Pythagoras was a Greek mathematician, philosopher, and the founder of the religion Pythagoreanism. He is of course, best known for the pythagorean theorem: a2=b2=c2.
The only writings about him were written many years after his death, so we are not sure of the validity of the facts about his life. It is known in Samos, Greece. It is said that a prophet told his pregnant mother that her child would be supremely wise, beautiful, and beneficial to mankind although this is debated. Pythagoras was not only a brilliant mathematician, but also made discoveries in the fields of mathematics, astronomy, and music. He was also said to have practiced divination and prophecy, often being associated with Apollo. According to some writings, he married a woman named Theano who bore him children including a son Telauges and three daughters Damo, Arignote, and Myia.
Source: http://en.wikipedia.org/wiki/Pythagoras
Math joke of the day:
I strongly dislike the subject of math, however I am partial to fractions.
Thursday, March 27, 2014
10.7: Graphs of Polar Equations
Today we learned how to make graphs of polar equations.
Symmetry tests for polar coordinates
1. Replace θ with -θ. If an equivalent equation results, the graph is symmetric with respect
to the polar axis.
2. Replace θ with -θ and r with -r. If an equivalent equation results, the graph is symmetric
with respect to θ = π .
3. Replace r with -r. If an equivalent equation results, the graph is symmetric with respect
to the pole.
Note: It is possible for a polar equation to fail a test and still exhibit that type of symmetry.
There are four main types of graphs of polar equations:
Circles
1. r = a cos θ is a circle where “a” is the diameter of the circle that has its left-most edge at
the pole.
2. r = a sin θ is a circle where “a” is the diameter of the circle that has its bottom-most edge
at the pole.
Limaçons (Snails)
1. r = a ± b sin θ, where a > 0 and b > 0
2. r = a ± b cos θ, where a > 0 and b > 0
-The limaçons containing sine will be above the horizontal axis if the sign between a and b is plus or below the horizontal axis if the sign if minus.
-If the limaçon contains the function cosine then the graph will be either to the right of the vertical axis if the sign is plus or to the left if the sign is minus.
-The ratio of a to b will determine the exact shape of the limaçon.
Rose Curves
A rose curve is a graph that is produced from a polar equation in the form of:
r = a sin nθ or r = a cos nθ, where a ≠ 0 and n is an integer > 1
The number of petals that are present will depend on the value of n. The value of a will determine the length of the petals.
Lemniscates
Lemniscates have the general polar equation of:
r2 = a2 sin 2θ or r2 = a2cos 2θ, where a ≠ 0
A lemniscate containing the sine function will be symmetric to the pole while the lemniscate containing the cosine function will be symmetric to the polar axis, to θ = 2π , and the pole.
Math joke of the day:
A team of engineers were required to measure the height of a flag pole. They only had a measuring tape, and were getting quite frustrated trying to keep the tape along the pole. It kept falling down, etc. A mathematician comes along, finds out their problem, and proceeds to remove the pole from the ground and measure it easily. When he leaves, one engineer says to the other: "Just like a mathematician! We need to know the height, and he gives us the length!"
Symmetry tests for polar coordinates
1. Replace θ with -θ. If an equivalent equation results, the graph is symmetric with respect
to the polar axis.
2. Replace θ with -θ and r with -r. If an equivalent equation results, the graph is symmetric
with respect to θ = π .
3. Replace r with -r. If an equivalent equation results, the graph is symmetric with respect
to the pole.
Note: It is possible for a polar equation to fail a test and still exhibit that type of symmetry.
There are four main types of graphs of polar equations:
Circles
1. r = a cos θ is a circle where “a” is the diameter of the circle that has its left-most edge at
the pole.
2. r = a sin θ is a circle where “a” is the diameter of the circle that has its bottom-most edge
at the pole.
Limaçons (Snails)
1. r = a ± b sin θ, where a > 0 and b > 0
2. r = a ± b cos θ, where a > 0 and b > 0
-The limaçons containing sine will be above the horizontal axis if the sign between a and b is plus or below the horizontal axis if the sign if minus.
-If the limaçon contains the function cosine then the graph will be either to the right of the vertical axis if the sign is plus or to the left if the sign is minus.
-The ratio of a to b will determine the exact shape of the limaçon.
Rose Curves
A rose curve is a graph that is produced from a polar equation in the form of:
r = a sin nθ or r = a cos nθ, where a ≠ 0 and n is an integer > 1
The number of petals that are present will depend on the value of n. The value of a will determine the length of the petals.
