In Advanced Math class right now, we are learning about series and sequences. Just like numbers, a number sequence can become well known and famous too. Most of the number sequences are famous because they are simple, but show some interesting properties.
There is one number sequence that is more famous than any other - Fibonacci Sequence. It is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathetics. He came up with this idea when he thought about how rabbits breed. Fibonacci’s number sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 44..
By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. The 0 is sometimes omitted and the sequence begins with two 1s. He considers the growth of a rabbit population, assuming that: 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; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The Fibonacci numbers and principle is used in the financial market and many other things.
I searched everywhere to find a famous series and sequence but I couldn’t find much. The only thing that really came up was Fibonacci but Brooke already did that one and you said we need to have different ones. I am not sure if this is what you are looking for but it’s all I could find.
Trigonometric Series:
Trigonometric Series are a series of trigonometric functions.
½ Ao + ∞ / ∑ / n = 1 (An cos nx + Bn sin nx)
The most important factor of a trigonometric series is the Fourier series of a function. A Fourier series decomposes any periodic signal into the sum of a set of simple oscillating functions (sines and cosines). Joseph Fourier introduced this series for the purpose of solving the heat equation in a metal plate. The study of Fourier series is a branch of Fourier analysis.
Chapter 13 in class this week has been all about arithmetic and geometric number sequences. There are many other famous sequences and series in math out there that we have not learned about. One famous one that I have found some information on is called the Fibonacci Sequence. It is named after the mathematician who invented it.
Oddly, this inventor came about this sequence by thinking about the way rabbits breed. The number sequence is: 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89,….. etc.
The pattern for this sequence is adding the last two terms of your number sequence. For example: 1+1 = 2, 1 + 2 = 3, 2 +3 = 5
This type of sequence can start with any number. You can use Fibonacci’s discovery to draw geometrical figures such as, the Golden Ratio, pentagon, logarithmic spiral, and more. It is actually a pretty simple discovery that was added into the math world, and it goes to show that you can make up almost any type of rule and form a sequence out of it.
We are currently discussing sequences and series in Advanced Math seventh hour. A sequence is simply a series of numbers. Two types of basic sequences are Arithmetic and Geometric. Ann arithmetic sequence is a sequence where a constant number (d) is being added. In a geometric sequence however, the constant number (r) is being multiplied. Another type of sequence is the Fibonacci sequence.
Leonardo Fibonacci was a mathematician born in the early twelfth century who studided sequences. His sequence, The Fibonacci Sequence, often shows up in nature, art and archeticture. The Fibonacci sequence is 0,1,1,2,3, 5,8. The sequence can be “solved” or figured out by adding the sum of the first two terms. Therefore t(n) would equal t(n-1)+t(n-2).
In advanced math this week we are learning how to figure out sequences and series. The main two sequences we talk about is arithmetic and geometric. I am going to talk about another sequence which is called an infinite sequence.
An infinite sequence is a sequence that most calculus students would know; it is continuously indefinite. An infinite series is used for finding antilogarithm, in physics, or used in some computer sciences. This formulas was founded by James Gregory.
This week in chapter 13 we learned about arithmetic and geometric sequences. I was researching about a famous sequences and I ran into Carl Friedrich Gauss.
He was born on April 17, 1777 to poor, working class parents in Brauschweig, Germany. His mother had been a maid and his father had been a laborer and handyman. His father, it is said, did not appreciate Gauss's abilities or his later accomplishments.
He is considered to be one of the three greatest mathematicians of all time. The two other great mathematicians were Archimedes and Sir Isaac Newton. Gauss showed indication in mathematic genius very early.
For example, there is a well known anecdote where a teacher gave his students an assignment to add up a series of 100 numbers. Gauss said that he had completed the exercise (the story goes that he had figured that 100 numbers could be determined by the equation n(a+b)(1/2)=50(a+b) where n=100, a = the first digit in the sequence and b = the last digit in the sequence.)
Gauss's teachers would lend him books and try to help him along with his independent studies of the subject. In 1788, at the age of 11, he entered the Gymnasium, a college preparatory school. In 1792, at the age of 14, he received a stipend from the Duke of Brunswick which made him financially independent for the next 16 years and it was that year that he entered Brunswick College.
He became famous in 1796 at the age of 19 with his discovery on March 30 of a method for constructing a heptadecagon (a regular 17-sided polygon) using only a compass and ruler. He was so excited by this that he requested that a heptadecagon be placed on his tombstone. For his doctoral thesis, he proved the fundamental theorem of algebra which states that all polynomials of degree n have n different solutions in the domain of complex numbers.
