# High School Math

MISSION STATEMENT: To encourage and promote a greater use of the internet and computer technology in the math classroom. For educators, students, parents and homeschoolers.

## Saturday, January 29, 2005

### Archimedes

DID YOU KNOW THAT :

Archimedes is considered one of the three greatest mathematicians of all. His works and inventions brought him fame that lasts to this very day. He was one of the last great Greek mathematicians. He was born in 287 B.C., in Syracuse, a Greek seaport colony in Sicily.

One day while wondering if a gold crown was pure gold "the wise one" entered his bathtub and recognized that the amount of water that overflowed the tub was proportional the amount of his body that was submerged. This observation is now known as Archimedes' Principle and gave him the means to solve the problem. He was so excited that he ran naked through the streets of Syracuse shouting "Eureka! eureka!" (I have found it!). He also had many other inventions including the Archimedes' watering screw and a miniature planetarium.

His greatest invention was integral calculus. To determine the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes broke the sections into an infinite number of rectangles and added the areas together. This is known as integration.

While Archimedes was drawing figures in the dust, a Roman soldier stepped on them. Archimedes responded, "Don't disturb my circles!" The soldier was so enraged that he pulled out his sword and slew the great geometer. When Archimedes was buried, they placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.

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### INFINITY.....

DID YOU KNOW THAT :

Infinity is from a Latin word which literally means that which is unlimited or unbounded. Originally it was applied to things that were unmeasureably large. It has come to mean the largest number imaginable. The origin of the theory of limits has also led to the need for a word that expressed the idea of things growing smaller and smaller without bound.

The symbol we now use for infinity, was first used by John Wallis (1616-1703) in 1655. Why he used it seems lost to history. The Late Romans used a symbol like two hooked together zeros, 00, for the number 1000. Or perhaps, it is a variant of the lowercase symbol for Omega, the last letter in the Greek alphabet, to symbolize the "final number" in a sense.

There was a young fellow from Trinity
Who took the square root of infinity
But the number of digits,
Gave him the fidgets;
He dropped Math and took up Divinity.
-- George Gamow

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## Friday, January 28, 2005

### Pi = 3.14159

DID YOU KNOW THAT :

That the ratio of the circumference to the diameter of a circle is constant (namely, pi) has been recognized for as long as we have written records.

A ratio of 3:1 appears in the following biblical verse: And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23; II Chronicles 4, 2.)

The ancient Babylonians generally calculated the area of a circle by taking 3 times the square of its radius (=3), but one Old Babylonian tablet (from ca. 1900-1680 BCE) indicates a value of 3.125 for pi.

The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE), one of the most brilliant mathematicians of the ancient world. Archimedes worked out that 223/71 < pi < 22/7. Archimedes's results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed. Beginning with a hexagon, he worked all the way up to a ploygon with 96 sides!

European mathematicians in the early modern period developed new arithmetical formulae to approximate the value of pi, such as that of James Gregory (1638-1675), which was taken up by Leibniz as :
pi/4 = 1 - 1/3 + 1/5 - 1/7 . . . .

The symbol for pi was introduced by the English mathematician William Jones in 1706. This symbol was adopted by Euler in 1737 and became the standard symbol for pi.

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### Islamic Calendar

DID YOU KNOW THAT :

The Islamic calendar has its starting point at the date of the flight of Mohammed from Mecca to Medina, known as the Hejira. The most widely accepted date for this event in the Gregorian calendar is sunset at July 16, 622 AD. This dating system is used in the Muslim world (except Turkey, which uses the Gregorian calendar) and based on a year of 12 months, each month beginning approximately at the time of the New Moon.

The Islamic calendar is tied to the lunar phase cycle, with each month alternatively having either 29 or 30 days. The calendar drifts by relative to the sun and uses an 11-year leap year cycle. The year has either 354 or 355 days. No months are ever added, so that the named months do not remain in the same seasons but move through all the seasons of the year (of about 365.25 days) every 32.5 solar years. The names of the Islamic months are Muharram, Safar, Rabia I, Rabia II, Jumada I, Jumada II, Rajab, Sha'ban, Ramadan, Shawwal, Dhu al-Qada, and Dhu al-Hijah.

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### Roman Calendar

DID YOU KNOW THAT :

The Roman calendar originally started the year with the vernal equinox and consisted of 10 months (Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September, October, November, and December) having a total of 304 days. The numbers still embedded in the last four months - September, October, November, and December, contain the Latin roots for the numerals seven, eight, nine, and ten, but now fall on the ninth, tenth, eleventh and twelfth months of the year.

The 304 days were followed by an unnamed, unnumbered period in winter. The second Roman king Numa Pompilius (715-673 BC) introduced February and January (in that order) between December and March, increasing the length of the year to 354 or 355 days. Then in 450 BC, February was moved to its current position. The Roman calendar was eventually changed to the more rational Julian calendar in 46 BC.

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### Julian Calendar

DID YOU KNOW THAT :

In the year 46 BC, Julius Caesar reformed the Roman calendar to a more manageable form. At this time, Julius changed the number of days in the months to achieve a 365 day year. In order to catch up with the seasons, Julius Caesar also added 90 days to the year 46 BC between November and February.

The Julian calendar consisted of cycles of three 365-day years followed by a 366-day leap year. Around 9 BC, it was found that the priests in charge of computing the calendar had been adding leap years every three years instead of the four decreed by Caesar. As a result of this error, no more leap years were added until 8 AD.

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### Gregorian Calendar

DID YOU KNOW THAT :

The calendar currently in worldwide use is the Gregorian Calendar. It has a year of length 365.2425 days. The Gregorian calendar is a modification of the Julian calendar in which leap years are omitted in years divisible by 100 but not divisible by 400. By this rule, the year 1900 was not a leap year (1900 is divisible by 100 and not divisible by 400), but the year 2000 was a leap year (2000 is divisible by 400). This gives an exact number of weeks per 400-year cycle.

