ARITHMETIC
TABLE OF CONTENTS INTRODUCTION 1 FIRST PART ARITHMETIC 6 I CALCULATING WITH NATURAL NUMBERS 7 II THE FIRST EXTENSION OF THE NOTION OF NUMBER 23 III CONCERNING SPECIAL PROPERTIES OF INTEGERS 37 IV COMPLEX NUMBERS 57 CONCERNING THE MODERN DEVELOPMENT AND THE GENERAL STRUCTURE OF MATHEMATICS 79 SECOND PART ALGEBRA 87 I REAL EQUATIONS WITH REAL UNKNOWNS 89 II EQUATIONS IN THE FIELD OF COMPLEX QUANTITIES 107 THIRD PART ANALYSIS 151 I LOGARITHMIC AND EXPONENTIAL FUNCTIONS 152 II THE GONIOMETRIC FUNCTIONS 172 III CONCERNING INFINITESIMAL CALCULUS PROPER 221 INTRODUCTION THE AIM OF THIS LECTURE COURSE IN RECENT YEARS5 A FAR REACHING INTEREST HAS ARISEN AMONG UNIVERSITY TEACHERS OF MATHEMATICS AND NATURAL SCIENCES DIRECTED TOWARD AN APPROPRIATE TRAINING OF FUTURE TEACHERS AT SECONDARY SCHOOLS SATISFYING ALL DESIRABLE NEEDS THIS IS REALLY QUITE A NEW PHENOMENON FOR A LONG TIME PRIOR TO ITS APPEARANCE UNIVERSITY MEN WERE PRACTICING EXCLUSIVELY RESEARCH OF OPTIMAL QUALITY WITHOUT GIVING A THOUGHT TO THE NEEDS OF THE SCHOOLS WITHOUT EVEN CARING TO ESTABLISH A CONNECTION WITH SCHOOL MATHEMATICS WHAT IS THE RESULT OF THIS PRACTICE THE YOUNG UNIVERSITY STUDENT NDS HIMSELF AT THE OUTSET CONFRONTED WITH PROBLEMS WHICH DO NOT REMEMBER IN ANY PARTICULAR THE THINGS WITH WHICH HE HAD BEEN CONCERNED AT SCHOOL NATURALLY HE FORGETS ALL THESE THINGS QUICKLY AND THOROUGHLY WHEN AFTER NISHING HIS COURSE OF STUDY HE BECOMES A TEACHER HE SUDDENLY NDS HIMSELF EXPECTED TO TEACH THE TRADI TIONAL ELEMENTARY MATHEMATICS ACCORDING TO SCHOOL PRACTICE AND SINCE HE WILL BE SCARCELY ABLE UNAIDED TO DISCERN ANY CONNECTION BETWEEN THIS TASK AND HIS UNIVER SITY MATHEMATICS HE WILL SOON FELL IN WITH THE TIME HONOURED WAY OF TEACHING AND HIS UNIVERSITY STUDIES REMAIN ONLY A MORE OR LESS PLEASANT MEMORY WHICH HAS NO IN UENCE UPON HIS TEACHING THERE IS NOW A MOVEMENT TO ABOLISH THIS DOUBLE DISCONTINUITY NEVER HAVING BEEN HELPFUL EITHER TO THE SCHOOL OR TO THE UNIVERSITY ON THE ONE HAND THERE IS AN EFFORT TO IMPREGNATE THE SUBJECT MATTER WHICH THE SCHOOLS TEACH WITH NEW IDEAS DERIVED FROM MODERN DEVELOPMENTS OF SCIENCE AND IN ACCORD WITH MODERN CULTURE WE SHALL OFTEN HAVE OCCASION TO GO INTO THIS ON THE OTHER HAND THE ATTEMPT IS MADE TO TAKE INTO ACCOUNT IN UNIVERSITY TEACHING THE NEEDS OF THE SCHOOL TEACHER AND IT IS PRE CISELY IN SUCH SUMMARISING LECTURE