He had a strong interest in astronomy. In 1801, the planetoid Ceres was first sighted. Gauss studied data abouts its orbit and was able to predict a second possible sighting which was confirmed on December 31, 1801. In 1805, he became Director of the Observatory in Gottingen. He died on February 23, 1855 leaving many great stuff that we use today in math.
To find the sum of the series, Euler started with a seemingly unrelated function: sin(x)/x
from this simple function he was able to expand it greatly into a famous sequence. A famous sequence is: 1-(1/n^2x1^2 + 1/n^2x2^2 + 1/n^2x3^2 + 1/n^2x4^2 +......)
This famous sequence was discovered by LEONHARD EULER. Before he obtained the answer to this sum sequence 2 other mathematicians had already tried. Leibniz and Jacob Bernoulli were 2 famous co-inventors of calculus that had already tried to solve this sequence. Euler was looking for the ratio of the circumference of a circle to its diameter.
I tried really hard to find someone that no one else had picked.
This week in advanced math we were introduced to two sequences, geometric and arithmetic. There are many different types of sequences and theories in trigonometry. Many of them become famous in mathematics history because of their simplicity and uniqueness.
1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+... ^Harmonic series
The series I have found is Harmonic series, which is the divergent infinite series in mathematics. The name comes from the concept of harmonies in music or overtones. Like the Harmonic series, the wavelengths overtone of a vibrating string are 1/2, 1/3, 1/4, 1/5, and so on. This series was first proven in the 14th century by Nicole Oresme but her achievment had fallen through, and was later further proved by Pietro Mengoli, Johann Bernoulli, and Jakob Bernoulli. The Harmonic series has been popular with many architects. The series was often derived to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.
The limit of the nth term as n goes to infinity is zero in a harmonic series. Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms after the first two and noting that each such has a sum larger than 1/2.
I picked the Harmonic series because i found it interesting just how math and music can have the same principles and follow alot of the same rhythms.
This week we are learning about sequence. Sequence is when you get a set of numbers that appear in a certain pattern. Their are two types, Arithmetic and Geometric. Arithmetic is addition and subtraction and Geometric is multiplication and division. Here is an example of both: Arithmetic 1,2,3,5,8,13...... This sequence you take the current number and add it to the previous number.
Geometric 2,6,18,54,..... This is simply multiplying the previous number by 3.
Vicken can kiss my behind because I found this first.
There is a story told about the famous mathematician Gauss when he was a child. Supposedly he and his classmates were challenged by their teacher to find the sum of all the integers from one to one hundred. Gauss had the answer almost immediately. When asked how he had done it he explained: I added the first number to the last to get 1 + 100 =101, then I added the second number to the next to last to get 2 + 99 = 101. I saw that I would continue to get a sum of 101 until I formed the last sum of 50 + 51 = 101. Thus I would have 50 sums of 101 for a total of 50(101) = 5050.
What Gauss discovered is a method for finding the sum of an arithmetic sequence: Add the first term to the last term and multiply by half the number of terms.
This week in advanced math we are learning chapter 13 which involves arithmetic and geometric sequences, our assignment is to find a famous sequence and series. I could not find any famous sequences that no one else had so here you go.
Mrs. Robinson's Famous Theories
Mrs. Robinson is a math genius who went to a boarding school for gifted people. She now teaches at Riverside Academy and has her own famous theories.
Theory 1 "If you do not know your formulas and the Trig Chart you will not pass this class". This theory has been proven by tests taken by her students.
Theory 2 "Only geniuses don't need to do homework and none of you are geniuses so do your homework". This theory is being inforced more and more everyday because zeros are now being handed out by the dozen for missing homework assignments. p.s. ( I'm sort of a genius in my own way(; )
Theory 3 "I rarely give extra credit assignments so when I do it would be wise to do them". This theory will be proven after the break when many of us, me not being one, will wish we had done the extra credit assignment.
Sources: Courtesy of John's Brain
A famous series is the infinite binomial series. Which is the binomial theorem for positive integers. This theorem states that you can expand the power (x+y)^n into a sum involving terms of the form ax^by^c.
In Advanced Math class right now, we are learning about series and sequences. Just like numbers, a number sequence can become well known and famous too. Most of the number sequences are famous because they are simple, but show some interesting properties.
ReplyDeleteThere is one number sequence that is more famous than any other - Fibonacci Sequence. It is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathetics. He came up with this idea when he thought about how rabbits breed. Fibonacci’s number sequence looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 44..
By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. The 0 is sometimes omitted and the sequence begins with two 1s. He considers the growth of a rabbit population, assuming that: 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; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The Fibonacci numbers and principle is used in the financial market and many other things.
http://en.wikipedia.org/wiki/Fibonacci_number#Applications
This comment has been removed by the author.
