Math Is Fun Forum

Hi guys,

If you plot the graphs for x^y and y^x for any particular y you can see some patterns.

0^x=x^0 has no solutions because x^0 is always 1 and 0^x is 0, x>0.

x=1 is a unique solution of 1^x=x^1 (you must set x^1=1).

For natural numbers higher than one we can find there are two solutions for odd numbers 3^x=x^3 etc. and three solutions, including a negative one, for even numbers 2^x=x^2 etc. The negative solution for x is able to produce a positive result for both sides in the case of even numbers (e.g. x^2 can be positive even if x is negative), but in the case of odd numbers a negative x will produce a negative RHS and positive LHS (e.g. x^3 is negative for negative x, but 3^x is always positive).

More generally, for positive y-values:
There are two or three solutions for x if y>1, depending on whether x^y can be positive if x is negative.
There are one or two solutions for x if 0<y≤1, again depending on whether x^y can be positive if x is negative.

For negative y-values the graph of y^x is only continuous in the complex plane. It looks like if x is an integer we get real values. Also every integer y will have solution x=y.

This is about as far as I would get with a plotter, with nothing to tie the different results together. If we can algebraically derive some of the non-obvious solutions that would be an improvement.

I'm going to list some solutions you can compute manually.
(1,1)
(2,4)
(2,-0.767...)
(3,2.478...)
(4,-0.767...)
(5,1.765...)
(6,-0.79...)
(6,1.624...)

I lack the knowledge to explain what is going on here in general terms. It apparently has something to do with the "product log function" and complex numbers.

There are 9*10=90 possible combinations. There are 3*4=12 possible prime number combinations: (2,3), (2,5), (2,7), (3,2), (3,5), (3,7), (5,2), (5,3), (5,7), (7,2), (7,3), (7,5). The probability is 12/90=2/15.

Hi (:

Agreed with bobbym, unless a and b are both zero. That just leaves y = (4-3x)/5

Hey;

1. I agree with bobbym

I verified the answer as well and like chen.aavaz only missed it because of the size of a. n = 36. Well done (:

Oh, I see. I will keep looking

Hi! I found three answers so far:

1. Set k=8, n=8, a=1234568, b=9876544,
c=123456809876544

2. Set k=10, n=12, a=8264462810, b=82644628100,
c=8264462810082644628100

3. Set k=40, n=10, a=5948840, b=237953600,
c=59488400237953600

Further answer:

4. Set k=188, n=6, a=28, b=5264,
c=28005264

#39. Michelangelo

Michelangelo di Lodovico Buonarroti Simoni (March 1475 – 18 February 1564), was an Italian sculptor, painter, architect, poet, and engineer of the High Renaissance who exerted an unparalleled influence on the development of Western art. Considered to be the greatest living artist during his lifetime, he has since also been described as one of the greatest artists of all time. Despite making few forays beyond the arts, his versatility in the disciplines he took up was of such a high order that he is often considered a contender for the title of the archetypal Renaissance man, along with contemporary rival and fellow Florentine Medici client, Leonardo da Vinci.

A number of Michelangelo's works in painting, sculpture, and architecture rank among the most famous in existence. His output in every field of interest was prodigious; given the sheer volume of surviving correspondence, sketches, and reminiscences taken into account, he is the best-documented artist of the 16th century.

Two of his best-known works, the Pietà and David, were sculpted before the age of thirty. Despite his low opinion of painting, Michelangelo also created two of the most influential frescoes in the history of Western art: the scenes from Genesis on the ceiling of the Sistine Chapel in Rome, and The Last Judgment on its altar wall. As an architect, Michelangelo pioneered the Mannerist style at the Laurentian Library. At the age of 74, he succeeded Antonio da Sangallo the Younger as the architect of St. Peter's Basilica. Michelangelo transformed the plan,with the western end being finished to his design, and the dome being completed after his death, with some modification.

In a demonstration of Michelangelo's unique standing, he was the first Western artist whose biography was published while he was alive. Two biographies were published of him during his lifetime; one of them, by Giorgio Vasari, proposed that he was the pinnacle of all artistic achievement since the beginning of the Renaissance, a viewpoint that continued to have currency in art history for centuries.