Lemniscates
Lemniscates have the general polar equation of:
r2 = a2 sin 2θ or r2 = a2cos 2θ, where a ≠ 0
A lemniscate containing the sine function will be symmetric to the pole while the lemniscate containing the cosine function will be symmetric to the polar axis, to θ = 2π , and the pole.
Math joke of the day:
A team of engineers were required to measure the height of a flag pole. They only had a measuring tape, and were getting quite frustrated trying to keep the tape along the pole. It kept falling down, etc. A mathematician comes along, finds out their problem, and proceeds to remove the pole from the ground and measure it easily. When he leaves, one engineer says to the other: "Just like a mathematician! We need to know the height, and he gives us the length!"
10.6: Polar Coordinates Introduction
Today we learned about a new way to graph. Instead of having (x, y), we now have (r, Ɵ).
The graph of polar coordinates will look something like this:
The r counts out from the center and the Ɵ on the corresponding labeled lines. Polar coordinates can also be negative. If a r or Ɵ is negative, then then move opposite. For instance, if the point is (1, pi/12), then the point would be on the first circle from the center and in quadrant 2 (top right), but if the point is (-1, -pi/12) then the point would be on the first circle in the center in quadrant 4 (bottom left) and on the line labeled 13pi/12.
Sometimes you will have to convert between polar equations and rectangular equations. To do this, use these equations.
When converting from polar to rectangular, use these equations:
x=rcosƟ
y=rsinƟ
When converting from rectangular to polar, use these equations.
tanƟ=y/x
r^2=x^2=y^2
If you need more help, check out Miss V's lecture:
https://docs.google.com/file/d/0B0qanadSJ9JgWlpnSEdVRGZya3c/edit
Math joke of the day:
A quote from Charles Darwin: "A mathematician is a blind man in a dark room looking for a black cat which isn't there."
The graph of polar coordinates will look something like this:
The r counts out from the center and the Ɵ on the corresponding labeled lines. Polar coordinates can also be negative. If a r or Ɵ is negative, then then move opposite. For instance, if the point is (1, pi/12), then the point would be on the first circle from the center and in quadrant 2 (top right), but if the point is (-1, -pi/12) then the point would be on the first circle in the center in quadrant 4 (bottom left) and on the line labeled 13pi/12.
Sometimes you will have to convert between polar equations and rectangular equations. To do this, use these equations.
When converting from polar to rectangular, use these equations:
x=rcosƟ
y=rsinƟ
When converting from rectangular to polar, use these equations.
tanƟ=y/x
r^2=x^2=y^2
If you need more help, check out Miss V's lecture:
https://docs.google.com/file/d/0B0qanadSJ9JgWlpnSEdVRGZya3c/edit
Math joke of the day:
A quote from Charles Darwin: "A mathematician is a blind man in a dark room looking for a black cat which isn't there."
Thursday, March 20, 2014
Interesting Math Stuff #11: Taming Infinity
Please Watch this Video: Taming Infinity Video
This video covers one of the topics we went over a couple weeks ago: convergent geometric infinite sequences. The way Miss V explained it was like this: If I take one half, than I add half of one half (one fourth), then I add half of one fourth (one eighth)....for all eternity, the answer will be a natural number.
This raises some confusion and also some awe. When I think of a situation like this, I think that however many you add, the number will keep approaching the natural number, but will never reach it. So if the presumed natural number that is the sum is N, than shouldn't the sum of any given amount of numbers by <N? I suppose that to fully understand this concept, I would have to completely understand the nature of infinity. Is infinity a number or a concept? Because adding any number of terms to this a convergent geometric infinite sequence would logically result it <N, then I guess infinity must be a concept.
Maybe by the time I finish AP Calculus, I will finally be able to wrap by head around the idea of infinity.
Math joke of the day:
Q: Did you hear about the statistician who invented a device to measure the weight of trees?
A: It is referred to as the log scale.
This video covers one of the topics we went over a couple weeks ago: convergent geometric infinite sequences. The way Miss V explained it was like this: If I take one half, than I add half of one half (one fourth), then I add half of one fourth (one eighth)....for all eternity, the answer will be a natural number.
This raises some confusion and also some awe. When I think of a situation like this, I think that however many you add, the number will keep approaching the natural number, but will never reach it. So if the presumed natural number that is the sum is N, than shouldn't the sum of any given amount of numbers by <N? I suppose that to fully understand this concept, I would have to completely understand the nature of infinity. Is infinity a number or a concept? Because adding any number of terms to this a convergent geometric infinite sequence would logically result it <N, then I guess infinity must be a concept.
