Transcript The Beauty in Numbers

The Beauty in Numbers
Danang Jaya
Math Club Monthly Seminar
January 2007
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Surprising Number Patterns(1)
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Surprising Number Patterns(2)
Now look at the product, 142,857 • 7 = 999,999. Surprised?
142,857 • 8 = 1,142,856.
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Surprising Number Patterns(3)
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Surprising Number Patterns(4a)
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Surprising Number Patterns(4b)
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Surprising Number Patterns(5)
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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.
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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)
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Surprising Number Patterns(7)
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Prime contest
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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.
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Prime triangle.
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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.
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Amazing Power Relationships
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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
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Amazing Power Relationships(2)
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Beautiful Number Relationships
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Unusual Number Relationships
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Strange Equalities
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The Irrepressible Number 1
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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.
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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.
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Friendly Numbers
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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.
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Number 7
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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.
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Number 7(2)
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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.”
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What next?
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The beauty of prime number
Algebraic Entertainments
Geometric Wonders
Mathematical Paradoxes
Big Numbers and Infinity
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Quiz
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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?
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Quiz
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Thank You
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