We finished off the semester by learning about the last three mathmaticians:

1. Sonya Kovalevskaya: Born on January 15, 1850. Her father was in the Russian Army and her mother was from a German family. She was interested in learning immediately as a child. Her father privately tutored her but he wouldn’t allow her to go study over seas which is what she wanted to do.  She entered  a marriage of convience with Vladimir Kovalensky. She left Russia in 1869 with her husband and sister. They went to Heidelberg Germany and she eventually admitted to University of Heifelberg.  Sonya earned her doctorate in 1874. Sonya won the Prix bordin from the French Academie Royale des Sciences. She also won a prize from Swedish Academy of Sciences in 1889. She was appointed to the chair at the university that same year, the first woman appointed to a chair.  She died in 1891 of influenza.

2. Grace Chisholm Young: Born March 15, 1868 London, England. Grace Chisholm Young, was a very intelligent women, as all the women we learned about are. At 17 she got permission to attend Cambridge Univeristy. She attended Griton College at age 21. She fell in love with her professor William Young. The two got married and left England to study mathematics in Europe. Grace was the first Mathematician to receive a doctrine when she was 27 years old. Grace Young died at the age of 76, 2 years after her husband, she suffered from a heart attack.

3. Emmy Noether: Born on March 23, 1882, her plan was to teach but eventually realized it was math she wanted to persue. It was then that she went to University of Erlangen where she received a PHD in 1907.  At the University of Gottengan she was recognized as a lecturer however because she was a woman, the university didn’t pay her.  She dealt with algebra and her work “Noetherian Ring”. Albert Einstein described Emmy as ” the most important women in mathematics”. Emmy died in 1935 at the age of 53.

At the end of class today, Professor Kim introduced The Fibonacci Rabbit Sequence. I thought this sequence seemed really interesting and it caught my attention so I had a chance to look it up more when I got home. I found a good website that went into depth about it and explain it in an easy way for me to understand: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#rabeecow

Rabbits, Cows and Bees Family Trees

Let’s look first at the Rabbit Puzzle that Fibonacci wrote about and then at two adaptations of it to make it more realistic. This introduces you to the Fibonacci Number series and the simple definition of the whole never-ending series.

Fibonacci’s Rabbits

The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was…

How many pairs will there be in one year?

1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

Can you see how the series is formed and how it continues? If not, look at the answer!

The first 300 Fibonacci numbers are here and some questions for you to answer.

Now can you see why this is the answer to our Rabbits problem? If not, here’s why.
Another view of the Rabbit’s Family Tree:

Both diagrams above represent the same information. Rabbits have been numbered to enable comparisons and to count them, as follows:

• All the rabbits born in the same month are of the same generation and are on the same level in the tree.
• The rabbits have been uniquely numbered so that in the same generation the new rabbits are numbered in the order of their parent’s number. Thus 5, 6 and 7 are the children of 0, 1 and 2 respectively.
• The rabbits labelled with a Fibonacci number are the children of the original rabbit (0) at the top of the tree.
• There are a Fibonacci number of new rabbits in each generation, marked with a dot.
• There are a Fibonacci number of rabbits in total from the top down to any single generation.

After learning about Mary Somerville, we breifly learned about a second mathmatician, Ada Byron Lovelace:

Ada Lovelace was the daughter, and only child, to Lord Byron. She was born December 10, 1815.  Ada was often sick as a child and didn’t end up living a very long life. She died at age 37. However for the short life she lived she had a good share of accomplishments!

She corresponded with Charles Babbage as a teenager, her are some of the accomplishments I found that the two had together: http://www.maxmon.com/1830ad.htm

We learned about two new female Mathmaticians:

The first is Mary Fairfax Greg Somerville (That’s a long name):