Maybe by the time I finish AP Calculus, I will finally be able to wrap by head around the idea of infinity.
Math joke of the day:
Q: Did you hear about the statistician who invented a device to measure the weight of trees?
A: It is referred to as the log scale.
10.3: Hyperbolas
Here is a graphic representation of all the parts of a hyperbola:
.jpg)
The equations of hyperbolas are as follows. Don't forget that when talking about hyperbolas, the a does not have to be bigger than b. a will always be in front of the subtraction sign.
.jpg)
Center
The center will be (h,k)
Vertices
The vertices will be a units from the center point.
When the hyperbola is vertical...subtract/add a to the y value.
When the hyperbola is horizontal...subtract/add a to the x value.
Foci
The foci will be c units from the center point.
When the hyperbola is vertical...subtract/add c to the y value.
When the hyperbola is horizontal...subtract/add c to the x value.
Asymptotes
Use this formula to find the asymptotes:
.jpg)
Graph the Equation.
1. graph the vertices
2. count b from the center and draw points.
When the hyperbola is vertical...subtract/add b to the x value
When the hyperbola is horizontal...subtract/add b to the y value.
3. Draw a rectangle connecting these points.
4. Draw diagonal lines through the corners of the rectangle. These are your asymptotes.
5. Draw a hyperbola through the vertex, but not touching any of the asymptotes.
Math joke of the day:
Q: What's 2n plus 2n?
A: I don't know, it's 4n to me!
The equations of hyperbolas are as follows. Don't forget that when talking about hyperbolas, the a does not have to be bigger than b. a will always be in front of the subtraction sign.
Center
The center will be (h,k)
Vertices
The vertices will be a units from the center point.
When the hyperbola is vertical...subtract/add a to the y value.
When the hyperbola is horizontal...subtract/add a to the x value.
Foci
The foci will be c units from the center point.
When the hyperbola is vertical...subtract/add c to the y value.
When the hyperbola is horizontal...subtract/add c to the x value.
Asymptotes
Use this formula to find the asymptotes:
Graph the Equation.
1. graph the vertices
2. count b from the center and draw points.
When the hyperbola is vertical...subtract/add b to the x value
When the hyperbola is horizontal...subtract/add b to the y value.
3. Draw a rectangle connecting these points.
4. Draw diagonal lines through the corners of the rectangle. These are your asymptotes.
5. Draw a hyperbola through the vertex, but not touching any of the asymptotes.
Math joke of the day:
Q: What's 2n plus 2n?
A: I don't know, it's 4n to me!
10.5: Eliminating the Parameter
To get a set of equations with three variables (usually x, y, and t) into a form where we can graph it, we have to follow these steps:
1. solve for t in one equation
2. Substitute into 2nd equation
3. Simplify and get rectangular equation (in form that we can graph in)
4. Graph using rules of conic sections (identify center, identify vertices, etc.)
Note that the two parametric equations will be given to you, usually in the form x=__ and y=__
Example problem:
Math joke of the day:
Q: How do we know that the following fractions are inEurope ? A/C, X/C and W/C ?
A: Because their numerators are all over C's
1. solve for t in one equation
2. Substitute into 2nd equation
3. Simplify and get rectangular equation (in form that we can graph in)
4. Graph using rules of conic sections (identify center, identify vertices, etc.)
Note that the two parametric equations will be given to you, usually in the form x=__ and y=__
Example problem:
Math joke of the day:
Q: How do we know that the following fractions are in
A: Because their numerators are all over C's
Thursday, March 13, 2014
Interesting Math Stuff #9: Priyanshi Somani
Priyanshi Somani is a fifteen year old girl who can perform amazing mental calculations. She was the youngest competitor in the Mental Calculation World Cup in 2010 at twelve years old and won the overall title. She is the only participant in the history of the competition who has achieved 100% accuracy in addition, multiplication, and square roots.
In this competition, Priyanshi Somani was required to extract square roots from six digit numbers up to eight digits and managed to do it in 6 minutes and 51 seconds. She also added 10 numbers with 10 digits and multiplied 2 numbers of 8 digits with 100% accuracy.
She also has a World Cup record in calculating square roots as she solved 10 square roots correctly in 6 minutes and 28 seconds on June 7, 2010.
Priyanshi Somani is featured in the Limca Book of World Records and the Guinness Book of World Records. She has also been named the Indian Ambassador for the World Maths Day in 2011.
Source: http://en.wikipedia.org/wiki/Priyanshi_Somani
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