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#1 Dark Discussions at Cafe Infinity » Avatar, Avenue, and Avenues Quotes » Today 17:30:50

ganesh
Replies: 0

Avatar, Avenue, and Avenues Quotes

1. The ear is the avenue to the heart. - Voltaire

2. I hate dentists. That's why my tooth fell out. I was in the middle of a root canal and wouldn't go back, so it just dropped out when I was in the middle of Fifth Avenue. - Kate Moss

3. The day will come - and it is not far off - when the legacy of Lincoln will finally be fulfilled at 1600 Pennsylvania Avenue, when a black man or woman will sit in the Oval Office. When that day comes, the most remarkable thing about it will be how naturally it occurs. - George H. W. Bush

4. I've successfully convinced others to let me redevelop the historic Old Post Office Pavilion on Pennsylvania Avenue in Washington, D.C. I also led the acquisition of the iconic 800-acre Doral Resort & Spa from my hospital bed after giving birth to my daughter, Arabella. - Ivanka Trump

5. I don't look at my work as an avenue for generating more work. For me, my work itself is sufficient. - Naseeruddin Shah

6. I'm glad I made business investments, because it gave me the confidence financially to make brave choices. If I hadn't done that, I'd still have been trying to play 19-year-olds in films. I know there are other avenues for me. - Preity Zinta

7. I lose around a couple of crores every year on the school, but even if I was to make profit from it, I would never use it for myself. I'd plough every penny back into improving facilities for the school. Just as I do with the cricket academies I run around the country. These are not for making money; for that, I have other avenues. - Virender Sehwag

8. In our effort to tackle terrorism, we should look into all possible avenues of cyber-crime, its linkages with the terrorist world, and how these could be dealt with. - Rajnath Singh

9. Building upon the world we created with 'Avatar' has been a rare and incredibly rewarding experience. In writing the new films, I've come to realize that 'Avatar's world, story and characters have become even richer than I anticipated, and it became apparent that two films would not be enough to capture everything I wanted to put on screen. - James Cameron

10. I believe in 3D for certain kinds of films. I certainly believe in using 3D for all things in animation because animation has such clarity and so much depth of focus. It worked great with 'Avatar' because 70 percent of that film is animated. - Steven Spielberg

11. The only sci-fi movie that I've ever been offered that, had circumstances been different, I would have definitely done, was 'Avatar.' And I literally couldn't do it because of my schedule. But listening to James Cameron talk about 'Avatar' was so fascinating. Because he literally invented the world in his mind - and it literally existed. - Matt Damon

12. I want to be in 'Avatar'. I want somebody to hire me to be Superman, a Chinese Superman or Spider-Man. - Jackie Chan

13. When I stopped my work in films and took over the management of the house, sometime after marriage, my new avatar was replicated hilariously by Dilip Sahab a few times having me and my mother in splits. - Saira Banu.

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#2 Re: Exercises » Compute the solution: » Today 16:21:39

Hi Denominator,

Neat work!

1392.

#3 Re: Ganesh's Puzzles » Doc, Doc! » Today 16:07:45

Hi,

#1944. What does the medical term 'Trismus' mean?

#4 Re: Ganesh's Puzzles » Doc, Doc! » Today 16:07:27

Hi,

#1944. What does the medical term 'Trismus' mean?

#7 Re: Ganesh's Puzzles » Coordinate Geometry » Today 15:12:20

Hi Denominator,

Excellent!

CG # 226.

#8 Re: This is Cool » Miscellany » Today 14:34:23

1442) Color blindness

Color blindness (color vision deficiency) is the decreased ability to see color or differences in color. It can impair tasks such as selecting ripe fruit, choosing clothing, and reading traffic lights. Color blindness may make some academic activities more difficult. However, issues are generally minor, and the colorblind automatically develop adaptations and coping mechanisms. People with total color blindness (achromatopsia) may also be uncomfortable in bright environments and have decreased visual acuity.

The most common cause of color blindness is an inherited problem or variation in the functionality of one or more of the three classes of cone cells in the retina, which mediate color vision. Males are more likely to be color blind than females, because the genes responsible for the most common forms of color blindness are on the X chromosome. Non-color-blind females can carry genes for color blindness and pass them on to their children. Color blindness can also result from physical or chemical damage to the eye, the optic nerve, or parts of the brain. Screening for color blindness is typically done with the Ishihara color test.

