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I wrote this note in 2008 to introduce complex numbers. It is posted for the Math Teachers at Play blog carnival.
What is the ?
In the set of Real Numbers, , the solution is undefined. We must consider the set of Complex Numbers, . Complex numbers are numbers of the form of , where and are real numbers: and is the imaginary unit. Note is defined by or equivalently .
Thus, we have:
We now have . So we must ask "What is the ?". To evaluate we will use Euler’s formula, So by substituting , giving
Note we have isolated and now arrived at the following equality: . Taking the square root of both sides gives
Thus, we have now have and following through application of Euler’s formula to , gives
So going back to the original problem and substituting , gives
Hence, the solution, , is a complex number!
Proofs from THE BOOK by Martin Aigner and Günter Ziegler begins by giving six proofs of the infinity of primes. We will go over the third proof. Before we go over this proof, lets cover some background.
— 1. Fermat Numbers —
Fermat numbers are defined by
so that and . They are of great interest in many ways: for example, it was proved by Gauss that, if , is a prime , then a regular polygon of sides can be inscribed in a circle by Euclidean methods. The property of the Fermat numbers which is relevant here is
We will prove this theorem later.
The first four Fermat numbers are prime, and Fermat conjectured that all were prime. Euler, however, found in 1732 that
In 1880 Landry proved that
It is currently known that , is composite for . Factoring Fermat numbers is extremely difficult as a result of their large size. has known factors with remaining (where denotes a composite number with digits). has known factors with remaining. has no known factors but is composite. There are currently four Fermat numbers that are known to be composite, but for which no single factor is known: and . In all the other cases proved to be composite a factor is known. No prime has been found beyond , and it seems unlikely that any more will be found using current computational methods and hardware.
— 2. Infinitude of Primes Theorem —
We are now ready to prove Euclid’s Second Theorem, also called the Infinitude of Primes Theorem using the third proof in Proofs from THE BOOK.
from which our assertion follows immediately. Indeed, if is a divisor of, say, and , then divides , and hence or . But is impossible since all Fermat numbers are odd.
To prove the recursion we use induction on . For we have and . With induction we now conclude
Proofs from THE BOOK by Martin Aigner and Günter Ziegler begins by giving six proofs of the infinity of primes. We will go over the second proof. Before we go over this proof, lets cover some background.
— 1. Algebraic Structures —
One of the themes of modern algebra is to compare algebraic structures. An algebraic structure refers to a nonempty set equipped with a binary operation or several binary operations (usually two). Also, we shall refer to a nonempty set equipped with two binary operations satisfying certain properties as a number system. First, we shall define algebraic structures called group, ring and field. A group consists of nonempty set with one binary operation satisfying several conditions. It is the simplest of the three.
The Euler formula, sometimes also called the Euler identity, states
where is the imaginary unit. Note that Euler’s polyhedral formula is sometimes also called the Euler formula, as is the Euler curvature formula.
The special case of the formula with gives the beautiful identity
an equation connecting the fundamental numbers and , the fundamental operations , , and exponentiation, the most important relation , and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician.