Friday, February 26, 2010

Irrational Numbers

Of the infinite set of numbers, the most known are the counting numbers, 1,2,3... The negative numbers and zero are combined with the counting numbers to form the wider set of integers. Integers can be divided to form fractions, therefore expanding the system to rational numbers.

All of the above is common knowledge, but the number system can be further expanded by taking the infinite sum of a convergent sequence of rationals, or, more commonly, simply taking the square root of a non-perfect square. Either method results in a number that has an infinite decimal expansion that never repeats. Mathematically, this means that the number cannot be expressed as a fraction. Numbers of this type are called irrational numbers.

Irrational numbers can also be subdivided further into two groups: the algebraic irrationals, and the transcendental irrationals. The algebraic irrationals are simply the ones that solve a polynomial equation with integer coefficients. This simply means any equation like x^2+3x+5=0, x^5-97x^3+4=0, etc. Other numbers can also be algebraic, and all rational numbers are. (for equations like 3x-2=0, the solution is 2/3. All fractions, and therefore all rationals, can be expressed this way) It is also obvious why square roots are algebraic, they all fit equations of the form x^2=n, where the solution will always be the square root of n. Cube roots, and all other roots come about in the same way. Finally, all numbers of the form a+square root of b can be reached by means of the quadratic formula. Some other irrationals are algebraic as well, but it is a lot less obvious.

Transcendental numbers are all irrational, and these are simply the remaining numbers that do not fit the above category. Among these are π (pi), e, and any nonzero powers of either of them. There are still some numbers, like the square root of two to the power of the square root of two, π-e, and others that it is not known whether they are algebraic or transcendental.

Another method for representing the irrational numbers is the continued fraction. A continued fraction looks like this:



In the above picture, the first constant a0 must be an integer, and the remaining numbers must be positive integers. If this conditions are filled, the expression is a continued fraction. Continued fractions can represent all real numbers, rational and irrational (technically, complex numbers are the sum of two continued fractions, one real, one imaginary). Unlike decimal expansions, where rational numbers can go on forever (Ex. 1/3=.333333...) every continued fraction for a rational number is finite, meaning that there are a limited number of constants in the continued fraction. Conveniently, all irrational numbers have infinite continued fractions. The continued fraction for the square root of 2 has the constants 1,2,2,2,2,2... with the two repeating indefinitely. Another interesting property of continued fractions is that the constants of a algebraic irrational number repeat (Ex. square root of three has constants 1,1,2,1,2,1,2,1,2) while transcendental irrationals have a non-repeating sequence of constants (Ex. π has the constants 3, 7, 15, 1, 292, 1, 1, 1, 2, 1...). In a sense, the number is a irrational irrational!

There are infinite irrational numbers, but three notable ones are listed below.

π (pi)
A detailed article on pi in available elsewhere on this blog, see here.

e
Type: Transcendental irrational
Decimal Expansion: 2.71828182845904523536...
Continued Fraction Constants: 2,1,2,1,1,4,1,1,8,1,1,16...(this sequence of constants does have a pattern, but it still does not repeat, making e a transcendental irrational)
Description: The constant e is important in many aspects of mathematics, particularly calculus, and it appears in many functions. e is the limit of the function (1+1/x)^x as x approaches infinity and is the base of the function e^x, which is the only function of x that when differentiated stays the same. e is also the base of the natural logarithm, denoted ln(x) which has many properties. e is related to the notable constants π, i, 1, and 0, through Leonhard Euler's famous eqaution e^πi+1=0. Overall, the constant e shows up in many unlikely places in various regions of mathematics.

φ (phi)
Type: Algebraic irrational
Decimal Expansion: 1.6180339887
Continued fraction constants: 1,1,1,1,1,1,1...
Description: φ is an irrational number that is also commonly known as the golden ratio. This is because it is the ration between two line segments that makes it also the ratio between the longer segment and the sum of the two segments. The image below illustrates this property.



φ is also the only number whose reciprocal is itself minus one (1/φ+1=φ). Interestingly, φ is also related to the Fibonacci Sequence (the sequence that starts 1,1 and each new number is the sum of the previous two). The ratio of two consecutive terms of the Fibonacci sequence approximates φ, and the approximation becomes more accurate with higher Fibonacci numbers. φ has numerous properties, and is the only number to satisfy many formulae throughout mathematics.

There are many other irrationals but the above are the most notable.

Sources: http://en.wikipedia.org/wiki/Continued_fraction (info and image), http://en.wikipedia.org/wiki/E_(mathematical_constant) (info)

Thursday, February 18, 2010

Notable Hurricanes: Longevity; Part 3

This is the conclusion of a three part post. For the first part see here. For the second part see here.

