REAL MATHEMATICS
CONTENTS THE SET OF REAL NUMBERS ARITHMETIC IN R THE ORDER RELATION IN R COMPLETENESS OF THE REAL NUMBER SYSTEM THE GEOMETRIC REPRESENTATION ALGEBRAIC IDENTITIES, EQUATIONS AND SYSTEMS ALGEBRAIC IDENTITIES THE MODULUS OF A REAL NUMBER A FIRST LOOK AT POLYNOMIAL EQUATIONS LINEAR SYSTEMS AND ELIMINATION MISCELLANEOUS ELEMENTARY SEQUENCES PROGRESSIONS LINEAR RECURRENCES OF ORDERS 2 AND 3 THE † AND … NOTATIONS INDUCTION AND THE BINOMIAL FORMULA THE PRINCIPLE OF MATHEMATICAL INDUCTION BINOMIAL NUMBERS THE BINOMIAL FORMULA ELEMENTARY INEQUALITIES THE AM-GM INEQUALITY CAUCHY’S INEQUALITY MORE ON INEQUALITIES THE CONCEPT OF FUNCTION DE?NITIONS AND EXAMPLES MONOTONICITY, EXTREMA AND IMAGE COMPOSITION OF FUNCTIONS INVERSION OF FUNCTIONS DE?NING FUNCTIONS IMPLICITLY GRAPHS OF FUNCTIONS TRIGONOMETRIC FUNCTIONS MORE ON REAL NUMBERS SUPREMUM AND IN?MUM LIMITS OF SEQUENCES KRONECKER’S LEMMA SERIES OF REAL NUMBERS CONTINUOUS FUNCTIONS THE CONCEPT OF CONTINUITY SEQUENTIAL CONTINUITY THE INTERMEDIATE VALUE THEOREM LIMITS AND DERIVATIVES SOME HEURISTICS I LIMITS OF FUNCTIONS BASIC PROPERTIES OF DERIVATIVES COMPUTING DERIVATIVES RÔLLE’S THEOREM AND APPLICATIONS THE FIRST VARIATION OF A FUNCTION THE SECOND VARIATION OF A FUNCTION SKETCHING GRAPHS RIEMANN’S INTEGRAL SOME HEURISTICS II THE CONCEPT OF INTEGRAL RIEMANN’S THEOREM AND SOME REMARKS OPERATING WITH INTEGRABLE FUNCTIONS THE FUNDAMENTAL THEOREM OF CALCULUS THE CHANGE OF VARIABLES FORMULA LOGARITHMS AND EXPONENTIALS IMPROPER INTEGRATION TWO IMPORTANT APPLICATIONS SERIES OF FUNCTIONS TAYLOR SERIES POWER SERIES SOME APPLICATIONS A GLIMPSE ON ANALYTIC FUNCTIONS CHAPTER THE SET OF REAL NUMBERS THIS ?RST CHAPTER RECALLS SOME DE?NITIONS AND RESULTS WHICH ARE ESSENTIAL TO ALL FURTHER DEVELOPMENTS WE ASSUME FROM THE READER A MODEST ACQUAINTANCE WITH THE MOST BASIC CONCEPTS OF SET THEORY; WE ALSO ASSUME THAT HE OR SHE IS FAMILIAR WITH THE SETS OF NATURALS, N D F1; 2; 3; 4; : : :G; INTEGERS, Z D F0; ?1; ?2; ?3; : : :G; AND RATIONALS, Q D NA BI A; B 2 Z; B ¤ O ; AS WELL AS WITH THE ELEMENTARY ARITHMETIC OPERATIONS WITHIN THESE SETS IN WHAT CONCERNS THE INTEGERS1, GIVEN A; B 2 Z, WITH A ¤ 0, WE SAY THAT A DIVIDES B IF THERE EXISTS AN INTEGER C SUCH THAT B D AC EQUIVALENTLY, TO SAY THAT A DIVIDES B IS THE SAME AS TO SAY THAT THE RATIONAL NUMBER B A IS AN INTEGER; FOR EXAMPLE, 13 DIVIDES 52, SINCE 13 D 4 IF A DIVIDES B, WE ALSO SAY THAT A IS A DIVISOR OF B, OR THAT B IS DIVISIBLE BY A; IN SUCH A CASE, WE DENOTE A J B IF A DOESN’T DIVIDE B (OR, WHICH IS THE SAME, IF B A … Z), WE DENOTE A ?B AN INTEGER N IS EVEN IF 2 J N; OTHERWISE, N IS SAID TO BE ODD HENCE, 0, ?2, ?4, ?6, ARE THE EVEN INTEGERS, WHILE ?1, ?3, ?5, ARE THE ODD ONES THE SET OF REAL NUMBERS GIVEN NATURAL NUMBERS A AND B, IT IS WELL KNOWN THAT THERE EXIST UNIQUE INTEGERS Q AND R SATISFYING THE FOLLOWING CONDITIONS: B D AQ C R AND 0 R < A: (1 1) THE ABOVE RELATION IS KNOWN AS THE DIVISION ALGORITHM, AND THE INTEGERS Q AND R ARE RESPECTIVELY CALLED THE QUOTIENT AND THE REMAINDER OF THE DIVISION OF B BY A CONDITIONS (1 1) ARE USUALLY CONDENSED IN THE DIAGRAM B A R Q IN PARTICULAR, FOR GIVEN NATURALS A AND B AND IN THE ABOVE NOTATIONS, A J B IS THE SAME AS R D 0 AND Q D B A TWO NONZERO INTEGERS A AND B ALWAYS HAVE A