There is no cure for color blindness. Diagnosis may allow an individual, or their parents/teachers to actively accommodate the condition. Special lenses such as EnChroma glasses or X-chrom contact lenses may help people with red–green color blindness at some color tasks, but they do not grant the wearer "normal color vision". Mobile apps can help people identify colors.

Red–green color blindness is the most common form, followed by blue–yellow color blindness and total color blindness. Red–green color blindness affects up to 1 in 12 males (8%) and 1 in 200 females (0.5%). The ability to see color also decreases in old age. In certain countries, color blindness may make people ineligible for certain jobs, such as those of aircraft pilots, train drivers, crane operators, and people in the armed forces. The effect of color blindness on artistic ability is controversial, but a number of famous artists are believed to have been color blind.

Details

Colour blindness is the inability to distinguish one or more of the three colours red, green, and blue. Most people with colour vision problems have a weak colour-sensing system rather than a frank loss of colour sensation. In the retina (the light-sensitive layer of tissue that lines the back and sides of the eyeball), humans have three types of cones (the visual cells that function in the perception of colour). One type absorbs light best in wavelengths of blue-violet and another in the wavelengths of green. The third type is most sensitive to longer wavelengths—more sensitive to red. Normal colour vision, when all three cone types are functioning correctly, is known as trichromacy (or trichromatism).

Types of colour blindness

There are several different types of colour blindness, which may be subdivided generally into dichromacy (dichromatism), when only two cone types are functional, and monochromacy (monochromatism), when none or only one type of cone receptor is functional. Dichromatic individuals are ordinarily unable to distinguish between red and green. Blindness to red is known as protanopia, a state in which the red cones are absent, leaving only the cones that absorb blue and green light. Blindness to green is known as deuteranopia, wherein green cones are lacking and blue and red cones are functional. Some persons experience anomalous dichromatic conditions, which involve only minor reductions or weaknesses in colour sensitivity. In protanomaly, for example, sensitivity to red is reduced as a result of abnormalities in the red cone photopigment. In deuteranomaly, in which sensitivity to green is reduced, the green cones are functionally limited. Two forms of blue-yellow colour blindness are known: tritanopia (blindness to blue, usually with the inability to distinguish between blue and yellow), which occurs when blue cones are absent; and tritanomaly (reduced sensitivity to blue), which arises from the abnormal function of blue cones.

Monochromacy, or complete colour blindness, is sometimes accompanied by deficiencies in visual acuity. Such conditions are rare and include achromatopsia (or rod monochromacy; the complete absence of functional cone photopigments) and cone monochromacy (when two of the three cone types are nonfunctional).

Inherited and acquired colour blindness

Hereditary red-green colour blindness occurs mainly in males and Caucasian persons, with about 8 percent of men and 0.5 percent of women of European ancestry inheriting the conditions. Its predominance in males is due to the fact that red-green colour blindness is a gender-linked recessive characteristic, carried on the X chromosome. Hence, the trait for red-green colour blindness is passed from mother to son, from mother to daughter, or from mother and father to daughter. A son who inherits the trait from a carrier mother will be red-green colour blind (males inherit only one X chromosome, directly from the mother). A daughter who inherits the trait from a carrier mother (with a normal father) will have normal colour vision but be a carrier of the trait. A daughter who inherits the trait from both her mother and her father will be red-green colour blind.

Blue-yellow colour blindness, by contrast, is an autosomal dominant disorder and therefore is not gender-linked and requires only one copy of the defective gene from either parent to be expressed. Achromatopsia is an autosomal recessive disorder, occurring only when two copies of the defective gene (one from each parent) have been inherited. Persons who inherit colour blindness may show symptoms at birth (congenital colour blindness), or they may become symptomatic later, in childhood or adulthood.