#3 Hurricane Inga (1969) September 20-October 15 24.75 days (from formation to final extratropical transition)

The depression that became Inga formed on September 20 from a tropical wave that moved off of Africa a few days before. The system quickly became a tropical storm, but then was a depression again until the 28th. Upon turning northward, Inga became a hurricane and made a very tight loop southeast of Bermuda, before making a bigger loop and becoming a major hurricane, reaching its peak intensity of 115 mph winds and a pressure of 964 millibars. Still moving slowly, Inga started to weaken halfway through its loop and dissipated on October 15. Although Inga was very long lived, it didn't affect land and no damage resulted. Inga had an ACE of 31.64.



Track of Inga.

#2 Hurricane Ginger (1971) September 5-October 3 27.25 days (from formation to final extratropical transition)

Hurricane Ginger originated from a very large trough of low pressure spanning from the Gulf of Mexico, into the Bahamas, and through parts of the Caribbean. A total of four tropical cyclones, one depression, one tropical storm, and two hurricanes, could be traced back to this low-pressure area. The second of these to form from the trough was Ginger. The tropical depression that became Ginger was recognized as such on September 5, and drifted for much of the next four days. Heading southeast, and then northeast, the depression became Tropical Storm Ginger on September 10. Ginger started to accelerate east-north-east, and rapidly gained intensity before becoming a hurricane. On September 14, now a Category 2 hurricane, Ginger reached its peak intensity of 105 mph winds and a minimum sea level pressure of 959 millibars. Then, a strong high pressure system to the northeast of Ginger slowed its movement, turned it around, and weakened it to a Category 1 hurricane. It stalled for a few days, making a very tight loop on the 19th and 20th of September, before adopting a sluggish, but steady, westward movement. On September 23, Ginger passed its closest to Bermuda. Ginger stalled again a day later and drifted from the 24th to the 29th. During this time, Project Stormfury tried to weaken the storm on September 26 and September 28. Project Stormfury is a program that attempted to dissipate the clouds associated with tropical cyclones by "seeding" them, or dropping chemicals into the eyewall to weaken it. Ginger was the last hurricane to be used for this experiment due to lack of financial support and the fact that it simply didn't seem to work. The idea is sound in theory, but execution isn't feasible for two reasons. Firstly, it costs too much to carry a significant amount of chemicals to high altitudes, and second that the hurricanes quickly redevelop due to the abundance of moisture in ocean air. Soon after Ginger turned to the northwest, and became a Category 2 hurricane again, before weakening to a Category 1. On September 30, Ginger struck North Carolina, still a hurricane, but barely. It weakened inland and was nearly stationary for the first few days of October before becoming extratropical and moving off the coast the east on October 3, joining the Atlantic Ocean once more. Ginger caused some damage, $53 million adjusted for inflation ($10 million at the time), but no deaths resulted. Most of the damage came from the central coast of North Carolina, where a wide area had received over ten inches of rain from Ginger. Ginger had an ACE of 44.19.



Track of Hurricane Ginger, spanning nearly a month.

#1 Hurricane Three (1899) August 3-September 4 28 days (from formation to final extratropical transition)

The tropical storm destined to be named Hurricane San Ciriaco for the place it damaged formed on August 3, 1899. It quickly intensified in the favorable region and was a hurricane by August 5 and a major hurricane soon afterward. It brushed the easternmost islands of the Caribbean shortly before striking Puerto Rico, and its namesake, San Ciriaco, at its peak intensity as a Category 4 of 150 mph winds and an approximate pressure of 930 millibars. After devastating Puerto Rico, the system moved onward to the Dominican Republic, passing north of the island. Over the next four days or so, the hurricane passed through the Bahamas, still a Category 3, before striking North Carolina on August 17. It emerged off of North Carolina as a minimal Category 1 hurricane, and sped off eastward into open waters becoming extratropical on August 22. However, the system was pushed south for a time, and regained enough tropical characteristics to be a tropical storm again on August 26. The tropical storm slowly moved north, and then slowly east. The storm intensified and accelerated east, briefly becoming a Category 1 hurricane on October 3rd before losing its tropical features and becoming extratropical for the last time the next day. Although the damage caused by this storm cannot be calculated in dollars, it can be measured as one of the deadliest tropical cyclones in history, causing 3,433 fatalities, most of them (about 3,200 of them) occurring in Puerto Rico. Hurricane San Ciriaco also had the highest ACE of any Atlantic hurricane, at 73.57.



