SUMMARY / RELATED TOPICS

Exponentiation

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: b n = b × ⋯ × b ⏟ n times; the exponent is shown as a superscript to the right of the base. In that case, bn is called "b raised to the n-th power", "b raised to the power of n", "the n-th power of b", "b to the n-th", or most as "b to the n". For any positive integers m and n, one has bn ⋅ bm = bn+m. To extend this property to non-positive integer exponents, b0 is defined to be 1, b−n with n a positive integer and b not zero is defined as 1/bn. In particular, b − 1 is equal to the reciprocal of b; the definition of exponentiation can be extended to allow any complex exponent. Exponentiation by integer exponents can be defined for a wide variety of algebraic structures, including matrices. Exponentiation is used extensively in many fields, including economics, chemistry and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, public-key cryptography.

The term power was used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios. Archimedes discovered and proved the law of exponents, 10a ⋅ 10b = 10a+b, necessary to manipulate powers of 10. In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms mal for a square and kahb for a cube, which Islamic mathematicians represented in mathematical notation as m and k by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī. In the late 16th century, Jost Bürgi used Roman numerals for exponents. Nicolas Chuquet used a form of exponential notation in the 15th century, used by Henricus Grammateus and Michael Stifel in the 16th century; the word "exponent" was coined in 1544 by Michael Stifel. Samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the terms square, zenzizenzic, zenzicube, second sursolid, zenzizenzizenzic. Biquadrate has been used to refer to the fourth power as well.

Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie. Some mathematicians used exponents only for powers greater than two, preferring to represent squares as repeated multiplication, thus they would write polynomials, for example, as ax + bxx + cx3 + d. Another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote "consider exponentials or powers in which the exponent itself is a variable, it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant." With this introduction of transcendental functions, Euler laid the foundation for the modern introduction of natural logarithm as the inverse function for the natural exponential function, f = ex. The expression b2 = b ⋅ b is called "the square of b" or "b squared" because the area of a square with side-length b is b2; the expression b3 = b ⋅ b ⋅ b is called "the cube of b" or "b cubed" because the volume of a cube with side-length b is b3.

When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, 35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243; the base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, 243 is the power or, more 3 raised to the 5th power; the word "raised" is omitted, sometimes "power" as well, so 35 can be read "3 to the 5th" or "3 to the 5". Therefore, the exponentiation bn can be expressed as "b to the power of n", "b to the nth power", "b to the nth", or most as "b to the n"; the exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations. Formally, powers with positive integer exponents may be defined by the initial condition b 1 = b and the recurrence relation b n + 1 = b n ⋅ b. From the associativity of multiplication, it follows that for any positive integers m and n, b m + n = b m ⋅ b n. Any nonzero number raised to the 0 power is 1: b 0 = 1. One interpretation of such a power is as an empty product.

The case of 00 is more complicated, the choice of whether to assign it a value and what value to assign may depend on context. For more details, see Zero to the power of zero; the following identity holds for any integer n and nonzero b: b − n = 1 b n. Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity; the identity above may be derived through a definition aimed at extending the range of exponents to negative integ

Margie Goldstein-Engle

Margie Goldstein-Engle is an American show jumping equestrian, a 10-time American Grandprix Association Rider of the Year. She was born in Wellington, Florida, to Mona and Irvin Goldstein, is Jewish, she grew up in her middle-class family in South Miami, with two older brothers. In third grade, she became passionate about horses. At the age of nine, she took jobs at horse barns and dog kennels as a way to pay for riding lessons. Less affluent than other riders, she said: "You're maybe not dressed like the other riders. You don't have the custom things, you don't have the top clothing, a lot of my stuff was hand-me-downs.... It was more cliquish than anything. They'd more snub you than tease you."She attended South Miami High School and North Miami Beach High School, graduated from Florida International University with a 4.0 GPA, majoring in business education. She married her husband, horse veterinarian Steve Engle, in 1995. Goldstein-Engle won 6 World Cups and 20 Nations Cups between 1984 and 2005.

The FEI ranked her as high as # 6 all-time. In 1987, she recorded a world-record-high jump of 7 feet 8 3⁄4 inches. Speaking of such high jump event, she said: "You have to figure the horse either has a lot of trust, or a lot of heart, because once the wall gets over six and a half feet, it looks more like the side of a building."In 1991, she suffered broken bones and nerve damage in her left foot as the result of a fall at a horse show. Doctors told her she would not walk again; the following week, she was again riding, 10 weeks she resumed competing. In 1992, a 1,200-pound horse fell on her at a show, opening a deep 12-inch cut on her back and breaking four of her ribs. In July 1998, she received injuries to her face as the result of a fall, she rode the next day. She has fractured her left shoulder, broken her collarbone twice, her arm, her wrist, two fingers. At the 1999 Pan American Games in Winnipeg, she won a silver medal with the U. S. jumping team. She competed for the U. S. 2000 Olympics team in Sydney, Australia.

She won a team gold medal and an individual bronze medal at the 2003 Pan American Games, a silver medal with the U. S. team in the 2006 World Equestrian Games. Goldstein-Engel was the American Grandprix Association’s only ten-time Rider of the Year, she won the award in 1989, 1991, 1994, 1995, 1996, 1999/2000, 2000/2001, 2003, 2005, 2006. She was the 1991 American Horse Shows Association Equestrian of the Year. Goldstein-Engle set a record with career show-jumping earnings of more than $4 million, she has more than 195 Grand Prix victories, as of October 2011 she was the all-time career leader in Grand Prix wins. She set a record with most Grand Prix wins in a single season. In 2001, she was honored by the U. S. Jewish Sports Hall of Fame, in 2009 she was inducted into the International Jewish Sports Hall of Fame. List of select Jewish equestrians No Hurdle Too High: The Story of Show Jumper Margie Goldstein Engle, by Mona Pastroff Goldstein, Margie Goldstein Engle, Globe Pequot, 2005, ISBN 1-59228-683-6

Gartan

Gartan is a parish in County Donegal, Ireland. It is best known for being the birthplace of Columba, one of the three patron saints of Ireland and one of the most revered saints in the Christian world. Here he founded a monastery in 521; the popular song "Gartan Mother's Lullaby" comes from the area, has been performed by many artists, including American actress Meryl Streep. The book Kenny, by Leona Dalrymple, which may be read online says: Parish of Gartan and Termon Gartan Outdoor Education Centre View of Lough Gartan from the burial ground at Gartan Rath Photo of birthplace of St. Columba at Gartan Photo of Lough Gartan seen from the burial ground at Gartan Rath