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The Beauty in Numbers Danang Jaya Math Club Monthly Seminar January 2007 Math Club Monthly Seminar 1 Surprising Number Patterns(1) Math Club Monthly Seminar 2 Surprising Number Patterns(2) Now look at the product, 142,857 • 7 = 999,999. Surprised? 142,857 • 8 = 1,142,856. Math Club Monthly Seminar 3 Surprising Number Patterns(3) Math Club Monthly Seminar 4 Surprising Number Patterns(4a) Math Club Monthly Seminar 5 Surprising Number Patterns(4b) Math Club Monthly Seminar 6 Surprising Number Patterns(5) Any six-digit number composed of two repeating sequences of three digits is divisible by 7, 11, and 13. Try: 643.643 A number with six repeating digits is always divisible by 3 7 11,and 13. Try: 111.111 Find some example. And try it. Math Club Monthly Seminar 7 Surprising Number Patterns(6) While playing with the number 9, find an eight-digit number in which no digit is repeated and which when multiplied by 9 yields a nine-digit number in which no digit is repeated. (81274365, 72645831, 58132764) Math Club Monthly Seminar 8 Surprising Number Patterns(7) Math Club Monthly Seminar 9 Prime contest 7299270072992700729927007299270072992700729927 0072992700729927007299270072992700729927007299 2700729927007299270072992700729927007299270072 9927007299270072992700729927007299270072992700 7299270072992700729927007299270072992700729927 0072992700729927007299270072992700729927007299 2700729927007299270072992700729927007299270072 9927007299270072992700729927007299270072992700 7 This number includes the first 371 digits of 1/137, with the first two zeros omitted. Math Club Monthly Seminar 10 Prime triangle. In the seventeenth century, mathematicians showed that the following numbers are all prime: 31 331 3331 33331 333331 3333331 33333331 At the time, some mathematicians were tempted to assume that all numbers of this form were prime; however, the next number in the pattern 333,333,331turned out not to be prime because 333,333,331 = 17 × 19, 607,843. Math Club Monthly Seminar 11 Amazing Power Relationships Our number system has many unusual features built into it. Discovering them can certainly be a rewarding experience. Most students need to be coaxed to look for these relationships. You might tell them about the famous mathematician Carl Friedrich Gauss (1777–1855), who had superior arithmetic abilities to see relationships and patterns that eluded even the brightest minds. What is going on here: 81 = (8 + 1)2 = 92, 4913 = (4 + 9 + 1 + 3)3 = 173 Math Club Monthly Seminar 12 Amazing Power Relationships(2) Math Club Monthly Seminar 13 Beautiful Number Relationships Math Club Monthly Seminar 14 Unusual Number Relationships Math Club Monthly Seminar 15 Strange Equalities Math Club Monthly Seminar 16 The Irrepressible Number 1 Begin by asking your students to follow two rules as they work with any arbitrarily selected number. If the number is odd, then multiply by 3 and add 1. If the number is even, then divide by 2. Regardless of the number they select, they will always end upwith 1, after continued repetition of the process. Math Club Monthly Seminar 17 The Irrepressible Number 1(2) Let’s try it for the arbitrarily selected number 12: 12 is even; therefore, we divide by 2 to get 6. 6 is also even, so we again divide by 2 to get 3. 3 is odd; therefore, we multiply by 3 and add 1 to get 3•3 + 1 = 10. 10 is even, so we simply divide by 2 to get 5. 5 is odd, so we multiply by 3 and add 1 to get 16. 16 is even, so we divide by 2 to get 8. 8 is even, so we divide by 2 to get 4. 4 is even, so we divide by 2 to get 2. 2 is even, so we divide by 2 to get 1. Math Club Monthly Seminar 18 Friendly Numbers Mathematicians have decided that two numbers are considered friendly if the sum of the proper divisors of one equals the second and the sum of the proper divisors of the second number equals the first number. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Their sum is 1+2+4+5+10+11+20+22+44+55+110 = 284. The proper divisors of 284 are 1, 2, 4, 71, and 142, and their sum is 1 + 2 + 4 + 71 + 142 = 220. This shows the two numbers are friendly numbers. Math Club Monthly Seminar 19 Number 7 There is an incredibly large number of occurrences of 7 in all religions. In the Old Testament, Lamech, the father of Noah and the son of the famous longlived Methuselah, is born 7 generations after Adam. Lamech lives for 777 years. Another Lamech should be avenged 77-fold (Genesis 4:24). Zechariah, a major biblical prophet, speaks of the 7 eyes of the Lord. The idea of 7 divine eyes occurs in Sufism in connection with 7 important saints who are the eyes of God. God is praised by creatures with 70,000 heads, each of which has 70,000 faces. Math Club Monthly Seminar 20 Number 7(2) There are 7 points in the body upon which mystics concentrate their spiritual power. Seven is important for Kabbalists. In fact, Trachtenberg, in his Jewish Magic and Superstition, mentions the following cure for tertian (malarial) fever: “Take 7 pickles from 7 palmtrees, 7 chips from 7 beams, 7 nails from 7 bridges, 7 ashes from 7 ovens, 7 scoops of earth from 7 door sockets, 7 pieces of pitch from 7 ships, 7 handfuls of cumin, and 7 hairs from the beard of an old dog, and tie them to the neck-hole of the shirt with a white twisted cord.” Math Club Monthly Seminar 21 What next? The beauty of prime number Algebraic Entertainments Geometric Wonders Mathematical Paradoxes Big Numbers and Infinity Math Club Monthly Seminar 22 Quiz What number gives the same result when it is added to 1/2 as when it is multiplied by 1/2? Donald Trumpet died, leaving a peculiar will. His will states that he will leave one million dollars to be split between his son William and his daughter Hillary. Hillary, his favorite child, gets four times the amount of William. If Hillary takes less than 30 seconds to determine how much William will get, the money is distributed immediately; otherwise, Hillary gets nothing. Can you help her? What did William get? Math Club Monthly Seminar 23 Quiz Math Club Monthly Seminar 24 Thank You Math Club Monthly Seminar 25