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ALGEBRAIC AND TRANSCENDENTAL NUMBERS

The Bolzano–Weierstrass theorem states that every bounded in?nite set has at least one limit point.

A proof of this is given in Problem 2.23, Chapter 2.

ALGEBRAIC AND TRANSCENDENTAL NUMBERS

A number x which is a solution to the polynomial equation

where a₀6¼ 0, a₁; a₂; . . . ; a_nare integers and n is a positive integer, called the degree of the equation, is

called an     algebraic number.          A number which cannot be expressed as a solution of any polynomial

equation with integer coe?cients is called a transcendental number.

EXAMPLES.²3^andp???₂which are solutions of 3x         2 ¼ 0 and x²2 ¼ 0, respectively, are algebraic numbers.

The numbers         and   e can be shown to be transcendental numbers.                   Mathematicians have yet to

determine whether some numbers such as e            or e ‏         are algebraic or not.

The set of algebraic numbers is a countably in?nite set (see Problem 1.23), but the set of transcen-

dental numbers is non-countably in?nite.

THE COMPLEX NUMBER SYSTEM

Equations such as         x²‏ 1 ¼ 0 have no solution within the real number system.                            Because these

equations were found to have a meaningful place in the mathematical structures being built, various

mathematicians of the late nineteenth and early twentieth centuries developed an extended system of

numbers in which there were solutions. The new system became known as the complex number system.

It includes the real number system as a subset.

We can consider a complex number as having the form a ‏ bi, where a and b are real numbers called

the real and imaginary parts, and i ¼                  1 is called the imaginary unit.           Two complex numbers a ‏ bi

and c ‏ di are equal if and only if a ¼ c and b ¼ d. We can consider real numbers as a subset of the set

of complex numbers with b ¼ 0.          The complex number 0 ‏ 0i corresponds to the real number 0.

The absolute value or modulus of a ‏ bi is de?ned as ja ‏ bij ¼                        a²‏ b². The complex conjugate of

a ‏ bi is de?ned as a             bi. The complex conjugate of the complex number z is often indicated by z or z                   .

The set of complex numbers obeys rules 1 through 9 of Page 2, and thus constitutes a ?eld.                                In

performing operations with complex numbers, we can operate as in the algebra of real numbers, replac-