Mary Fairfax Greg Somerville was born on December 26, 1780 in Jedburgh Scotland, the daughter of Margaret Charters and Lieutenant William George Fairfax. Her father was is the British Navy and was gone for long periods of time. She attended a boarding school for girls in Musselburgh, rather miserable and unhappy, she wasn’t given a good eduation, as most girls of this time. Mary studied her first simple arithmetic and algebra at the age of thirteen. She married her cousin, Samuel Greig when she was 24 years old. Greig was a member of the Russian Navy and had no interest in math. Together the couple had 2 sons, Woronzow and William George. After only 3 years of being married, Samuel Greig passed away. This gave Mary more freedom to do as she pleased, and continue to study. She mastered J. Ferguson’s Astronomy and became a student of Isaac Newton’s Principia. She corresponded frequently with Scotsman William Wallace, who was mathematics master at a military college. In 1812 she married William Somerville who was also her cousin. Sommerville was very supportive of his wife’s studies. They went on to have four children together. In the summer of 1825, Mary began to investigate Magnetism. By 1826 she came out with a paper, “The Magnetic Properties of the Violet Rays of the Solar Spectrum”.  In 1834, she published her second book, “The Connection of the Physical Sciences”. In 1835, she and another woman, Caroline Herschel were the first women to be elected to the Royal Astronomical Society. As Mary became older, it didn’t stop her from her studies. She eventually became deaf and lived to be 92 years old!

Our first exam

1. Name the four conic sections:

a. parabola

b. ellipse

c. hyperbola

d. cone

2. An ellipse is formed by cutting through the cone at an angle.

3. A circle is formed by cutting the cone parallel to the cone’s base.

4. A hyperbola is the path of points whose total distances from two fixed points are equal.

5. A parabola is a path of points which are the same distance from a fixed point and a fixed line.

6. A hyperbola is the path of points whose distances from two fixed points have equal differences.

7. What is Fermat’s theorem? Why is it named after Pierre de Fermat?

X to the n + Y to the n= Z to the n

It cannot be proven

If n>2, it doesn’t work.

x,y,z,n are all whole positive integers

8. What are Diophantine equations?

Diophantine equations are equations with real natural number solutions. ax+by=1.

9. Does Fermat’s theorem deal with Diophantine equations? Why or why not?

No it does not deal with Diophantine equations because it can’t be proven and doesn’t deal with natural number solutions: X to the n+ Y to the n=  Z to the n

if n >2, i doesn’t work

10. State the Pythagorean theorem and prove it.

In algebraic terms, a² + b² = c² where c is the hypotenuse while a and b are the legs of the triangle.

I had trouble proving this but after re looking online I found:

Proof #2

We start with two squares with sides a and b, respectively, placed side by side. The total area of the two squares is a²+b².

The construction did not start with a triangle but now we draw two of them, both with sides a and b and hypotenuse c. Note that the segment common to the two squares has been removed. At this point we therefore have two triangles and a strange looking shape.

As a last step, we rotate the triangles 90°, each around its top vertex. The right one is rotated clockwise whereas the left triangle is rotated counterclockwise. Obviously the resulting shape is a square with the side c and area c². This proof appears in a dynamic incarnation.

(A variant of this proof is found in an extant manuscript by Thâbit ibn Qurra located in the library of Aya Sofya Musium in Turkey, registered under the number 4832. [R. Shloming, Thâbit ibn Qurra and the Pythagorean Theorem, Mathematics Teacher 63 (Oct., 1970), 519-528]. ibn Qurra’s diagram is similar to that in proof #27. The proof itself starts with noting the presence of four equal right triangles surrounding a strangely looking shape as in the current proof #2. These four triangles correspond in pairs to the starting and ending positions of the rotated triangles in the current proof. This same configuration could be observed in a proof by tessellation.)

I thought that this segment from http://www.cut-the-knot.org/pythagoras/index.shtml, made it a lot easier for me to understand how to prove it.

11. Use a drawing (and some words) to prove that (a+b)2 = a2+2ab+b2

When taking the test, I went into a lot of words and not so much a visual proof. I talked about FOILing the numbers to come up with the final solution. I didn’t get any credit for it so I went to look up a visual proof. I looked at other people’s blogs and found Amada’s to be very helpful. In her’s she says:

(a+b)²=a²+2ab+b²

(a-b)=a² – 2ab+b²

a²-b²=2ab+(a-b)²

Here is my take on them:

.gallery { margin: auto; } .gallery-item { float: left; margin-top: 10px; text-align: center; width: 33%; } .gallery img { border: 2px solid #cfcfcf; } .gallery-caption { margin-left: 0; }

The large square is made up of squares b&a, both with an area of a²&b². There are also two rectangels with sides a&b (2ab).