Track of Hurricane San Ciriaco, a long-lived and deadly hurricane.

Through this and two other posts, the top ten Atlantic hurricanes it terms of longevity have been listed. However, one record remains, the longest lived tropical cyclone in the world. Hurricane San Ciriaco is still second in the world, behind only a remarkable cyclone that was both a hurricane and a typhoon in the Pacific Ocean, due to the fact that it crossed the International Date Line. A summary of the cyclone is shown below.

#1 (world) Hurricane/Typhoon John (1994) August 11-September 10 31 days (from formation to final extratropical transition)

Due to the advanced reanalysis that occurs for all tropical cyclones after (sometimes years after) they form, this particularly powerful hurricane that formed in the Pacific, can have its wave origins trace back to the Atlantic. On July 25, 1994, a tropical wave moved off Africa, and moved westward into unfavorable conditions for development. Therefore, the wave didn't become a low pressure system until it entered the East Pacific Ocean through Central America and became a tropical depression on August 11. It quickly became a tropical storm, but wind sheer was ripping at the system, and Tropical Storm John couldn't strengthen until August 19, when John had nearly passed out of the reach of the sheering system. Upon entering this favorable zone, John became a hurricane, and soon, a major hurricane. On August 20, John passed out of the East Pacific, into the Central Pacific. Unlike the Atlantic, whose boundaries are easily measured by land and the equator, the Pacific is split into three separate basins, the East, the Central, and the West. The East Pacific goes from the west coast of Mexico and the U.S. out to a line. In this area there is an alphabetic name list, similar to the Atlantic. A cyclone that passes from the East to the Central Pacific is still called a hurricane, and retains the same name. However, cyclones that form in this basin follow a different name list that does not go in any alphabetical order, because of the relative lack of cyclones in the region. Therefore, when Hurricane John entered the Central Pacific and switched basins, it was still was Hurricane John, and was still gaining intensity. By the time John was at its closest approach to the Hawaiian Islands it was a Category 5 hurricane, but, fortunately for the islands, John was 345 miles away. Despite this, John's outer rain bands caused a significant amount of damage. However, the system couldn't turn towards Hawaii because the usual high pressure system over the islands was blocking all cyclones from turning north. Later that same day of its closest approach, on August 22, John reached its peak intensity of 175 mph winds and a pressure of 929 millibars. However, the pressure was probably much lower than this, because the pressure had last been taken when John was not at its peak intensity. Now, John took a northwest turn, and steadily weakened to Category 1 hurricane strength. Over the next days, however, John regained some of its intensity and reached Category 4 status. It was at a secondary peak with 135 mph winds when it crossed the International Date Line and entered the West Pacific. The West Pacific had another name list still, with another major difference. The scale of intensity was also different. Instead of going tropical depression, tropical storm, hurricane, it went tropical depression, tropical storm, typhoon, super typhoon. Therefore, when John crossed the date line on August 27, it was still John, but a typhoon, not a hurricane. However, upon crossing the date line, John had nearly stopped moving and by September 1, John was a stationary tropical storm. John went through a slow loop, first east, and then back west, still a tropical storm. By the time this loop ended, a trough was moving in, forcing John quickly to the northeast. On September 8, John had crossed back into the Central Pacific and was still a tropical storm. John attained hurricane strength again and reached 90 mph winds before finally becoming extratropical on September 10. John hadn't caused much damage and the only damage was to Hawaii and to Johnston Atoll in the north-central Pacific. John's ACE was not accurate because it had crossed three basins, but it is estimated to be in the 50's.



Hurricane John at primary peak intensity as a strong Category 5 hurricane.



Track of John.

Some of the above cyclones are the most notable in the world, and many have some of the most erratic tracks of any cyclones ever.

The information on this post can be credited mostly to wikipedia, which, in turn, got its information from the National Hurricane Center that keeps records of Atlantic tropical cyclones dating back to 1850. Any cyclones before that time cannot be on the above list, due to lack of information.

Wednesday, February 10, 2010

Notable Hurricanes: Longevity; Part 2

This is the second part of a three part post. For the first part, see here.

Most of the hurricanes so far on the longevity countdown have made a loop. This usually is a result of a blocking high pressure, which steers the system, but doesn't always weaken it. However, blocking low pressure systems steer storms and weaken them or are too weak to affect them, and are therefore are less common.

