 | Complex Numbers (part 2) Dividing complex numbers. Complex conjugates. |
 | Complex Numbers Conjugate and Magnitude A short introduction to Complex Numbers. Check out www.gaussianmath.com for related math topics and more AMC problems. |
 | The Swallowtail Consider the univariate polynomial p(x)=x^4+ax^2+bx+c with parameters a,b,c (the cyan-colored curve in the plane which is located in the upper part of our film). It has a double root where both p(x) and its derivative p'(x) vanish. To determine for which a, b, c this happens, we eliminate the variable x from these two equations and obtain the so-called discriminant of p. Many articles have been written about discriminants; here, we focus on the example p(x) above which is well-known and even appears in many text books. The discriminant s(a,b,c) which is a polynomial of degree 5 in the variables a, b, c, is called the swallowtail. This is the pink surface in our film. Clearly, p(x) has a double root for those a, b, c for which s(a,b,c) vanishes. But s(a,b,c) holds much more information than that! Depending on the position of the point (a,b,c) w.r.t. the swallowtail, we can see how many real roots the corresponding polynomial p(x) has and which multiplicity they have. Of course, the point (0,0,0) corresponds to the case for which the polynomial p(x) has a 4-tuple root 0. On the swallowtail, this is the most singular point. For smooth points (a,b,c) on the swallowtail, the corresponding polynomial p(x) has exactly one double root and either 0 or 2 simple roots depending on the location of the smooth point. Where the left and the right smooth part intersect transversally (i.e. in the lower middle part of our film), this happens twice and p(x) has two distinct double roots. On the real isolated half-parabola, which can be seen in the part of the film where the swallowtail rotates, these two distinct double roots are both complex (and thus complex conjugate). For points inside the triangular-shaped part of the swallowtail (i.e. in the center part of our film) p(x) has exactly four distinct real roots. On a smooth point of the part of the swallowtail which is the boundary of this triangular-shaped region p(x) has four real roots, two of which coincide. On the cuspidal edge of the triangular-shaped region, this happens twice but not symmetrically which means that p(x) has one tripled root and one simple root. This film was made by Oliver Labs using surfex. We thank Frank-Olaf Schreyer for useful comments. |
 | Imaginary Numbers and Roots of a Quadratic Equation Why the roots of a quadratic equation are either both real or a complex conjugate pair. |
 | Watch Video on Complex Numbers - Algebra 2 Help - High School Math - College Algebra Access full lesson containing this video at: http://www.yourteacher.com/algebra2/multiplyingcomplexnumbers.php Students learn to add, subtract, multiply, and divide complex numbers that contain radicals. For example, to divide (3 + 5i root 11) / (6 + 2i root 11), the first step is to multiply both the numerator and denominator of the fraction by the conjugate of the denominator, which is (6 -- 2i root 11), then FOIL in both the numerator and denominator, and combine like terms. |
 | Watch Video on Complex Numbers - Algebra 2 Help Access full lesson containing this video at: http://www.yourteacher.com/algebra2/complexnumbers.php Students learn that a complex number is the sum or difference of a real number and an imaginary number and can be written in a + bi form. For example, 1 + 2i and -- 5 -- i root 7 are complex numbers. Students then learn to add, subtract, multiply, and divide complex numbers that do not contain radicals, such as (5 + 3i) / (6 -- 2i). To divide (5 + 3i) / (6 -- 2i), the first step is to multiply both the numerator and denominator of the fraction by the conjugate of the denominator, which is (6 + 2i), then FOIL in both the numerator and denominator, and combine like terms. |
 | ZSAI 2008 Zaytuna Summer 2008 Graduation Ceremony Brother Dawud's Poem Al-Weird Latif / Dawud McCarthy The young Muslim can feel abused His brothers make him feel so confused They point the finger at his point of views Asks for help but gets refused Sometimes you want leave the deen entirely When the Imam dismissed your entire society So much pressure you're ready to pop Can't you do your dhikh in the form of hip hop? For years we've struggled through this phase Of trying to find the American ways To know what's up But still lower our gaze The shades of gray only grew more dense Does Islam in this country even make sense? Of your jeans and t-shirts you're in constant defense Can't the yard of the mosque have a white picket fence? From my circumstances I had to get free And I heard there was shade under the olive tree Made up my mind to make a new start Going to California with an aching in my heart So each night I prayed and prayed for Allah to send me to meet Imam Zaid Before I end up completely reclusive I had to talk to Hamza Yusef Prayers were answered, bags to pack Go West young man and Don't Look Back Find if its true that I'm a