I thought she did a good job visual with some words to help me understand, so thanks Amanda!

12. True or False?

a. F Differentiation involves an infinite sum of infinitely thin rectangles.

b. T Integration involves an infinite sum of infinitely small rectangles.

c. F Differentiation is the calculus operation used for calculating area under a curve.

d. T Integration is the calculus operation used for calculating area under a curve.

e. T Differentiation and integration both deal with infinitesimally small quantities.

Next we had 4 essays:

#13. talked about Hypatia’s life and background

#14. Emilie du Chetelet’s life and background

#15. Professor Blanton’s lecture

#16. Professor Kim’s lecture.

• From Italy
• Taught to pray as a child
• edicate and religious
• 1 of 21 siblings
• father’s parents were wealthy merchants
• father was a mathmatician
• wrote speeches and recited them
• Age 13- Greek, Hebrew, French, Italian, Latin, Spanish
• Reading Newton’s ideas in Latin
• had a remarkable childhood
• her father pushed her and wanted her to have opportunity
• she was a genius, mathmatician and philosopher
• She has debates with Newton
• 1732- father died while giving birth to her 8th child
• father remarried twice, Maria took care of all her brothers and sisters
• she wanted to become a nun but her father wouldnt allow it.
• in 1752 her father passed away.This was devistating to Maria who at the time was 34. She lost her father, friend, colleague, mentor.
• She continued to work in the field but it was very hard for her.
• She withdrew from all mathamatical activities
• She moved into a rented house and sheltered poor people
• Became ill (blind and deaf)
• died at 81 from heart failure, had no money when she died
• it’s hard to get a hold of Maria’s work

(1)

where is an integer. The Mersenne numbers consist of all 1s in base-2, and are therefore binary repunits. The first few Mersenne numbers are 1, 3, 7, 15, 31, 63, 127, 255, … (Sloane’s A000225), corresponding to , , , , … in binary.

The Mersenne numbers are also the numbers obtained by setting in a Fermat polynomial. They also correspond to Cunningham numbers .

The number of digits in the Mersenne number is

(2)

where is the floor function, which, for large , gives

(3)

The number of digits in is the same as the number of digits in , namely 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, … (Sloane’s A034887). The numbers of decimal digits in for , 1, … are given by 1, 4, 31, 302, 3011, 30103, 301030, 3010300, 30103000, 301029996, … (Sloane’s A114475), which correspond to the decimal expansion of (Sloane’s A007524).

The numbers of prime factors of for , 2, … are 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, … (Sloane’s A046051), and the first few factorizations are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

From     http://mathworld.wolfram.com/MersenneNumber.html

A perfect number is a number whose sum of divisors is equal to itself.

Conjecture: there are no odd perfect numbers

4= 2+2

6= 3+3

8= 5+3

10= 7+3 or 5+5

12= 7+5

14= 7+7 or 11+3

16= 13+13 or 11+5

18= 11+7

20= 7+13

22= 19+3 or 11+11

24= 19+5 or 11+13

Goldblach’s Conjecture:

-Every even number greater than or equal to 4 can be written as the sum of two primes.

-Can’t prove it or check it because there are an infinite amount of numbers

example: (2,3,5,7,11,13,17,19,23…)

-There are an infinite number of prime numbers.

For more information on Prime Numbers, I asked Dr. Math: http://mathforum.org/dr.math/faq/faq.prime.num.html

A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That’s six factors. If we list the factors of 29, we only have 1 and 29. That’s two factors. So we say that 29 is a prime number, but 28 isn’t.

Another way of saying this is that a prime number is a positive integer that is not the product of two smaller positive integers.

Note that the definition of a prime number doesn’t allow 1 to be a prime number: 1 only has one factor, namely 1. Prime numbers have exactly two factors, not “at most two” or anything like that. When a number has more than two factors it is called a composite number.

Here are the first few prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.

Chart

p                      2p+1                           Both prime?

2                      5                                           yes

3                     7                                             yes

5                    11                                            yes

7                     15                                          not prime

2,3, and 5 are all sofie germain primes. The highest Germain prime that was ever found was 48047305725 × 2¹⁷²⁴⁰³−1 which was found by David Underbakke