COURSES AS I AM ABOUT TO DELIVER TO YOU THAT I SEE ONE OF THE MOST IMPORTANT TOOLS I SHALL BY NO MEANS ADDRESS MYSELF TO BEGINNERS BUT I SHALL TAKE FOR GRANTED THAT YOU ARE ALL ACQUAINTED WITH THE MAIN FEATURES OF THE MOST IMPORTANT DISCIPLINES OF MATHEMATICS I SHALL OFTEN HAVE TO TALK OF PROBLEMS OF 5 ATTENTION IS AGAIN DRAWN TO THE FACT THAT THE WORDING OF THE TEXT IS ALMOST THROUGHOUT THAT OF THE LITHOGRAPHED VOLUME OF 1908 AND THAT COMMENTS WHICH REFER TO LATER YEARS HAVE BEEN PUT INTO THE FOOTNOTES AND APPENDICES 1 ALGEBRA OF NUMBER THEORY OF FUNCTION THEORY ETC WITHOUT BEING ABLE TO GO INTO DETAILS YOU MUST THEREFORE BE MODERATELY FAMILIAR WITH THESE ELDS IN ORDER TO FOLLOW ME MY TASK WILL ALWAYS BE TO SHOW YOU THE MUTUAL CONNECTION BETWEEN PROBLEMS IN THE VARIOUS DISCIPLINES THESE CONNECTIONS USE NOT TO BE SUF CIENTLY CONSIDERED IN THE SPECIALISED LECTURE COURSES AND I WANT MORE ESPECIALLY TO EMPHA SIZE THE RELATION OF THESE PROBLEMS TO THOSE OF SCHOOL MATHEMATICS IN THIS WAY I HOPE TO MAKE IT EASIER FOR YOU TO ACQUIRE THAT ABILITY WHICH I LOOK UPON AS THE REAL GOAL OF YOUR ACADEMIC STUDY THE ABILITY TO DRAW IN AMPLE MEASURE FROM THE GREAT BODY OF KNOWLEDGE TAUGHT TO YOU HERE AS VIVID STIMULI FOR YOUR TEACHING LET ME NOW PUT BEFORE YOU SOME DOCUMENTS OF RECENT DATE WHICH GIVE EVIDENCE OF WIDESPREAD INTEREST IN THE TRAINING OF TEACHERS AND WHICH CONTAIN VALUABLE MATE RIAL FOR US ABOVE ALL I THINK HERE OF THE TALKS GIVEN AT THE LAST MEETING OF NATURALISTS HELD SEPTEMBER 16 1907 IN DRESDEN TO WHICH BODY WE SUBMITTED THE PROPOSALS FOR THE SCIENTI C TRAINING OF PROSPECTIVE TEACHERS OF MATHEMATICS AND SCIENCE OF THE COMMITTEE ON INSTRUCTION OF THE SOCIETY OF GERMAN NATURALISTS AND PHYSICIANS YOU WILL ND THESE PROPOSALS AS THE LAST SECTION IN THE COMPLETE REPORT OF THIS COMMITTEE6 WHICH SINCE 1904 HAS BEEN CONSIDERING THE ENTIRE COMPLEX OF QUES TIONS CONCERNING TEACHING OF MATHEMATICS AND NATURAL SCIENCES AND HAS NOW ENDED ITS ACTIVITY I URGE YOU TO TAKE NOTICE NOT ONLY OF THESE PROPOSALS BUT ALSO OF THE OTHER PARTS OF THIS VERY INTERESTING REPORT SHORTLY AFTER THE DRESDEN MEETING THERE OCCURRED A SIMILAR DEBATE AT THE MEETING OF GERMAN PHILOLOGISTS AND SCHOOLMEN IN BASEL SEPTEMBER 25 IN WHICH TO BE SURE THE MATHEMATICAL SCIENTI C REFORM MOVEMENT WAS DISCUSSED ONLY AS ONE LINK IN THE CHAIN OF PARALLEL MOVEMENTS OC