Nature of Mathematics PDF

MATHEMATICS IN HISTORY  HUMAN MIND AND CULTURE HAVE DEVELOPED A FORMAL SYSTEM OF THOUGH FOR RECOGNIZING, CLASSIFYING, AND EXPLOITING PATTERNS CALLED MATHEMATICS BY USING MATHEMATICS TO ORGANIZE AND SYSTEMATIZE OUT IDEAS ABOUT PATTERN, WE HAVE DISCOVERED GREAT SECRET: NATURE’S PATTERNS A...

MATHEMATICS IN HISTORY  HUMAN MIND AND CULTURE HAVE DEVELOPED A FORMAL SYSTEM OF THOUGH FOR RECOGNIZING, CLASSIFYING, AND EXPLOITING PATTERNS CALLED MATHEMATICS BY USING MATHEMATICS TO ORGANIZE AND SYSTEMATIZE OUT IDEAS ABOUT PATTERN, WE HAVE DISCOVERED GREAT SECRET: NATURE’S PATTERNS ARE NOT JUST THERE TO BE ADMIRED, THEY ARE VITAL CLUES TO THE RULES THAT GOVERNS NATURAL PROCESSES. MATHEMATICS IN EGYPT  COUNTING IN EARLY EGYPT 10 IS A HOBBLE FOR CATTLE 10,000 IS REPRESENTED BY A FINGER 100 IS REPRESENTED BY A COILED ROPE 100,000 IS REPRESENTED BY A FROG 1000 IS REPRESENTED BY A LOTUS FLOWER MILLION WAS REPRESENTED BY A GOD WITH HIS HANDS RAISED IN ADORATION MATHEMATICS IN EGYPT MATHEMATICS IN ANCIENT EGYPT IS COMPOSED OF FOUR MAIN OPERATION. ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION WHICH IS ALSO USED NOWADAYS. THE ONLY DIFFERENCE IS INSTEAD OF NUMBERS THEY USE SYMBOLS CALLED HIEROGLYPHICS/COUNTING GLYPHS.  NUMBER PATTERNS IN MATHEMATICS, SEQUENCES ARE DEFINING THE COLLECTIONS OF NUMBERS (OR OTHER OBJECTS) THAT FOLLOWS A PATTERNS. THE DIFFERENT ELEMENTS IN A SEQUENCE ARE CALLED TERMS AND IT CAN BE A FINITE OR INFINITE SEQUENCE. EXAMPLE: FINITE SEQUENCES: {1, 2, 3, 4, 5} , {2, 4,6, 8, 10} INFINITE SEQUENCES: {1, 2, 3, …} NUMBER PATTERNS ARITHMETIC SEQUENCE AN ARITHMETIC SEQUENCE HAS A CONSTANT DIFFERENCE BETWEEN TERMS. THE FIRST TERM IS A1, THE COMMON DIFFERENCE IS D, AND THE NUMBER OF TERMS IS N. EXPLICIT FORMULA: FIND THE NEXT TWO TERMS IN THE SEQUENCE: {1, 4, 7, 10, 13, ___, ___} WE CAN SOLVE THE DIFFERENCE (D) BY SUBTRACTING THE TWO CONSECUTIVE NUMBERS. 4-1=3 7-4=3 10-7=3 13-10=3 BY REPEATING THIS PROCESS, WE CAN SAY THAT THE NEXT TWO TERMS ARE 16 AND 19.  USING THE FORMULA, 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅, WE CAN ALSO FIND THE OTHER SEQUENCES. B. FIND THE 25TH TERM IN THE SEQUENCE: {1, 4, 7, 10, 13, …𝑎25 } 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 𝑎25 = 1 + 25 − 1 3 𝑎25 = 73 C. FIND THE 10TH TERM IN THE SEQUENCE: {5, 10, 15, 20, …𝑎10 } 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 𝑎10 = 5 + 10 − 1 5 𝑎10 = 50 NUMBER PATTERNS GEOMETRIC SEQUENCE A GEOMETRIC SEQUENCE HAS A CONSTANT RATIO BETWEEN TERMS. THE FIRST TERM IS A1, THE COMMON RATIO IS R, AND THE NUMBER OF TERMS IS N. EXPLICIT FORMULA: a. FIND THE NEXT TWO TERMS IN THE SEQUENCE: {2, 8, 32, ___, ___} WE CAN SOLVE THE RATION (R) BY DIVIDING THE TWO CONSECUTIVE NUMBERS. 