villain If I eat hot dogs and sing Bob Dylan Make salah and go to class I hope I find the truth at last Would learning Arabic make it easy To translate the Shariah to Engleezy? I must have thought my eyes had lied The girls and boys sat side by side! And I almost had a heart attack When I heard Imam Zaid quote the Man in Black! How refreshing it is to see Islam with gender equality American born spiritual reformers Who used to skate board and watch "Transformers" And what a beautiful surprise A culture I could recognize Muslims who can mix between Seen, sheen and Rage Against the Machine Finally my heart felt free With 'Abdullah 'Ali and Ustadh D They took the complex to simplicity Walking Alif Baa Taa to Run DMC (or Mork and Mindy) And did Huey Lewis hear the news That our Imam wears Adidas shoes? And accepts the point of different views And marches to the beat of this song Maybe we can all get along What do you think the muftis would say If they heard the idea That both Sunni and Shia Feel that its OK to pray Side by side in the city by the Bay? Amazing that a little tolerance now Can slaughter the old sacred cow A little Abdul Latif will correct false belief An application oh so vital Use a Public Enemy to smash a Billy Idol A middle way that can be found Educated Islam in a college town In halaqas we'd gather 'round Quoting 'Omar, 'Ali, and James Brown The Berkeley scene is great for joking But I don't know what that guy was smoking It really is an incredible town If only the frat house could keep it down So make our spirits feel at peace We'd burn some Ood with Rhamis And if our evenings ever needed a spark I'm pretty sure his shoes glow in the dark And if we needed room to pray During San Fransisco stay We had Sevim to plan our day If our hunger made us mad We looked no further than Ebad Who knew where tasty meals are had Because safe inside Zaytuna school We take exception to the rule That you must choose between Islam and cool Between the 4th of July and Fiqh and Usool Only an accusative tool could be so cruel We on the A-Team pity the fool And then you woke up to these realities East Coast Shafi'is meet West Side Malikis Each day in class we get our fix Verb conjugation and politics If I ask not what my country can do for me The grammar makes its plain to see That I would trade Barack Obama For candidate Sidi Usama Like the Monkees, I'm believin' That a truly inspired resident Would make Eric B the president And stop the bleedin' Do you really think they would come from Sweden For anything less than the Garden of Eden? For an Arabic Summer dream come true From around the world on planes they flew From Australia to Great Britain Trying to find where we all fit it To find religion at its best Revival coming from the West Lessons we can teach the rest For these two months Allah I praise I'll remember them the rest of my days God bless Zaytuna Institute For the misconceptions it does refute Incredibly we've come so far Filled with pride for who we are Amazing Grace how sweet the sound I once was lost but now I'm found And if you know what I mean Make du'a and say "Ameen" Say Ameen! |
 | Parameciums Here is a video of a drop of water from my pond through a microscope. Theres loads of Parameciums swimming around, the specks are single celled algae (I think) and the clumps of plant matter are tiny specks to the naked eye. Here's some info about Parameciums... Movement The outer surface of the cell is covered with many hundreds of tiny hair-like structures called cilia. These act like microscopic oars to push through the water, enabling the organism to swim. The speed of motion is about four times its own length per second. It moves so quickly that microscopists have to add a thickening agent to the water to slow it down to study it. As it moves through the water it rotates on its axis and small particles of debris and food are collected and swept into the gullet. If Paramecium comes across an obstacle, it stops, reverses the beating of the cilia, swims backwards, turns through an angle and moves forward again on a slightly different course. Feeding Paramecium has a permanent feeding mechanism, consisting of a funnel-shaped gullet into which food is drawn by the combined action of cilia which cover the body and other cilia lining the gullet. They feed on small organisms such as bacteria and even other smaller protozoa. Reproduction Paramecium has two means of reproduction, simple division and conjugation. A. Division. In favourable conditions the cell divides in two by a process called binary fission. This forms two new cells, each of which rapidly grows any new structures required and increases in size. This whole process make take place two or three times a day if conditions were right. B. Conjugation. This is a more complicated method. It involves two cells coming together to exchange nuclear material. The two cells then separate and continue to reproduce by simple division. It is similar in some ways to sexual reproduction in more complex animals. |
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