# Algebraic Symmetries

**Algebraic Symmetries I**

Just as we can factor

we can factor

It follows that each of the four numbers

satisfies the equation

Thus our algebraic interpretation of the five vertices of
the regular pentagon

as the five fifth roots of unity has destroyed the five-fold symmetry. We have

distinguished one vertex, placing it at the point 1=1+0i. So we now have to

look for a different kind of symmetry, that among the four remaining vertices,

or better the four remaining roots.

There is a one obvious symmetry, that which interchanges
and as

well as and
. This is an algebraic as well as an geometric symmetry

because it is just a matter of replacing each of the numbers by its complex

conjugate

and

I take it as obvious that the complex conjugate of the sum
or the difference of

two complex numbers is the sum or the difference of their complex conjugates.

In other words the operation of complex conjugation that
interchanges and

as well as and

is like reflection in a mirror. All arithmetic properties are faithfully preserved.

**Small remarks**

is the same as

Why is, for example, ?

Take and . Then and this becomes

One shows in the same way that, for example, = .
Observe

also that, along the same lines,

and

This is because the angle
is equal to
and
the cosine

and sine do not change when 2π is added to or subtracted from an

angle. Indeed, in some respects, the angle itself does not change!

(Note: this statement is correct, but calls for some reflection!)

**Algebraic Symmetries II**

This is because i is just a symbol that stands for the
square root of −i

and −i is then introduced and defined by the condition that

But −i is also just a symbol and can be taken as the
primary symbol. Then

i is a second symbol that functions as −(−i). Even if i is taken to have some

meaning beyond that of a mere symbol, it cannot have a different meaning

than −i, so that the two have to be regarded as perfectly interchangeable.

**Are there other symmetries of this kind?**

Whether there are other symmetries of this kind affecting
all complex

numbers is not a question for us, but we can ask whether there are symmetries

of this kind affecting just , , and . Before we do, we make use of the

symmetry at hand. Since ,

= 1 and so on, and since in addition

the numbers

where a, b, c and d are arbitrary ordinary fractions form
a collection closed

under addition, subtraction, multiplication, and even as it turns out division.

The numbers that are equal to their own reflections can be singled out. These

are the numbers

**The appearance of.**

Let w be the number .
It is equal to its own reflection. So is its

square. Thus

Thus w satisfies a quadratic equation

We calculate this equation

Thus

Since w is a positive number,

In other words, w can as we know be constructed with ruler
and compass.

Since

we have

Since , this is

Since lies above the axis of abscissas,

**Symmetries III**

Having found , we can easily find , its complex conjugate, and we can
certainly

find by squaring . We can also
find by working with + rather than w.
This is,

however, straightforward algebra. As Descartes insisted, the algebra often turns
a problem

into an almost unthinking manipulation of symbols, a turn that it can indeed
often take,

but we prefer another direction. This is the direction taken by Gauss.

Let be the number . Then , and

because

Our numbers can also be expressed as

**Are these numbers all different?
**

This is an important question! The answer is yes and I shall give a proof following

Gauss. We first note the consequences. Since the numbers are all different, we cannot have

an equation

unless a = b = c = d = 0. This means that satisfies the equation

and essentially only this equation, because if we have any other such as

then by the process of long division we will have

so that

We have just seen that this implies P = Q = R = S = 0. Thus

and (II) is a consequence of (I).