GREATEST COMMON DIVISOR, WHICH WILL BE DENOTED GCD A; B/; MOREOVER, A AND B ARE SAID TO BE RELATIVELY PRIME IF GCD A; B/ D 1; IN PARTICULAR, IF B D KA C 1, THEN A AND B ARE RELATIVELY PRIME IF R D M N , WITH M AND N INTEGERS, IS A REPRESENTATION OF THE NONZERO RATIONAL R, THEN, BY CANCELLING OUT FACTORS COMMON TO M AND N (I E , CANCELLING GCD M; N/ OUT OF M AND N), WE GET AN IRREDUCIBLE REPRESENTATION FOR R FOR EXAMPLE, SINCE GCD 12; 18/ D 6, THE RATIONAL NUMBER 18 HAS 3 AS AN IRREDUCIBLE REPRESENTATION, WHICH WAS OBTAINED BY CANCELLING OUT A FACTOR 6 FROM BOTH 12 AND 18 AN INTEGER P > 1 IS PRIME IF 1 AND P ARE ITS ONLY POSITIVE DIVISORS; IN ANOTHER WAY, AN INTEGER P > 1 IS PRIME IF, FOR A 2 N, WE HAVE THAT P A 2 N ) A D 1 OR P: IT IS A WELL KNOWN FACT (CF CHAP 6 OF [5]) THAT THE SET OF PRIME NUMBERS IS IN?NITE BELOW, WE LIST ALL PRIME NUMBERS LESS THAN 100: 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71; 73; 79; 83; 89; 97: AN INTEGER GREATER THAN 1 AND WHICH IS NOT PRIME IS SAID TO BE COMPOSITE IT IS ALSO A WELL KNOWN FACT (CF CHAP 6 OF [5]) THAT EVERY NATURAL NUMBER N > 1 CAN BE WRITTEN (OR DECOMPOSED) AS A PRODUCT OF A ?NITE NUMBER OF POWERS2 OF PRIME NUMBERS (ITS PRIME FACTORS); ALSO, SUCH A REPRESENTATION OF N IS UNIQUE, UP TO THE ORDER OF THE POWERS FOR INSTANCE, 9000 D 233253 IS THE DECOMPOSITION OF 9000 AS A PRODUCT OF POWERS OF PRIMES THE EXISTENCE AND UNIQUENESS OF SUCH A DECOMPOSITION OF N > ARITHMETIC IN R WE USE TO REPRESENT RATIONAL NUMBERS IN DECIMAL NOTATION FOR THE RATIONAL NUMBER 8, FOR EXAMPLE, WE WRITE 8 D 0:125 AS A SHORTHAND FOR THE EQUALITY 8 D 10 C 102 C 103 ; AND SAY THAT 0:125 IS THE DECIMAL REPRESENTATION OF 8 SOME RATIONAL NUMBERS HAVE MORE COMPLICATED DECIMAL REPRESENTATIONS WE TAKE AS AN EXAMPLE THE RATIONAL 12, FOR WHICH WE USUALLY WRITE 12 D 0:08333 : : : : WHAT DOES THIS EQUALITY MEAN? ARGUING AS IN THE CASE OF 8, WE ARE TEMPTED TO SAY THAT THIS EQUALITY IS A SHORTHAND FOR 12 D 103 C 104 C 105 C : (1 2) THIS IS ACTUALLY SO, PROVIDED WE CORRECTLY INTERPRET THE SUM WITH AN IN?NITE NUMBER OF SUMMANDS AT THE RIGHT HAND SIDE RIGOROUSLY, (1 2) MEANS THAT, IF WE ?X THE A PRIORI MAXIMUM ERROR 10N D 0: 00 „ ?‚ … N , THEN WE HAVE 0 < 104 C C 10K 10N FOR EVERY NATURAL NUMBER K N; YET IN ANOTHER WAY, (1 2) MEANS THAT ALL OF THE NUMBERS 10K , WITH K N, APPROXIMATE 12 BY DEFECT WITH ERROR LESS THAN OR EQUAL TO 0: 00 THAT 10K D 3 10K ; SO THAT THE ERROR IN THE DEFECT APPROXIMATION 10K OF 12 EQUALS 310K , WHICH IS ALWAYS LESS THAN OR EQUAL TO THE MAXIMUM ERROR 10N , WHENEVER K N IT IS PRECISELY IN THIS SENSE THAT WE SHOULD THINK OF THE EQUALITY 12 D 0:08333 THE SET OF REAL NUMBERS IN THE LIGHT OF THE ABOVE DISCUSSION, ONE QUESTION SUGGESTS ITSELF NATURALLY: FOR AN ARBITRARY SEQUENCE3 A1; A2; A3; : : :/ OF DECIMAL DIGITS, CAN WE THINK OF 0:A1A2A3 : : : AS THE DECIMAL REPRESENTATION OF SOME RATIONAL NUMBER? IT IS POSSIBLE TO PROVE (AND WE WILL DO SO IN PROBLEM 2, PAGE 7) THAT THE ANSWER TO THIS QUESTION IS YES IF AND ONLY IF THE LIST A1; A2; A3; : : :/ IS PERIODIC FROM A CERTAIN POINT ON, I E , IF AND ONLY IF IT IS OF THE FORM A1; A2; : : : ; AL; B1; B2; : : : ; BP „ ?‚ … P ; B1; B2; : : : ; BP „ ?‚ … P ; B1; B2; : : : ; BP „ ?‚ … P ; : : :/: (1 3) (IN PARTICULAR, FOR THE RATIONAL NUMBER