Preface		xi
PART 1 JUDAISM AND MATHEMATICS
	Foreword     by Rabbi Dr. Morris Pickholz	3
	Introduction References	5
1	NUMBERS IN JUDAISM AND MATHEMATICS	8
	Introduction	8
	One	10
	Six and Seven	14
	Two through Five	19
	Ten, Twelve, and Eighteen	22
	Prime Numbers	28
2	INFINITY IN JUDAISM AND MATHEMATICS	31
	Attributes of God	31
	Names of God	34
	Mathematics and Infinity	36
	Transfinite Numbers	38
	The Omnipresence of God	40
	Creation of the Universe	42
	The Eternity of God	45
	The Omnipotence of God	46
3	OTHER TOPICS IN JUDAISM AND MATHEMATICS	50
	Deductive Systems	50
	Does Mathematics Represent True Wisdom?	52
	Does Science Represent True Wisdom?	54
	Proofs of God's Existence	56
	Higher Dimensions	60
	Computers: A Blessing Or A Curse?	64
	Talmudic Mathematics	66
4	GEMATRIA OF THE TORAH AND THE PRAYER BOOK	71
	Introduction	71
	Types of Gematria	73
	Preliminary Examples	75
	A Note of Caution	77
	Bereisheyt	79
	First Words of the Torah	81
	God's Ineffable Name	87
	Biblical Families	93
	The Priestly Blessings	103
	Other Topics	107
	Table 4.1   Gematria and Reverse Gematria Correspondences	117
PART 2 MATHEMATICS OF THE HEBREW CALENDAR
	Foreword by Herbert A. Hauptman, Ph.D., 1985 Nobel Laureate	121
	Introduction	122
	References	124
1	INTRODUCTION TO THE HEBREW CALENDAR	125
	Hebrew/Civil Calendar Comparisons	126
	Hebrew Months and Days per Month	127
	Leap Years and Regular Years	129
	The Fourteen Types of Calendars	132
	Lunar Months and Years	134
	Review and a Look Ahead	137
	Computing the NYM	138
	The First Day of the Year	139
	Construction of Hebrew Calendars II	141
	Hebrew Calendar Time	142
	Days, Hours, Parts	145
	Construction of Hebrew Calendars II	147
	Where Do We Go from Here?	149
	Importance of the Hebrew Calendar	150
	Table 1.1   Regular Years	152
	Table 1.2   Leap Years	153
	Table 1.3   Number of Days in Year	154
2	FOUNDATIONS OF THE HEBREW CALENDAR	155
	Introduction	155
	Historical Background	157
	The Hebrew Months	158
	The Nineteen-Year Cycles	159
	Days, Hours, Parts	163
	Months and Years	165
	Arithmetic of Ordered Triples	167
	The Set S and Standard Form	168
	Group S and a Coset Decomposition of S (Optional)	170
	A Preview of Future Chapters	170
3	CONSTRUCTION OF HEBREW CALENDARS	172
	Introduction	174
	First Day of the Hebrew Year	174
	Postponement Rules 1 and 2	175
	Constructing Hebrew Calendars	176
	Postponement Rules 3 and 4	179
	Q(D, H, P) and Q()	182
	The First Nineteen Years	185
	The New Moon of (Month, )	187
	Table 3.1   New Year Moons of Years 1 to 19	189
	Table 3.2   Multiples of (1,12,793) in Standard Form	190
4	FURTHER ANALYSIS OF ORDERED TRIPLES	191
	Introduction	191
	Linear Diophantine Equations in Two Variables (Optional)	193
	N(Month, ) and N(D, H, P)	197
	The Set of New Moons	200
	The Set of New Year Moons	203
	A Partition of Set S: The New Year Moons	206
	The Partition of Set S: All New Moons	211
	Table 4.1 Month Numbers	218
5	FURTHER ANALYSIS OF POSTPONEMENT RULES	220
	Introduction	220
	Postponement Intervals (Optional)	223
	A Partition of the Years 19N + K (Optional)	225
	Number of Days in a Year (Optional)	226
	Applications of Tables 5.1 to 5.4	228
	Counting Ordered Triples	230
	Tables 5.1 to 5.4   Calendar Information for Four Sets of Years	234
6	PROBABILITIES ASSOCIATED WITH THE HEBREW CALENDAR	242
	Introduction	242
	Introduction to Probability	245
	Hebrew Calendar Probabilities	249
	Holiday Probabilities	256
	Rosh Chodesh / Shabbat Probabilities	259
	Table 6.1   Probabilities: Leap Years, Regular Years, All Years	263
	Table 6.2   Days of Jewish Holidays	265
	Table 6.3   Percentage of Years Holiday (starts) on Given Day	266
	Table 6.4   Rosh Chodesh / Twenty-ninth Day of Month on Saturday	267
	Table 6.5   Percentages of Rosh Chodesh / 29th Day on Saturday	269
7	HEBREW DATES AND CIVIL DATES	270
	Introduction	270
	Notation for Hebrew/Civil Dates	271
	Number of Days between Two Dates	273
	The Gregorian Civil Date of Tishri 1	278
	The Julian Civil Date of Tishri 1	285
	Table 7.1   The Years 5701 to 5719	287
	Table 7.2   Number of Days Prom (September 1, C)	288
	Table 7.3(a) to 7.3(f)   Set of Six Perpetual Hebrew Calendars	292
PART 3 SUPPLEMENTARY SECTIONS
1	SOLUTIONS TO EXERCISES IN PART 2	319
2	PROOFS OF THEOREMS IN PART 2	359
3	APPENDIX FOR PART 2	373
	1. 400 Year Repetition of the Civil (Gregorian) Calendar	373
	2. Analysis Leading to Postponement Rules 3 and 4	374
	3. Years 1 to 19 (Table 3.1)	375
	4. Equations 5x + 3y = 38 and 2x - 7y = 1	376
	5. Equation 39673x - 181440y = 1 (Theorem 4.1)	377
	6. Equation 13943n - 36288y = j (Theorem 4.3)	378
	7. Completion of Proof of Theorem 4.7	379
	8. Number of Days in a Leap Year (Table 5.1)	381
	9. Number of Days in a Regular Year (Tables 5.2, 5.3, 5.4)	385
	10. Number of Triples in [Tishri, k] (Tables 5.1 to 5.4)	392
	11. Probabilities in Table 6.1 394	394
	12. Holiday Percentages in Table 6.3	396
	13. Rosh Chodesh / 29th Day Percentages in Table 6.5	397
	14. Years 5701 to 5719 (Table 7.1)	399
4	PART I   OUTLINES OF TOPICS AND BIBLICAL REFERENCES	404
5	PART 2   OUTLINES OF TOPICS AND BIBLICAL REFERENCES	412
Index		416