In his lifetime he was also often called Il Divino ("the divine one"). One of the qualities most admired by his contemporaries was his terribilità, a sense of awe-inspiring grandeur, and it was the attempts of subsequent artists to imitate Michelangelo's impassioned and highly personal style that resulted in Mannerism, the next major movement in Western art after the High Renaissance.

Michelangelo, with Leonardo da Vinci and Raphael, is one of the three giants of the Florentine High Renaissance. Although their names are often cited together, Michelangelo was younger than Leonardo by 23 years, and older than Raphael by eight. Because of his reclusive nature, he had little to do with either artist and outlived both of them by more than forty years. Michelangelo took few sculpture students. He employed Francesco Granacci, who was his fellow pupil at the Medici Academy, and became one of several assistants on the Sistine Chapel ceiling. Michelangelo appears to have used assistants mainly for the more manual tasks of preparing surfaces and grinding colours. Despite this, his works were to have a great influence on painters, sculptors and architects for many generations to come.

While Michelangelo's David is the most famous male nude of all time and destined to be reproduced in order to grace cities around the world, some of his other works have had perhaps even greater impact on the course of art. The twisting forms and tensions of the Victory, the Bruges Madonna and the Medici Madonna make them the heralds of the Mannerist art. The unfinished giants for the tomb of Pope Julius II had profound effect on late-19th- and 20th-century sculptors such as Rodin and Henry Moore.

Michelangelo's foyer of the Laurentian Library was one of the earliest buildings to utilise Classical forms in a plastic and expressive manner. This dynamic quality was later to find its major expression in Michelangelo's centrally planned St Peter's, with its giant order, its rippling cornice and its upward-launching pointed dome. The dome of St Peter's was to influence the building of churches for many centuries, including Sant'Andrea della Valle in Rome and St Paul's Cathedral, London, as well as the civic domes of many public buildings and the state capitals across America.

Artists who were directly influenced by Michelangelo include Raphael, who imitated Michelangelo's prophets in two of his works, including his depiction of the great master in the School of Athens. Other artists, such as Pontormo, drew on the writhing forms of the Last Judgement and the frescoes of the Capella Paolina.

The Sistine Chapel ceiling was a work of unprecedented grandeur, both for its architectonic forms, to be imitated by many Baroque ceiling painters, and also for the wealth of its inventiveness in the study of figures. Vasari wrote: "The work has proved a veritable beacon to our art, of inestimable benefit to all painters, restoring light to a world that for centuries had been plunged into darkness. Indeed, painters no longer need to seek for new inventions, novel attitudes, clothed figures, fresh ways of expression, different arrangements, or sublime subjects, for this work contains every perfection possible under those headings."

#37. Napoleon

Napoléon Bonaparte (born Napoleone di Buonaparte; 15 August 1769 – 5 May 1821) was a French military and political leader who rose to prominence during the French Revolution and led several successful campaigns during the Revolutionary Wars. As Napoleon I, he was Emperor of the French from 1804 until 1814, and again in 1815. Napoleon dominated European and global affairs for more than a decade while leading France against a series of coalitions in the Napoleonic Wars. He won most of these wars and the vast majority of his battles, building a large empire that ruled over continental Europe before its final collapse in 1815. Often considered one of the greatest commanders in history, his wars and campaigns are studied at military schools worldwide. He also remains one of the most celebrated and controversial political figures in Western history. Napoleon had an extensive and powerful impact on the modern world, bringing liberal reforms to the numerous territories that he conquered and controlled, especially the Low Countries, Switzerland, and large parts of modern Italy and Germany. He implemented fundamental liberal policies in France and throughout Western Europe. His lasting legal achievement, the Napoleonic Code, has been adopted in various forms by a quarter of the world's legal systems, from Japan to Quebec.

Napoleon was born in Corsica to a relatively modest family of noble Tuscan ancestry. Napoleon supported the French Revolution from the outset in 1789 while serving in the French army, and he tried to spread its ideals to Corsica but was banished from the island in 1793. Two years later, he saved the French government from collapse by firing on the Parisian mobs with cannons. The Directory rewarded Napoleon by giving him command of the Army of Italy at age 26, when he began his first military campaign against the Austrians and their Italian allies, scoring a series of decisive victories that made him famous all across Europe. He followed the defeat of the Allies in Europe by commanding a military expedition to Egypt in 1798, invading and occupying the Ottoman province after defeating the Mamelukes and launching modern Egyptology through the discoveries made by his army.