Acquired colour blindness is usually of the blue-yellow type and ranges from mild to severe. Often it is associated with chronic disease, such as macular degeneration, glaucoma, diabetes mellitus, retinitis pigmentosa, or Alzheimer disease. Certain drugs and chemicals can also cause acquired colour blindness.

visiblecolorspectrumcomparison-a6ad132979ca421fb0de7952a3cc2e2c.jpg

#9 Jokes » One liners - 46 » Today 00:49:23

ganesh
Replies: 0

My IQ came back negative.
* * *
I used to be a banker, but then I lost interest.
* * *
A wise man once said... Nothing, he only listened.
* * *
The reason grandchildren and grandparents get along so well is because they have a common "enemy".
* * *
When I found out that my toaster wasn't waterproof, I was shocked.
* * *

#10 Re: Maths Teaching Resources » Quantitative Aptitude and Higher Mathematics » Today 00:23:13

Pascal's Triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

The rows of Pascal's triangle are conventionally enumerated starting with row n=0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k=0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

Factorial

Combinations and Permutations

Pascal's Triangle

Binomial Theorem.

#11 Re: Dark Discussions at Cafe Infinity » crème de la crème » Today 00:17:26

1122) Michael Collins

Summary

Michael Collins (October 31, 1930 – April 28, 2021) was an American astronaut who flew the Apollo 11 command module Columbia around the Moon in 1969 while his crewmates, Neil Armstrong and Buzz Aldrin, made the first crewed landing on the surface. He was also a test pilot and major general in the U.S. Air Force Reserves.

Born in Rome, Italy, Collins graduated in the Class of 1952 from the United States Military Academy. He joined the United States Air Force, and flew F-86 Sabre fighters at Chambley-Bussières Air Base, France. He was accepted into the U.S. Air Force Experimental Flight Test Pilot School at Edwards Air Force Base in 1960, also graduating from the Aerospace Research Pilot School (Class III).

Selected as part of NASA's third group of 14 astronauts in 1963, Collins flew in space twice. His first spaceflight was on Gemini 10 in 1966, in which he and Command Pilot John Young performed orbital rendezvous with two spacecraft and undertook two extravehicular activities (EVAs, also known as spacewalks). On the 1969 Apollo 11 mission, he became one of 24 people to fly to the Moon, which he orbited thirty times. He was the fourth person (and third American) to perform a spacewalk, the first person to have performed more than one spacewalk, and, after Young, who flew the command module on Apollo 10, the second person to orbit the Moon alone.

After retiring from NASA in 1970, Collins took a job in the Department of State as Assistant Secretary of State for Public Affairs. A year later, he became the director of the National Air and Space Museum, and held this position until 1978, when he stepped down to become undersecretary of the Smithsonian Institution. In 1980, he took a job as vice president of LTV Aerospace. He resigned in 1985 to start his own consulting firm. Along with his Apollo 11 crewmates, Collins was awarded the Presidential Medal of Freedom in 1969 and the Congressional Gold Medal in 2011.

Details

Michael Collins, (born October 31, 1930, Rome, Italy—died April 28, 2021, Naples, Florida, U.S.), was an U.S. astronaut who was the command module pilot of Apollo 11, the first crewed lunar landing mission.

A graduate of the U.S. Military Academy at West Point, New York, Collins transferred to the air force, becoming a test pilot at Edwards Air Force Base in California. He joined the space program in 1963.

Gemini 10, crewed by Collins and command pilot John W. Young, was launched on July 18, 1966. After a rendezvous with an Agena target vehicle, the two men used the Agena’s engines to propel them to a record altitude of 764 km (475 miles), where Collins left the spacecraft to remove equipment needed for a micrometeorite experiment from the aft end of the Gemini and attempted unsuccessfully to attach similar equipment to the Agena. He succeeded in retrieving an instrument from the Agena, but his activity was cut short because the Gemini craft was low on fuel. Gemini 10 returned to Earth on July 21.

On July 16, 1969, Collins was launched to the Moon in the Apollo 11 mission with commander Neil A. Armstrong and lunar module pilot Edwin E. Aldrin, Jr. Armstrong and Aldrin landed on the Moon in the lunar module Eagle on July 20 while Collins remained in the command module Columbia, circling the Moon at an altitude of 97–121 km (60–75 miles). On July 21 Armstrong and Aldrin rejoined him, and the following day the astronauts left lunar orbit. They splashed down in the Pacific Ocean on July 24. The three astronauts spent 18 days in quarantine to guard against possible contamination by lunar microbes. During the days that followed and during a tour of 21 nations, they were hailed for their part in the opening of a new era in humankind’s exploration of the universe.