CURRING IN THE SPHERES OF PHILOLOGISTS AFTER A REPORT BY ME CONCERNING OUR AIMS IN MATHEMATICAL NATURAL SCIENCE REFORM THERE WERE TALKS BY PAUL WENDLAND BRESLAU ON QUESTIONS IN CLASSICAL AND ANCIENT STUDIES ALMUT BRANDL BERLIN ON MODERN LANGUAGES AND NALLY ADOLF HARNACK BERLIN ON HISTORY AND RELIGION THESE FOUR TALKS APPEARED TOGETHER IN ONE BROCHURE7 TO WHICH I PARTICULARLY REFER YOU I APPRECI ATE THE THUS INITIATED JOINT PROCEEDING OF OUR SCIENCES WITH THE PHILOLOGISTS AS HIGHLY FRUITFUL SINCE IT WILL BRING ABOUT FRIENDLY FEELING AND MUTUAL UNDERSTANDING BETWEEN TWO GROUPS WHO FACE EACH OTHER AS ALIENS OR EVEN ADVERSELY LET US ENDEAVOUR AL WAYS TO FOSTER SUCH GOOD RELATIONS EVEN IF WE DO AMONG OURSELVES OCCASIONALLY DROP A CRITICAL WORD ABOUT THE PHILOLOGISTS JUST AS THEY MAY ABOUT US BEAR IN MIND THAT YOU WILL LATER BE CALLED UPON IN THE SCHOOLS TO WORK TOGETHER WITH THE PHILOLOGISTS FOR THE COMMON GOOD AND THAT THIS REQUIRES MUTUAL UNDERSTANDING AND APPRECIATION ALWAYS BEYOND THE PARTICULARISM FOR THE PROPER DISCIPLINE 6 DIE T TIGKEIT DER UNTERRICHTSKOMMISSION DER GESELLSCHAFT DEUTSCHER NATURFORSCHER UND RZTE EDITED BY AUGUST GUTZMER LEIPZIG AND BERLIN 1908 THE GERMAN TITLE OF THE LAST SECTION IS VORSCHL GE F R DIE WISSENSCHAFTLICHE AUSBILDUNG DER LEHRAMTSKANDIDATEN DER MATHEMATIK UND NATURWISSENSCHAFTEN 7 UNIVERSIT T UND SCHULE VORTR GE GEHALTEN VON F KLEIN PAUL WENDLAND ALMUT BRANDL ADOLF HARNACK LEIPZIG 1907 2 ARITHMETIC WE NEED TO GRADE 3 RELEVANT LITERATURE ALONG WITH THIS EVIDENCE OF EFFORTS WHICH REACH BEYOND THE BORDERS OF OUR ELD I SHOULD LIKE TO MENTION A FEW BOOKS WHICH AIM IN THE SAME DIRECTION IN PARTICULAR IN THE MATHEMATICAL ELD AND WHICH WILL THEREFORE BECOME VERY IMPORTANT FOR THESE LECTURES THREE YEARS AGO I GAVE FOR THE RST TIME A COURSE OF LECTURES WITH A SIM ILAR PURPOSE MY ASSISTANT AT THAT TIME RUDOLF SCHIMMACK WORKED THE MATERIAL UP AND THE RST PART HAS RECENTLY APPEARED IN PRINT8 IN IT ARE CONSIDERED THE DIF FERENT KINDS OF SCHOOLS INCLUDING HIGHER EDUCATION THE GENERAL METHODS TO TEACH MATHEMATICS IN THEM THEIR MUTUAL INTERACTING AND OTHER SIMILAR MATTERS IN WHAT FOLLOWS I SHALL FROM TIME TO TIME REFER TO THINGS WHICH APPEAR THERE WITHOUT REPEAT ING THEM THE MORE EXTENSIVE I WILL DISCUSS HERE SOMEHOW AS CONTINUATION OF THOSE OBSERVATIONS ON THE ORGANISATION OF MATHEMATICS TEACHING THE MATHEMATICAL