Math Club Monthly Seminar1 The Beauty in Numbers Danang Jaya Math Club Monthly Seminar January 2007.

Math Club Monthly Seminar 1

The Beauty in Numbers

Danang Jaya

Math Club Monthly Seminar

January 2007


Surprising Number Patterns(1)



Now look at the product, 142,857 • 7 = 999,999. Surprised?142,857 • 8 = 1,142,856.


Surprising Number Patterns(4a)


Surprising Number Patterns(4b)



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.



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)


Prime contest

729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007

This number includes the first 371 digits of 1/137, with the first two zeros omitted.


Prime triangle.

In the seventeenth century, mathematicians showed that the following numbers are all prime:

31331

333133331

3333313333331

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.


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


Amazing Power Relationships(2)


Beautiful Number Relationships


Unusual Number Relationships


Strange Equalities


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.


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.


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.


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 long-lived 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.


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.”


What next?

The beauty of prime number Algebraic Entertainments Geometric Wonders Mathematical Paradoxes Big Numbers and Infinity


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?


Quiz


Thank You

Math Club Monthly Seminar1 The Beauty in Numbers Danang Jaya Math Club Monthly Seminar January 2007.

Documents

Transcript of Math Club Monthly Seminar1 The Beauty in Numbers Danang Jaya Math Club Monthly Seminar January 2007.