#7 (tied) Hurricane Inez (1966) September 21-October 11 20.75 days (from formation to last extratropical transition)

The tropical wave that was to become Hurricane Inez emerged off Africa on September 18. On September 21, the system became a tropical depression, and on the 24th, it became Tropical Storm Inez. Inez strengthened rapidly into a hurricane and slammed into the most eastern islands in the Caribbean. Strengthening still further, Inez reached its peak intensity of 150 mph winds and a pressure of 929 millibars before making landfall in the Dominican Republic on September 29. The system weakened and emerged over water shortly before striking Cuba. After traveling over Cuba, Inez was barely holding on to minimal hurricane strength and became a tropical storm upon entering the southwestern Bahamas on October 2. Over the Bahamas, Inez became a hurricane again and was forced to the southwest by a subtropical ridge on the 4th. Soon after, Inez crossed the Florida Keys, and began to intensify once again. By this time, Inez skimmed the northern Yucatan Peninsula on October 7, it was a Category 3 and still getting stronger. Then, on October 10, after briefly reaching Category 4 status, the system made landfall in Mexico as a Category 3. Inez weakened rapidly, and dissipated on the 11th of October. Over 1,000 people died form Inez and $222 million in damage resulted ($1.5 billion adjusted for inflation since 1966). Because of this, the name Inez was retired from the circulating name list. Inez had an ACE of 54.58.



Track of Inez.

#7 (tied) Hurricane Carrie (1957) September 2-24 20.75 days (from formation to last extratropical transition)

The tropical depression that eventually became Carrie formed near Africa on September 2. Moving slowly westward, the system gained intensity, first becoming a tropical storm, then a hurricane, and then a major hurricane. Soon after, on September 8, Carrie reached its peak intensity of 155 mph winds and a pressure of 945 millibars. The system turned north weakening to a Category 1, and then turned west strengthening to a Category 4. On September 16, the system passed Bermuda at distance of about 100 miles, not causing any damage. Soon after, the system turned east as a minimal hurricane and became extratropical on the 23rd. The only impact of Carrie was a sunken ship off the Azores on September 21. 86 were aboard the ship and only 6 survived, despite a rescue effort. Carrie had an ACE of 62.59.



Track of Carrie.

#5 Hurricane Nadine (2012) September 11-October 4 21.75 days (spent as a tropical cyclone)
A post on this hurricane is located elsewhere on this blog.

#4 Hurricane Kyle (2002) September 20-October 12 22 days (from formation to final extratropical transition)

The low pressure system that later became Kyle was originally part of a front that moved over the Atlantic Ocean on October 15. A low formed over the dissipating front on September 18 and gained some characteristics of a tropical cyclone. On September 20, the system was midway between tropical and non-tropical features, and was therefore classified as a subtropical depression. The depression, moving slowly northward in the Central Atlantic, was hindered in movement by a blocking trough to its north and another low to its south. Despite all these factors, the depression strengthened into Subtropical Storm Kyle on September 21. Then, getting close to the blocking force in the north, Kyle was forced to make a loop, first going east, then south, and then west. While making this loop, Kyle started to exhibit more tropical features, and Kyle was reclassified as Tropical Storm Kyle on September 22. The system continued generally west over the next few days, intensifying as it went, and soon becoming a hurricane. Soon after, Kyle reached its peak intensity of 85 mph winds and a pressure of 980 millibars on September 26. The next day, however, Kyle started to weaken due to wind sheer. Before long, Kyle was a tropical storm again. Kyle later weakened into a depression, although only briefly, and it regained tropical storm status on October 1. Kyle did not move for the next few days, but soon continued its trek to the west. On October 7, Kyle was stalled again, and was forced southwest, again becoming a depression. It followed a slow arc, curving southwest, then west, and finally northwest due to an approaching front, becoming a tropical storm yet again before striking the coast of South Carolina, and then North Carolina on October 11. The system emerged over water and became extratropical on October 12. The system caused minor damage and one fatality. Kyle had an ACE of 14.44.



Kyle at minimal hurricane status.



The odd-looking track of Kyle.

For the last part of this post, see here.

Tuesday, February 2, 2010

Notable Hurricanes: Longevity; Hurricane ACE

Among the many Atlantic hurricanes that have formed in the past 150 years or so, most are notable because of their intensity or for their impact. In this post, starting with the 12th, the top twelve hurricanes in terms of longevity will be listed. But first, a little word about ACE, or Accumulated Cyclone Energy, which combines the intensity and longevity of a cyclone into one statistic.