ReplyDeleteTrigonometric Series and Fourier Series
ReplyDeleteI searched everywhere to find a famous series and sequence but I couldn’t find much. The only thing that really came up was Fibonacci but Brooke already did that one and you said we need to have different ones. I am not sure if this is what you are looking for but it’s all I could find.
Trigonometric Series:
Trigonometric Series are a series of trigonometric functions.
½ Ao + ∞ / ∑ / n = 1 (An cos nx + Bn sin nx)
The most important factor of a trigonometric series is the Fourier series of a function.
A Fourier series decomposes any periodic signal into the sum of a set of simple oscillating functions (sines and cosines). Joseph Fourier introduced this series for the purpose of solving the heat equation in a metal plate. The study of Fourier series is a branch of Fourier analysis.
http://en.wikipedia.org/wiki/Fourier_series
http://en.wikipedia.org/wiki/Series_(mathematics)#Dirichlet_series
Chapter 13 in class this week has been all about arithmetic and geometric number sequences. There are many other famous sequences and series in math out there that we have not learned about. One famous one that I have found some information on is called the Fibonacci Sequence. It is named after the mathematician who invented it.
ReplyDeleteOddly, this inventor came about this sequence by thinking about the way rabbits breed. The number sequence is:
1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89,….. etc.
The pattern for this sequence is adding the last two terms of your number sequence. For example:
1+1 = 2, 1 + 2 = 3, 2 +3 = 5
This type of sequence can start with any number. You can use Fibonacci’s discovery to draw geometrical figures such as, the Golden Ratio, pentagon, logarithmic spiral, and more. It is actually a pretty simple discovery that was added into the math world, and it goes to show that you can make up almost any type of rule and form a sequence out of it.
http://www.mathsisgoodforyou.com/topicsPages/number/famousequences.htm
We are currently discussing sequences and series in Advanced Math seventh hour. A sequence is simply a series of numbers. Two types of basic sequences are Arithmetic and Geometric. Ann arithmetic sequence is a sequence where a constant number (d) is being added. In a geometric sequence however, the constant number (r) is being multiplied. Another type of sequence is the Fibonacci sequence.
ReplyDeleteLeonardo Fibonacci was a mathematician born in the early twelfth century who studided sequences. His sequence, The Fibonacci Sequence, often shows up in nature, art and archeticture. The Fibonacci sequence is 0,1,1,2,3, 5,8. The sequence can be “solved” or figured out by adding the sum of the first two terms. Therefore t(n) would equal t(n-1)+t(n-2).
http://en.wikipedia.org/wiki/Fibonacci_number
http://www.geom.uiuc.edu/~demo5337/s97b/fibonacci.html
okkay wow, so i just blogged about a topic that was already donee :/
ReplyDeleteIn advanced math this week we are learning how to figure out sequences and series. The main two sequences we talk about is arithmetic and geometric. I am going to talk about another sequence which is called an infinite sequence.
ReplyDeleteAn infinite sequence is a sequence that most calculus students would know; it is continuously indefinite. An infinite series is used for finding antilogarithm, in physics, or used in some computer sciences. This formulas was founded by James Gregory.
http://www.math.wpi.edu/IQP/BVCalcHist/calc3.html
This week in chapter 13 we learned about arithmetic and geometric sequences. I was researching about a famous sequences and I ran into Carl Friedrich Gauss.
ReplyDeleteHe was born on April 17, 1777 to poor, working class parents in Brauschweig, Germany. His mother had been a maid and his father had been a laborer and handyman. His father, it is said, did not appreciate Gauss's abilities or his later accomplishments.
He is considered to be one of the three greatest mathematicians of all time. The two other great mathematicians were Archimedes and Sir Isaac Newton.
Gauss showed indication in mathematic genius very early.
For example, there is a well known anecdote where a teacher gave his students an assignment to add up a series of 100 numbers.
Gauss said that he had completed the exercise (the story goes that he had figured that 100 numbers could be determined by the equation n(a+b)(1/2)=50(a+b) where n=100, a = the first digit in the sequence and b = the last digit in the sequence.)
Gauss's teachers would lend him books and try to help him along with his independent studies of the subject. In 1788, at the age of 11, he entered the Gymnasium, a college preparatory school. In 1792, at the age of 14, he received a stipend from the Duke of Brunswick which made him financially independent for the next 16 years and it was that year that he entered Brunswick College.
He became famous in 1796 at the age of 19 with his discovery on March 30 of a method for constructing a heptadecagon (a regular 17-sided polygon) using only a compass and ruler. He was so excited by this that he requested that a heptadecagon be placed on his tombstone. For his doctoral thesis, he proved the fundamental theorem of algebra which states that all polynomials of degree n have n different solutions in the domain of complex numbers.
