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A set containing all its limit points is called a closed set. The set of rational numbers is not a closed p set since, for example, the limit point 2 is not a member of the set (Problem 1.5). However, the set of all real numbers x such that 0 @ x @ 1 is a closed set.
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BOUNDS If for all numbers x of a set there is a number M such that x @ M, the set is bounded above and M is called an upper bound. Similarly if x A m, the set is bounded below and m is called a lower bound. If for all x we have m @ x @ M, the set is called bounded. If M is a number such that no member of the set is greater than M but there is at least one member which exceeds M  for every  > 0, then M is called the least upper bound (l.u.b.) of the set. Similarly " " if no member of the set is smaller than m but at least one member is smaller than m  for every  > 0, " then m is called the greatest lower bound (g.l.b.) of the set.
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BOLZANO WEIERSTRASS THEOREM 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, 2.
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ALGEBRAIC AND TRANSCENDENTAL NUMBERS A number x which is a solution to the polynomial equation a0 xn a1 xn 1 a2 xn 2 an 1 x an 0 1
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where a0 6 0, a1 ; a2 ; . . . ; an are 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.
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p 2 which are solutions of 3x 2 0 and x2 2 0, respectively, are algebraic numbers.
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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 transcendental numbers is non-countably in nite.
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THE COMPLEX NUMBER SYSTEM Equations such as x2 1 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 having the form a bi, where a and b are real numbers called pas the real and imaginary parts, and i 1 is called the imaginary unit. Two complex numbers a bi and c di are equal if and only if a c and b d. We can consider real numbers as a subset of the set of complex numbers with b 0. The complex number 0 0i corresponds to the real number 0. p The absolute value or modulus of a bi is de ned as ja bij 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, replacing i2 by 1 when it occurs. Inequalities for complex numbers are not de ned.
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