Apollo 11 was his last space mission; later in 1969 Collins was appointed assistant secretary of state for public affairs. In 1971 he became the first director of the National Air and Space Museum in Washington, D.C., and in 1978 he became undersecretary of the Smithsonian Institution. From 1980 to 1985 he was vice president for field operations for Vought Corporation, an American aerospace firm. He wrote four books, including an account of the Apollo 11 mission, Carrying the Fire (1974), and a history of the American space program, Liftoff (1988).

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#13 Re: Ganesh's Puzzles » English language puzzles » Today 00:15:21

Hi,

#4671. What does the noun legatee mean?

#4672. What does the noun legion mean?

#14 Re: Ganesh's Puzzles » General Quiz » Today 00:15:05

Hi,

#8543. What does the term 'Logorrhea' (psychology) mean?

#8544. What does the term 'Ludomania' mean?

#15 Dark Discussions at Cafe Infinity » Avail, and Available Quotes » Yesterday 17:46:57

ganesh
Replies: 0

Avail, and Available Quotes

1. There is no substitute for talent. Industry and all its virtues are of no avail. - Aldous Huxley

2. It will be of little avail to the people that the laws are made by men of their own choice if the laws be so voluminous that they cannot be read, or so incoherent that they cannot be understood. - James Madison

3. So long as you do not achieve social liberty, whatever freedom is provided by the law is of no avail to you. - B. R. Ambedkar

4. I am an enthusiast, but not a crank in the sense that I have some pet theories as to the proper construction of a flying machine. I wish to avail myself of all that is already known and then, if possible, add my mite to help on the future worker who will attain final success. - Wilbur Wright

5. The lack of financial strength to avail healthcare is a major challenge. - N. R. Narayana Murthy

6. Whenever senior players are missing, its ideal for youngsters to avail of these opportunities. - Virender Sehwag

7. What this power is I cannot say; all I know is that it exists and it becomes available only when a man is in that state of mind in which he knows exactly what he wants and is fully determined not to quit until he finds it. - Alexander Graham Bell

8. Time can be an ally or an enemy. What it becomes depends entirely upon you, your goals, and your determination to use every available minute. - Zig Ziglar

9. The wealth of information now available at the click of a finger amazes me. - A. P. J. Abdul Kalam

10. It's difficult to believe that people are still starving in this country because food isn't available. - Ronald Reagan

11. One thing that is great about India is the freedom to speak and the spaces available in our democracy to protest which doesn't exist in many places in the world. - Arvind Kejriwal

12. Be of service. Whether you make yourself available to a friend or co-worker, or you make time every month to do volunteer work, there is nothing that harvests more of a feeling of empowerment than being of service to someone in need. - Gillian Anderson

13. Since being in India, I am more convinced than ever before that the method of nonviolent resistance is the most potent weapon available to oppressed people in their struggle for justice and human dignity. - Martin Luther King, Jr.

14. Rhetoric may be defined as the faculty of observing in any given case the available means of persuasion. This is not a function of any other art. - Aristotle

15. Surveillance technologies now available - including the monitoring of virtually all digital information - have advanced to the point where much of the essential apparatus of a police state is already in place. - Al Gore

16. I've gotten very good at scheduling my life, scheduling the scene and preparing myself for knowing, saving the energy, consuming the energy, knowing when to go for it and having the available reserves to be able to do that. You have to think about that, because it's endurance. - Tom Cruise

17. As an educator, I have always striven to see that the fruits of the American Dream are available to all. - James D. Watson

18. That's what I'd like to do on the President's Council. Make sports and athletics available to every youth in America, not just one day a week like it was for me, but every day. - Florence Griffith Joyner

19. If you have 50 different plug types, appliances wouldn't be available and would be very expensive. But once an electric outlet becomes standardized, many companies can design appliances, and competition ensues, creating variety and better prices for consumers. - Bill Gates

20. I was always longing to do, emotionally and physically, what my male counterparts always got to do. I just felt envious, every time I saw a movie that I was in awe of, and it was usually a male lead. And those kinds of roles weren't available. They just weren't being written. - Sandra Bullock

21. I'm not the kind of person who likes to shout out my personal issues from the rooftops, but with my bipolar becoming public, I hope fellow sufferers will know it's completely controllable. I hope I can help remove any stigma attached to it, and that those who don't have it under control will seek help with all that is available to treat it. - Catherine Zeta-Jones