He had a strong interest in astronomy. In 1801, the planetoid Ceres was first sighted. Gauss studied data abouts its orbit and was able to predict a second possible sighting which was confirmed on December 31, 1801. In 1805, he became Director of the Observatory in Gottingen.
He died on February 23, 1855 leaving many great stuff that we use today in math.
http://fermatslasttheorem.blogspot.com/2005/06/carl-friedrich-gauss.html
To find the sum of the series, Euler started with a seemingly unrelated function:
ReplyDeletesin(x)/x
from this simple function he was able to expand it greatly into a famous sequence.
A famous sequence is:
1-(1/n^2x1^2 + 1/n^2x2^2 + 1/n^2x3^2 + 1/n^2x4^2 +......)
This famous sequence was discovered by LEONHARD EULER. Before he obtained the answer to this sum sequence 2 other mathematicians had already tried. Leibniz and Jacob Bernoulli were 2 famous co-inventors of calculus that had already tried to solve this sequence. Euler was looking for the ratio of the circumference of a circle to its diameter.
I tried really hard to find someone that no one else had picked.
http://ptrow.com/articles/Infinite_Series_Sept_07.htm
This comment has been removed by the author.
ReplyDeleteThis week in advanced math we were introduced to two sequences, geometric and arithmetic. There are many different types of sequences and theories in trigonometry. Many of them become famous in mathematics history because of their simplicity and uniqueness.
ReplyDelete1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+...
^Harmonic series
The series I have found is Harmonic series, which is the divergent infinite series in mathematics. The name comes from the concept of harmonies in music or overtones. Like the Harmonic series, the wavelengths overtone of a vibrating string are 1/2, 1/3, 1/4, 1/5, and so on. This series was first proven in the 14th century by Nicole Oresme but her achievment had fallen through, and was later further proved by Pietro Mengoli, Johann Bernoulli, and Jakob Bernoulli. The Harmonic series has been popular with many architects. The series was often derived to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.
The limit of the nth term as n goes to infinity is zero in a harmonic series. Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms after the first two and noting that each such has a sum larger than 1/2.
I picked the Harmonic series because i found it interesting just how math and music can have the same principles and follow alot of the same rhythms.
www.wikipedia.com/harmonicseries
http://mathworld.wolfram.com/HarmonicSeries.html
This week we are learning about sequence. Sequence is when you get a set of numbers that appear in a certain pattern. Their are two types, Arithmetic and Geometric. Arithmetic is addition and subtraction and Geometric is multiplication and division. Here is an example of both:
ReplyDeleteArithmetic
1,2,3,5,8,13...... This sequence you take the current number and add it to the previous number.
Geometric
2,6,18,54,..... This is simply multiplying the previous number by 3.
Vicken can kiss my behind because I found this first.
There is a story told about the famous mathematician Gauss when he was a child. Supposedly he and his classmates were challenged by their teacher to find the sum of all the integers from one to one hundred. Gauss had the answer almost immediately. When asked how he had done it he explained: I added the first number to the last to get 1 + 100 =101, then I added the second number to the next to last to get 2 + 99 = 101. I saw that I would continue to get a sum of 101 until I formed the last sum of 50 + 51 = 101. Thus I would have 50 sums of 101 for a total of 50(101) = 5050.
What Gauss discovered is a method for finding the sum of an arithmetic sequence: Add the first term to the last term and multiply by half the number of terms.
This week in advanced math we are learning chapter 13 which involves arithmetic and geometric sequences, our assignment is to find a famous sequence and series. I could not find any famous sequences that no one else had so here you go.
ReplyDeleteMrs. Robinson's Famous Theories
Mrs. Robinson is a math genius who went to a boarding school for gifted people. She now teaches at Riverside Academy and has her own famous theories.
Theory 1
"If you do not know your formulas and the Trig Chart you will not pass this class". This theory has been proven by tests taken by her students.
Theory 2
"Only geniuses don't need to do homework and none of you are geniuses so do your homework". This theory is being inforced more and more everyday because zeros are now being handed out by the dozen for missing homework assignments. p.s. ( I'm sort of a genius in my own way(; )
Theory 3
"I rarely give extra credit assignments so when I do it would be wise to do them". This theory will be proven after the break when many of us, me not being one, will wish we had done the extra credit assignment.
Sources: Courtesy of John's Brain
A famous series is the infinite binomial series. Which is the binomial theorem for positive integers. This theorem states that you can expand the power (x+y)^n into a sum involving terms of the form ax^by^c.
http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/fpure_ch6.pdf