After returning from Egypt, Napoleon engineered a coup in November 1799 and became First Consul of the Republic. Another victory over the Austrians at the Battle of Marengo in 1800 secured his political power. With the Concordat of 1801, Napoleon restored the religious privileges of the Catholic Church while keeping the lands seized by the Revolution. The state continued to nominate the bishops and to control church finances. He extended his political control over France until the Senate declared him Emperor of the French in 1804, launching the French Empire. Intractable differences with the British meant that the French were facing a Third Coalition by 1805. Napoleon shattered this coalition with decisive victories in the Ulm Campaign and a historic triumph at the Battle of Austerlitz, which led to the elimination of the Holy Roman Empire. In October 1805, however, a Franco-Spanish fleet was destroyed at the Battle of Trafalgar, allowing Britain to impose a naval blockade of the French coasts. In retaliation, Napoleon established the Continental System in 1806 to cut off continental trade with Britain. The Fourth Coalition took up arms against him the same year because Prussia became worried about growing French influence on the continent. Napoleon knocked out Prussia at the battles of Jena and Auerstedt, then turned his attention towards the Russians and annihilated them in June 1807 at Friedland, which forced the Russians to accept the Treaties of Tilsit.

Hoping to extend the Continental System, Napoleon invaded Iberia and declared his brother Joseph the King of Spain in 1808. The Spanish and the Portuguese revolted with British support. The Peninsular War lasted six years, noted for its brutal guerrilla warfare, and culminated in an Allied victory. Fighting also erupted in Central Europe, as the Austrians launched another attack against the French in 1809. Napoleon defeated them at the Battle of Wagram, dissolving the Fifth Coalition formed against France. By 1811, Napoleon ruled over 70 million people across an empire that had domination in Europe, which had not witnessed this level of political consolidation since the days of the Roman Empire. He maintained his strategic status through a series of alliances and family appointments. He created a new aristocracy in France while allowing the return of nobles who had been forced into exile by the Revolution.

Tensions over rising Polish nationalism and the economic effects of the Continental System led to renewed confrontation with Russia. To enforce his blockade, Napoleon launched an invasion of Russia in the summer of 1812. The resulting campaign witnessed the catastrophic collapse of the Grand Army, forcing the French to retreat, as well as leading to the widespread destruction of Russian lands and cities. In 1813, Prussia and Austria joined Russian forces in a Sixth Coalition against France. A chaotic military campaign in Central Europe eventually culminated in a large Allied army defeating Napoleon at the Battle of Leipzig in October. The next year, the Allies invaded France and captured Paris, forcing Napoleon to abdicate in April 1814. He was exiled to the island of Elba. The Bourbons were restored to power and the French lost most of the territories that they had conquered since the Revolution. However, Napoleon escaped from Elba in February 1815 and took control of the government once again. The Allies responded by forming a Seventh Coalition, which ultimately defeated Napoleon at the Battle of Waterloo in June. The Royal Navy then thwarted his planned escape to the United States in July, so he surrendered to the British after running out of other options. The British exiled him to the remote island of Saint Helena in the South Atlantic. His death in 1821 at the age of 51 was received with shock and grief throughout Europe. In 1840, a million people witnessed his remains returning to Paris, where they still reside at Les Invalides.

Agreed. Or... it refers to displacement and the minimum is 60 metres while the maximum is the highest speed the flea can achieve while changing direction.

Hi salem_ohio

You are right about the four cases and you understand the method But I agree with thickhead that you happened to put the wrong fractions of station B's probabilities: The denominator is 10. When you fix that, you get:
1) 18/25
2) 1/50
3) 2/25
4) 9/50

And also:
A) 4/13
B) 9/13

What you calculated, which happened to result in a 1/3:2/3 split, was for:
-Station A is right 80%, but
-Station B is right 88.8888...%