22. Even if a media of a TV is not available in a home, there's this concept of community homes, where a reasonably well-off villager will have a TV - and a nice TV - and he'll keep it outside the house in the evenings. - Azim Premji

23. The world's poorest people use the cheapest available fuels - dung and twigs and even leaves. - Abhijit Banerjee

24. I am always available to share my experiences. I am happy to talk to people in Indian sports administration, maybe teach them a few things, and eventually help out an athlete. That would make me happiest. - Abhinav Bindra

25. It's difficult to imagine the power that you're going to have when so many different sorts of data are available. - Tim Berners-Lee

26. I urge researchers to make use of the opportunities that are available to them and to do all they can to fulfill the promise that stem cell research offers. - Nancy Reagan

27. The Mobile Web Initiative is important - information must be made seamlessly available on any device. - Tim Berners-Lee

28. Socialism never arises in the earlier phases of capitalism, as, for instance, among the pioneers of civilisation in a country where there is plenty of land available for private appropriation by the last comer. - George Bernard Shaw

29. Towards the end of February 1928, I took the decision of using brilliant monochromatic illumination obtained by the aid of the commercially available mercury arcs sealed in quartz tubes. - C. V. Raman

30. Don't avoid extremes, and don't choose any one extreme. Remain available to both the polarities - that is the art, the secret of balancing. - Rajneesh.

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#16 Re: Ganesh's Puzzles » Doc, Doc! » Yesterday 15:43:15

Hi,

#1943. What does the medical term 'Neurogenic placode' mean?

#19 Re: This is Cool » Miscellany » Yesterday 14:09:36

1441) Scandinavian literature

Summary

Scandinavian literature or Nordic literature is the literature in the languages of the Nordic countries of Northern Europe. The Nordic countries include Denmark, Finland, Iceland, Norway (including Svalbard), Sweden, and Scandinavia's associated autonomous territories (Åland, Faroe Islands and Greenland). The majority of these nations and regions use North Germanic languages. Although majority of Finns speak Uralic languages, Finnish history and literature are clearly interrelated with those of both Sweden and Norway who have shared control of various areas and who have substantial Sami populations/influences.

These peoples have produced an important and influential literature. Henrik Ibsen, a Norwegian playwright, was largely responsible for the popularity of modern realistic drama in Europe, with plays like The Wild Duck and A Doll's House. Nobel prizes for literature, itself a Scandinavian award, have been awarded to Selma Lagerlöf, Verner von Heidenstam, Karl Adolph Gjellerup, Henrik Pontoppidan, Knut Hamsun, Sigrid Undset, Erik Axel Karlfeldt, Frans Eemil Sillanpää, Johannes Vilhelm Jensen, Pär Lagerkvist, Halldór Laxness, Nelly Sachs, Eyvind Johnson, Harry Martinson, and Tomas Tranströmer.

Details

Scandinavian literature, also called Nordic literature, is the body of works, both oral and written, produced within Scandinavia in the North Germanic group of languages, in the Finnish language, and, during the Middle Ages, in the Latin language.

Scandinavian literature traditionally consists of works in modern Swedish, Norwegian, Icelandic, Danish, and Faroese, all members of the North Germanic group of languages. The literary works written in these languages show deep-seated common linguistic ties. The Finnish language is unrelated to the North Germanic languages; it belongs instead to the Baltic-Finnic branch of the Finno-Ugric language family and is most closely related to Estonian and Karelian. Because Sweden ruled Finland for more than six centuries, Finnish literature, despite its linguistic differences, became closely intertwined with Swedish literature.

The term Scandinavia traditionally designates the two countries of the Scandinavian Peninsula—Norway and Sweden—and Denmark. Finland and Iceland are frequently called Scandinavian countries on geographic, political, and cultural grounds. The term Nordic is often used today to refer collectively to the Åland Islands, Denmark, Finland, the Faroe Islands, Greenland, Iceland, Norway, and Sweden.

Although the Scandinavian literatures exhibit similarities stemming from close cultural ties, they manifest differences reflective of distinct national institutions and historical and geographic conditions. They are therefore discussed separately under Danish literature, Faroese literature, Icelandic literature, Norwegian literature, and Swedish literature. Works written in Finland in the Swedish language (Finland-Swedish literature) and in the Finnish language are discussed under Finnish literature.