8 =4 2 32 =4 8 BY REPEATING THIS PROCESS, WE CAN SAY THAT THE NEXT TWO TERMS ARE 128 AND 512 USING THE FORMULA, 𝒂𝒏 = 𝒂𝟏 𝒓 𝒏−𝟏 , WE CAN ALSO FIND THE OTHER SEQUENCES. B. FIND THE 10TH TERM OF A GEOMETRIC SEQUENCE: {1, 5, 25, 125, …𝑎10 } 𝒂𝒏 = 𝒂𝟏 𝒓 𝒏−𝟏 𝑎10 = 1 𝟓 𝟏𝟎−𝟏 𝑎10 = 1,953,125 C. FIND THE 12TH TERM OF A GEOMETRIC SEQUENCE: {4, 8,16, 32, …𝑎12 } 𝒏−𝟏 𝒂𝒏 = 𝒂𝟏 𝒓 𝑎12 = 4 𝟐 𝟏𝟐−𝟏 𝑎12 = 8,192 NUMBER PATTERNS FIBONACCI SEQUENCE THE SEQUENCE OF NUMBERS WHICH THE NEXT TERM IS FOUND BY ADDING THE PREVIOUS TWO TERMS. WHAT IS THE GOLDEN RATIO?  FLOWER PETALS: THE NUMBER OF PETALS ON SOME FLOWERS FOLLOWS THE FIBONACCI SEQUENCE. IT IS BELIEVED THAT IN THE DARWINIAN PROCESSES, EACH PETAL IS PLACED TO ALLOW FOR THE BEST POSSIBLE EXPOSURE TO SUNLIGHT AND OTHER FACTORS.  SEED HEADS: THE SEEDS OF A FLOWER ARE OFTEN PRODUCED AT THE CENTER AND MIGRATE OUTWARD TO FILL THE SPACE. FOR EXAMPLE, SUNFLOWERS FOLLOW THIS PATTERN.  TREEBRANCHES: THE WAY TREE BRANCHES FORM OR SPLIT IS AN EXAMPLE OF THE FIBONACCI SEQUENCE. ROOT SYSTEMS AND ALGAE EXHIBIT THIS FORMATION PATTERN.  SHELLS: MANY SHELLS, INCLUDING SNAIL SHELLS AND NAUTILUS SHELLS, ARE PERFECT EXAMPLES OF THE GOLDEN SPIRAL.  SPIRAL GALAXIES: THE MILKY WAY HAS A NUMBER OF SPIRAL ARMS, EACH OF WHICH HAS A LOGARITHMIC SPIRAL OF ROUGHLY 12 DEGREES. THE SHAPE OF THE SPIRAL IS IDENTICAL TO THE GOLDEN SPIRAL, AND THE GOLDEN RECTANGLE CAN BE DRAWN OVER ANY SPIRAL GALAXY.  HURRICANES: MUCH LIKE SHELLS, HURRICANES OFTEN DISPLAY THE GOLDEN SPIRAL.  FINGERS: THE LENGTH OF OUR FINGERS, EACH SECTION FROM THE TIP OF THE BASE TO THE WRIST IS LARGER THAN THE PRECEDING ONE BY ROUGHLY THE RATIO OF PHI. ANIMAL BODIES: THE MEASUREMENT OF THE HUMAN NAVEL TO THE FLOOR AND THE TOP OF THE HEAD TO THE NAVEL IS THE GOLDEN RATIO. BUT WE ARE NOT THE ONLY EXAMPLES OF THE GOLDEN RATIO IN THE ANIMAL KINGDOM; DOLPHINS, STARFISH, SAND DOLLARS, SEA URCHINS, ANTS AND HONEYBEES ALSO EXHIBIT THE PROPORTION.  DNA MOLECULES: A DNA MOLECULE MEASURES 34 ANGSTROMS BY 21 ANGSTROMS AT EACH FULL CYCLE OF THE DOUBLE HELIX SPIRAL. IN THE FIBONACCI SERIES, 34 AND 21 ARE SUCCESSIVE NUMBERS. APPLICATIONS OF MATHEMATICS IN THE WORLD SEATWORK: LET FIB(N) BE THE NTH TERM OF THE FIBONACCI SEQUENCE, WITH FIB(1) = 1,FIB(2) = 2, FIB(3)=3, AND SO ON. 1.FIND FIB (8) 2.IF FIB (22) =17,711 AND FIB (24) = 46,368, WHAT IS FIB(23)?