ACE is a formula that adds together the intensities of a cyclone every six-hour period throughout the total history of the cyclone. For each period of six hours, the maximum sustained wind speed is estimated, squared, and divided by one thousand. This is the ACE figure for that six hour interval. The sum of all the intervals determines the ACE of a storm. However, there are a few extra rules associated with ACE. For one, the ACE is only taken when a cyclone is tropical (tropical meaning that the cyclone has hurricane-like characteristics). When a cyclone is extratropical or subtropical (midway between extratropical and tropical, but there is no given line between these categories) ACE is not taken. Also, when the tropical cyclone has winds less than 40 mph, i.e. a tropical depression, the ACE figure drops to zero. Most long-lived hurricanes have high ACE's, but this isn't always true. Since the statistic applies to longevity and intensity, an intense long-lived storm will have the highest ACE.

#12: Hurricane Ivan (2004) September 2-24 18.75 days (as a tropical cyclone)

Hurricane Ivan was a Cape Verde hurricane that became a tropical depression on September 2, 2004. The system became a tropical depression and then a hurricane quickly. Ivan also set the record for most southerly Atlantic major hurricane when it became a Category 3 on September 5 at 10.6 N latitude. It continued west-north-westward and became a Category 5 hurricane on September 9. Hurricane Ivan, continuing through the Caribbean and fluctuating between Category 4 and 5 once or twice, devastated the Cayman Islands, Cuba, Jamaica, and Grenada. Billions of dollars in damage and over 50 deaths occurred during this rampage, despite the fact that Hurricane Ivan did not make landfall in any of these places, but passed by them. On September 13, the storm entered the Gulf of Mexico and approached the coast of Alabama. The storm made landfall in the United States at Category 3 intensity on September 16 and became a remnant low on the 18th. Causing 54 deaths in the region, the low continued northeastward The low was disregarded while it slowly turned to the south, and emerged back into the Gulf from Florida. On September 22, the system became a tropical depression again, and the National Hurricane Center, recognizing the depression as the remnants of Ivan, called it Tropical Depression Ivan. It strengthened into a tropical storm shortly before making its second landfall in Louisiana and causing minor damage. The system dissipated on the 24th. Although the storm was only the 10th most long lived, it had the second highest ACE of any storm in the Atlantic: 70.38.



Ivan as a Category 5 in the Caribbean.



Track of Ivan. The circles denote the storm as tropical with blue the weakest and red the strongest. The triangles show the remnants of Ivan when it was not tropical.

#11 Hurricane Nine (1893) September 25-October 15 19.25 days (as a tropical cyclone)

Due to the lack of radar and other tracking instruments in 1893, it is not known whether the storm actually originated on the 25th of September. However, it is known that a Cape Verde hurricane formed in late September of this year, and very slowly made its way westward. The system quickly strengthened into a hurricane and then a major hurricane. Since names weren't introduced until 1950, the system was simply called Hurricane Nine. It took a turn north and then once again followed a westerly path as a Category 3 hurricane through the Bahamas. On October 13, the hurricane struck Myrtle Beach, South Carolina, and accelerated northward, dissipating on the 15th. The system caused 28 deaths and had an ACE of 63.5.



Track of Nine.

#10 Hurricane Four (1926) September 2-24 19.5 days (as a tropical cyclone)

A tropical storm formed on September 2 and moved northwest very slowly for the next two weeks. During this time, the system became a Category 4 hurricane, and its movement slowed even more. Then, it turned northeast and made a tight loop. It became extratropical shortly after and dissipated on the 24th. Although the system caused no damage to land, it had the third highest ACE of any Atlantic hurricane, at 67.59.



Track of Four.

#9 Hurricane Alberto (2000) August 3-23 19.75 days (as a tropical cyclone)

The tropical depression that was to become Hurricane Alberto formed very close to the African coast on August 3. The system moved generally westward, and strengthened into a tropical storm and then a hurricane. However, on August 9, the system experienced wind sheer, and weakened to a tropical storm, and turned northeast. After moving out of the influence of the wind sheer, the system strengthened to a hurricane and then reached its peak intensity of 125 mph and 950 millibars on August 12. Then, due to the influence of a subtropical ridge, Alberto began a large loop in the north-Central Atlantic, and once again weakened to a tropical storm on the 14th of August. Then, it began to turn south, and then west again, before becoming a hurricane for the third time on August 18. It became a Category 2 hurricane again, and reached a secondary peak intensity of 105 mph winds Then, Alberto finally turned northward once again, starting slow, and then accelerating, and finally becoming extratropical late on August 23. Although no damage resulted from Alberto, it was notable for its odd track, and it had an ACE of 36.9, which is fairly low for Alberto's longevity, but since it never went beyond Category 3, it wasn't intense for much of its lifetime.



Hurricane Alberto at its primary peak intensity as a major hurricane.



Track of Alberto.

The top ten is continued on Notable Hurricanes: Longevity; Part 2.