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#20 Re: This is Cool » Important Laws/Principles in Physics » Yesterday 01:11:33

27) Stokes' law

In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

Statement of the law

The force of viscosity on a small sphere moving through a viscous fluid is given by:

where:

is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle
is the dynamic viscosity (some authors use the symbol
)
R is the radius of the spherical object
v is the flow velocity relative to the object.

In SI units,

is given in newtons
in Pa·s
, R in meters, and v in m/s.

Stokes' law makes the following assumptions for the behavior of a particle in a fluid:

* Laminar flow
* Spherical particles
* Homogeneous (uniform in composition) material
* Smooth surfaces
* Particles do not interfere with each other.

Particles do not interfere with each other.

For molecules Stokes' law is used to define their Stokes radius and diameter.

The CGS unit of kinematic viscosity was named "stokes" after his work.

Applications

Stokes' law is the basis of the falling-sphere viscometer, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches terminal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid, Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameters are normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses glycerine or golden syrup as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. Several school experiments often involve varying the temperature and/or concentration of the substances used in order to demonstrate the effects this has on the viscosity. Industrial methods include many different oils, and polymer liquids such as solutions.

The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least three Nobel Prizes.

Stokes' law is important for understanding the swimming of microorganisms and sperm; also, the sedimentation of small particles and organisms in water, under the force of gravity.

In air, the same theory can be used to explain why small water droplets (or ice crystals) can remain suspended in air (as clouds) until they grow to a critical size and start falling as rain (or snow and hail). Similar use of the equation can be made in the settling of fine particles in water or other fluids.


For molecules Stokes' law is used to define their Stokes radius and diameter.

The CGS unit of kinematic viscosity was named "stokes" after his work.

#21 Re: Ganesh's Puzzles » English language puzzles » Yesterday 00:52:07

Hi,

#4669. What does the adjective unwonted mean?

#4670. What does the verb (used with object) upbraid mean?

#22 Re: Ganesh's Puzzles » General Quiz » Yesterday 00:38:09

Hi,

#8541. What does the term 'Graphomania' mean?

#8542. What does the term 'Hypomania' mean?

#23 Re: Maths Teaching Resources » Quantitative Aptitude and Higher Mathematics » Yesterday 00:12:45

Square and Square Roots

Cube and Cube Roots

nth Root

Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field.

General definition

An nth root of unity, where n is a positive integer, is a number z satisfying the equation

Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number -1 if n is even, which are complex with a zero imaginary part), and in this case, the nth roots of unity are

However, the defining equation of roots of unity is meaningful over any field (and even over any ring) F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either complex numbers, if the characteristic of F is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details.

An nth root of unity is said to be primitive if it is not an mth root of unity for some smaller m, that is if

If n is a prime number, then all nth roots of unity, except 1, are primitive.

In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers.

Subsequent sections of this article will comply with complex roots of unity.

Trigonometric expression

De Moivre's formula, which is valid for all real x and integers n, is

Setting

gives a primitive nth root of unity – one gets

but

for k = 1, 2, …, n - 1. In other words,

is a primitive nth root of unity.

This formula shows that in the complex plane the nth roots of unity are at the vertices of a regular n-sided polygon inscribed in the unit circle, with one vertex at 1 (see the plots for n = 3 and n = 5 on the right.) This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).

Euler's formula

which is valid for all real x, can be used to put the formula for the nth roots of unity into the form

It follows from the discussion in the previous section that this is a primitive nth-root if and only if the fraction

is in lowest terms; that is, that k and n are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as

is called a trigonometric number.

#24 Jokes » One liners - 45 » Yesterday 00:12:04

ganesh
Replies: 0

So apparently RSVP'ing back to a wedding invite 'maybe next time' isn't the correct response.
* * *
Did you hear about these new reversible jackets? I'm excited to see how they turn out.
* * *
Politics is the art of looking for trouble, finding it, misdiagnosing it and then misapplying the wrong remedies.
* * *
The light at the end of the tunnel has been turned off due to budget cuts.
* * *
I think we should get rid of democracy. All in favor raise your hand.
* * *

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