s focus on our goal in this video. To see what these
two seemingly identical expressions are equal to, and which one is bigger. Right out of
the gate, we should realize that although the two expressions look nearly identical,
the position of those parentheses plays an important role. Let’s just see what happens
when we use numbers in the real world. If I replace i with, say, 3, the expression on
the left would become 27 cubed, or 19,683. The one on the right would be 3 to the power
of 27. And that is equal to...hang on. Let’s give ourselves a little more room. 7 trillion
625 billion 597 million 484 thousand 987. In other words, a much bigger number. So it’s
clear that the placement of the parentheses is extremely important. Maybe the same thing
will happen with our two imaginary competitors. Let’s find out! We’ll start with the expression
on the left since it’s the easier of the two. First we can use an exponent rule. A
power to a power means we can multiply the powers. So we get i the power i times i. But
i times i is just negative one, so we get i to the minus 1. That would be one over i.
Now typically we want to write complex numbers in standard a plus bi form, so we can just
multiply top and bottom by i. Since i over i is just one, multiplying by this quantity
won’t change the value, only the form. We’ll get i over negative 1, or negative i. So the
left expression is just negative i. Now let’s work on the one on the right. This one is
going to be a bit more difficult. Let’s focus on the expression in parentheses. What
exactly is i to the i? We need Euler’s formula, to help us out. This amazing formula is the
gateway to higher mathematics, including complex analysis. But first let’s take a tiny tangent
and let me quickly show you one of the most celebrated equations in all of math which
we get using Euler’s formula. If we just let theta equal pi, then we get e to the i
pi equals cosine pi plus i sine pi. Notice that cos pi is negative 1 and sin pi is 0.
And we get this magical little identity, widely regarded as one of the most beautiful equations.
Now let’s return to Euler’s formula once more to figure out i to the i. Now the trick
for figuring out i to the power of i is to replace theta with pi over 2 this time. Since
cos pi over 2 is 0 and sin pi over 2 is 1, this simplifies nicely to i. Next let’s
raise both sides to the power of i. Once again, a power to a power allows us to multiply the
exponents, so i times i is negative one, and we get e to the negative pi over 2. That expression,
amazingly, computes to a real number, approximately 0.207. So we can now replace the blue expression
with e to the minus pi over 2. Which looks worse than what we started with. But wait!
I’m going to use the i over i trick again and insert it into the exponent. Let’s bring
an i out of the parentheses. But we know what i to the i is! We just calculated it. So let’s
replace i to the i with e to the negative pi over 2. What now? I want to get this back
into an e to the i theta expression to use Euler’s formula again. So let’s simplify
the exponent in blue by writing 1 over i as negative i. And now just rearrange the exponent
a little more to get this. This is exactly what we want. An expression in the form of
e to the i theta, where we’ll set theta equal to the stuff in blue. Here’s Euler’s
formula once again. And now I’ll replace theta with the blue stuff, namely pi over
2 times e to the negative pi over 2. And this is an exact expression for i to the i to the
i. Pretty awesome. Now, we can calculate this to get a numerical expression in a plus b
i form. That comes out to approximately 0.947 plus 0.321 i. You can easily verify this on
your calculator. That’s pretty remarkable, if you think about it. An expression purely
in terms of i is somehow able to spawn a real component. So it’s now pretty clear these
two opponents are definitely NOT equal to each other. But they are both complex. So
how do we figure out who’s bigger in the complex world? We go to the complex plane.
Recall that we plot the real part along the horizontal axis which we call the real axis
and the Imaginary part along the vertical axis, which is called the Imaginary axis.
The first expression we evaluated to be negative i, shown by the red dot. It has real component
zero and imaginary component negative 1. The second expression we evaluated to be approximately
0.947 plus 0.321i, which is shown by the blue dot. So how do we know which is bigger? A
reasonable approach to assessing their size would be to find their distance from the origin.
This makes perfect sense. After all, on a regular one-dimensional number line, the distance
of a number from zero represents its size. When working with complex numbers, we call
this the modulus. The formula is easy. For a complex number z = a+bi, the modulus, denoted
by the absolute value symbol, is the square root of the sum of the squares of the real
and imaginary parts. In other words it’s just the Pythagorean Theorem. You can confirm
for yourself by plugging in the values of a and b, that in both cases, their respective
moduli are equal to 1. Let’s draw a circle of radius 1 from the origin to see if this
is true, at least visually. Yup – both complex numbers have a modulus of 1. So, to summarize,
although the two expressions have different values, the modulus of each is identical.
As we just showed, it’s equal to one. And that means they are the same distance away
from the origin on the complex plane, so that’s going to bring this epic battle to a close.
Which means that our dueling powers of i will have to settle for a peaceful tie. Let me
know in the comments below if you know some other amazing applications of Euler’s formula.
And be sure to subscribe for a new video every week featuring puzzles, problems, applications,
and explorations. Thanks for watching and see you soon!