The Julian calendar was replaced by the Gregorian Calendar in 1582. Pope Gregory XIII decreed that the day after October 4, 1582 would be October 15, 1582, the Catholic countries of France, Spain, Portugal, and Italy complied. Various Catholic German countries (Germany was not yet unified), Belgium, the Netherlands, and Switzerland followed suit within a year or two, and Hungary followed in 1587.

Because of the Pope's decree, the reform of the Julian calendar came to be known as the Gregorian calendar. However, the rest of Europe did not follow suit for more than a century. The Protestant German countries adopted the Gregorian reform in 1700. By this time, the calendar trailed the seasons by 11 days. England (and the American colonies) finally followed suit in 1752, and Wednesday, September 2, 1752 was immediately followed by Thursday, September 14, 1752.

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## Wednesday, January 19, 2005

### Mixed Up

A) I had a jumbled mess in my dark closet.
There were 6 shoes of 3 types.
There were 24 socks, 12 brown & 12 black.
How many shoes and socks did I have to take out
with me to be sure I had a matching pair of shoes
and a matching pair of socks?

B) Three kinds of apples are mixed up in a bag.
How many apples must you take out to be sure
that you have at least 2 apples of one kind?
At least 3 apples of one kind?

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### What's Wrong?

`Let                    a = b   Multiply by a        a.a = a.bSubtract b.b   a.a - b.b = a.b - b.bFactor        (a+b)(a-b) = b(a-b)Divide by (a-b)      a+b = b      Now since a=b         2b = bWhich means that       2 = 1   `

What is wrong with this solution?

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## Monday, January 17, 2005

### Two Children

(A) I have 2 children. They are not both boys.
What is the probability that both children are girls?

(B) An artist has 2 children. The older one is a boy.
What is the probability that both children are boys?

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## Saturday, January 15, 2005

### Think !

a) What arithmetic symbol can we place
between 5 and 6 to make a number that
is greater than 5 but smaller than 6?

b) A book costs \$20 plus half its price.
How much does it cost?

c) How far can a horse walk with
a 50 foot rope tied around its neck?

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## Friday, January 14, 2005

### Camel & Bananas

A camel used to transport bananas must travel 1000 miles
across a desert to reach customers living in an exotic city. At any
given time, the camel can carry up to 1000 bananas and must eat
one banana for every mile it walks.
Challenge: Assuming an initial stock of 3000 bananas, what is the
maximum number of bananas that the camel can transport across
the desert and into the eager hands of waiting customers who live
in the exotic city?

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## Monday, January 03, 2005

### Euler

DID YOU KNOW THAT :

Leonhard Euler (1707-1783) was arguably the greatest mathematician of the eighteenth century and one of the most prolific of all time; his publication list has 886 papers and books. Euler's complete works fill about 90 volumes. Remarkably, much of this output dates from the the last two decades of his life, when he was totally blind. Euler's study of the bridges of Königsberg can be seen as the beginning of combinatorial topology.

Though born and educated in Basel, Switzerland, Euler spent most of his career in St. Petersburg and Berlin. He joined the St. Petersburg Academy of Sciences in 1727. In 1741 he went to Berlin at the invitation of Frederick the Great. In 1766 he returned to St. Petersburg, where he remained until his death. Euler's powers of memory and concentration were legendary. He was not troubled by interruptions or distractions; in fact, he did much of his work with his young children playing at his feet. He was able to do prodigious calculations in his head, a necessity after he went blind.

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### Newton

DID YOU KNOW THAT :

Certainly one of the greatest scientists who ever lived, Isaac Newton (1642-1727) had a profound impact on astronomy, physics, and mathematics. Born prematurely and after his father's death, Newton had a difficult childhood. His mother remarried when he was just three, and he was then sent to live with his grandparents. After his stepfather died, his mother brought him home to Woolsthorpe in Lincolnshire, where she wanted him to become a farmer. An uncle recognized his scholarly talents, however, and he eventually made it to Trinity College in Cambridge.

Many of his great ideas came in 1665-66, when he spent time back at Woolsthorpe while Cambridge was closed because of the plague. Among his many achievements were the invention of the reflecting telescope, the invention of a branch of mathematics known as calculus, a critical tool throughout science; the elucidation of the three laws of motion; and the development of the law of universal gravitation.

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### Gauss

DID YOU KNOW THAT :

Karl Friedrich Gauss was born in Brunswick, Germany, to a poor, unknown family. His father worked at various jobs as a stone cutter, gardener, canal worker, and foreman for a masonry firm. He was to follow in his father's foot steps, but eventually his father conceded to allow his gifted son to receive an education.Born with a natural knack for math and reasoning, he was a child prodigy. At his first arithmetic class, he instantly solved the sum of numbers from one to one-hundred. Gauss knew as much as his teachers at age ten, so there was nothing more they could teach him. Influential people were aroused by his genius. He went on to the University of Gottingen for the three years between 1795 and 1798.

Gauss made many mathematical discoveries, and finally published them in 1801 in the book that achieved him the recognition as a mathematical genius when he was only 24 years old: Disquisitiones Arithmeticae.Gauss's achievements were extraordinary in theoretical astronomy rather than mathematics. He created a theory of planetary and cometary orbit that included the gravitational affect of the planets as well as of the sun. With this he calculated the orbits of planetoids between Mars and Jupiter. Neptune was the first planet discovered through the use of this method. He studied other subjects as well: physics, mechanics, and theoretical astronomy, and made many discoveries in these areas.

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