SUBJECT MATTER WHICH IN WHATEVER WAY CAN BE RELEVANT FOR TEACHING IN SCHOOLS IF I FREQUENTLY REFER TO THE ACTUAL METHOD OF TEACHING IN THE SCHOOLS MY REMARKS WILL BE BASED NOT MERELY UPON INDE NITE PICTURES OF HOW THE THING MIGHT BE DONE OR EVEN UPON DIM RECOLLECTIONS OF MY OWN SCHOOL DAYS FOR I AM CONSTANTLY IN TOUCH WITH SCHIMMACK WHO IS NOW TEACHING IN THE G TTINGEN GYMNASIUM AND WHO KEEPS ME INFORMED AS TO THE PRESENT STATE OF TEACHING WHICH HAS IN FACT ADVANCED SUB STANTIALLY BEYOND WHAT IT WAS IN EARLIER YEARS DURING THIS WINTER SEMESTER I SHALL DISCUSS THE THREE GREAT A S THAT IS ARITHMETIC ALGEBRA AND ANALYSIS WITHHOLDING GEOMETRY FOR A CONTINUATION OF THE LECTURE COURSE DURING THE COMING SUMMER TERM LET ME REMARK THAT IN THE LANGUAGE OF THE SECONDARY SCHOOLS THESE THREE SUBJECTS ARE CLASSED TOGETHER AS ARITHMETIC AND THAT WE SHALL MORE OFTEN NOTE DEVIATIONS IN THE MATHEMATICAL TERMINOLOGY OF THE SCHOOLS AS COMPARED WITH THAT IN HIGHER EDUCATION THIS SMALL ILLUSTRATION SHOWS YOU THAT ONLY VIVID INTERLOCKING CAN BRING ABOUT AGREEMENT AS A SECOND REFERENCE I SHALL MENTION THE THREE VOLUME ENZYKLOP DIE DER ELE MENTARMATHEMATIK BY HEINRICH WEBER AND JOSEF WELLSTEIN THE WORK WHICH AMONG RECENT PUBLICATIONS MOST NEARLY ACCORDS WITH MY OWN TENDENCIES FOR THIS SEMESTER THE RST VOLUME ENZYKLOP DIE DER ELEMENTAREN ALGEBRA UND ANALYSIS PREPARED BY H WEBER9 WILL BE THE MOST IMPORTANT I SHALL INDICATE AT ONCE CERTAIN DIFFERENCES BETWEEN THIS WORK AND THE PLAN OF MY LECTURE COURSE IN WEBER WELLSTEIN THE EN TIRE STRUCTURE OF ELEMENTARY MATHEMATICS IS BUILT UP SYSTEMATICALLY AND LOGICALLY IN THE MATURE LANGUAGE ACCESSIBLE TO THE ADVANCED STUDENT NO ACCOUNT IS TAKEN OF HOW THESE THINGS ACTUALLY MAY COME UP IN SCHOOL TEACHING THE PRESENTATION IN THE SCHOOLS HOWEVER SHOULD BE PSYCHOLOGICAL TO USE A CATCH WORD AND NOT SYSTEMATIC THE TEACHER SO TO SPEAK MUST BE A DIPLOMAT HE MUST TAKE ACCOUNT OF THE PSYCHIC PROCESSES IN THE BOY IN ORDER TO GRIP HIS INTEREST AND HE WILL SUCCEED 8 KLEIN F VORTR GE BER DEN MATHEMATISCHEN UNTERRICHT AN H HEREN SCHULEN PREPARED BY RUDOLF SCHIMMACK PART 1 VON DER ORGANISATION DES MATHEMATISCHEN UNTERRICHTS LEIPZIG 1907 THIS BOOK IS REFERRED TO LATER AS KLEIN SCHIMMACK 9 SECOND EDITION LEIPZIG 1906 FOURTH EDITION 1922 REVISED BY P EPSTEIN REFERRED TO AS WEBER WELLSTEIN I 3 ARITHMETIC WE NEED TO GRADE 3