# Powers of Pascal’s Triangle

Uncategorized

Pascal’s Triangle

One day a man’s boss needed help gambling.  The boss asked his employee how he could win in gambling.  This employee made Pascal’s Triangle.

Blaise Pascal (1623-1662) studied mathematics, philosophy, and physics.  His father collected taxes in Rouen, France.  Pascal always loved creating calculators.  After 3 years, Pascal made 20 of the first calculators.  He called these calculators Pascalines.

Nevertheless, of all of Pascal’s discoveries and inventions, humanity most commonly remembers him for making Pascal’s Triangle.

Pascal’s triangle consists of a one on the first line, then two ones on the second line, then a one, a two, and a one on the third line.  Each line is formed of numbers which are the sum of the two numbers above it, as shown below.

As you can see, each row is formed of numbers which are the sum of the two numbers diagonally on top of the number.  Pascal’s triangle continues forever, always containing increasingly large numbers.  The border of Pascal’s triangle is always one.

Powers of 11

1 = 1                                                                  110
11 = 11                                                              111
11 * 11 = 121                                                    112
11 * 11 * 11 = 1331                                          113
11 * 11 * 11 * 11 = 14641                               114
11 * 11 * 11 * 11 * 11 = 161051                     115
11 * 11 * 11 * 11 * 11 * 11 = 1771561           116

The zeroth row of Pascal’s triangle is one.  11is one.
The first row of Pascal’s triangle is one one, or 11.  11is one.
The second row of Pascal’s triangle is one two one, or 121.  112 is 121.

This pattern progresses until we get to the fifth row of Pascal’s Triangle.  The fifth row of Pascal’s triangle is one five ten ten five one, or 15,101,051.  115 is 161,051.  So does that mean that Pascal’s triangle only tells us the powers of 11 up to 114?  Of course not! One just has to know how to read Pascal’s Triangle.

Whenever you have a two digit number produced by adding together the two numbers above it in Pascal’s triangle, you keep the unit digit and take all the digits after it and add it to the next number’s one’s digit.  For example, in the fifth row of Pascal’s triangle, you have:

1   5   10   10   5   1
Take the one in the first ten and carry it over to the next ten, you are left with:
1 5 11 0 5 1
then you take the tens digit one in 11 and carry it over to the leftmost 5. You get
1 6 1 0 5 1
11is 161,051.

This pattern works for every power of 11 greater than 4.  Lets try 116. The sixth row of Pascal’s Triangle is 1 6 15 20 15 6 1.

1 6 15 20 15 6 1
Take the one from the leftmost 15 and carry it over to the twenty.  You get:
1 6 15 21 5 6 1
Then take the 2 from the 21 and bring it over to the 15. You should now have:
1 6 17 1 5 6 1
Finally, take the 1 from the 17 and move it over to the 6.  You end up with:
1 7 7 1 5 6 1
11is 1771561

Now you know how to read the powers of 11 from Pascal’s Triangle!

Prime Numbers

In pascal’s triangle, the uppermost row is called the 0th row.  The second highest row is called the first row.  The leftmost number in each row is called the 0th number.  The second number to the left is called the first number.

Now examine the first number of each row.  If it is a prime number, then try dividing that number into each of the other numbers in that row except for the last number and the 0th number.  You should get a whole number for each answer.

Whenever the first number in a row is prime, all the preceding numbers except for the last one are multiples of that first number.

Powers of 2

1 = 1                                                       20
2 = 2                                                      21
2 * 2 = 4                                               22
2 * 2 * 2 = 8                                        23
2 * 2 * 2 * 2 = 16                               24
2 * 2 * 2 * 2 * 2 = 32                        25
2 * 2 * 2 * 2 * 2 * 2 = 64                 26

The zeroth row of Pascal’s triangle consists of the number one.   2is 1.
The first row of Pascal’s triangle consists of 2 ones.  1 + 1 = 2.  2is 2
The second row of Pascal’s triangle consists of 2 ones and a two.  1 + 2 + 1 = 4. 2is 4.
The third row of Pascal’s triangle consists of 2 ones and 2 threes.  1 + 3 + 3 + 1 = 8. 2is 8.

This pattern continues in Pascal’s triangle forever.

Hockey Sticks

Pick a number, any number.  That is, a number on Pascal’s triangle.  Then choose a diagonal direction going downwards.  Continue however many rows down you want.  On the last number, change the direction of the diagonal.  That number will be the sum of all the numbers in the first diagonal.

Diagonals

Triangular numbers are the number of units in an equilateral triangle.  In an equilateral triangle with a side length of 3, there are 6 units in the triangle.

The third diagonal going down on Pascal’s triangle contains all triangular numbers.

Counting numbers are positive integers such as 1, 2, 3, 4, &c.  The second row of Pascal’s triangle has every counting number.

Tetrahedral numbers are “numbers that can be represented that can be represented by a regular geometric arrangement of equally spaced points. Tetrahedral numbers correspond to placing discrete points in the configuration of a tetrahedron.”

Sources:

# Roberto Devereux

by Eugene Toth

For the New York Metropolitan Opera’s March 24, 2016 gala opening of “Roberto Devereux,” the eyes of opera enthusiasts sparkled.  Not only was the performance a new production.   For the first time ever, on the 413th anniversary of Queen Elizabeth I’s death, the Met staged Gaetano Donizetti’s “Roberto Devereux”

Donizetti set several operas in Britain. After Anna Bolena and Maria Stuarda Roberto Devereux was Donizetti’s third opera about British Queens.  “Roberto Devereux,” depicts the golden age of Elizabeth’s reign and the Tudor era, when poetry, music and theater flowered.

In 1599, Roberto Devereux, a former lover of Elizabeth I, returned from an unsuccessful war in Ireland, to England.   The opera tells why Elizabeth I executed him for treason.

With historical detail, exquisite costumes invoked the splendor of the Elizabethan era.  In

his first appearance, Devereux wore a black overcoat with silver lining over black plate armor, based upon a 1590 Portrait by William Segar of  Roberto Devereux, Earl of Essex.

In the Metropolitan Opera’s new production, the stage wore no curtains.  Between two balconies, a wall approached and receded from the audience.  In different scenes the wall represented a palace of Elizabeth I, the Palace of the Duke of Nottingham, and the Tower of London.  At the sides, in two galleries, chorus members acted as an audience and witnesses focusing attention upon the four soloists – soprano Elizabeth, the mezzo soprano Sara, the tenor Robert Devereux, and baritone Duke of Nottingham.

Elizabeth I’s passion for her lover  Robert, Earl of Essex, drives the plot.  In scene 1, Elizabeth displayed the character of an imperious, fearsome, and proud monarch.  She held more power than anyone else in England.  Parliament sought to execute Essex as a rebel for treason.  The elderly Queen loved a younger man.  The Queen confided to Sara, a beautiful lady in waiting, that the Queen would pardon Devereux of the treason charges, if he still loved the Queen.

Devereux loved Sara.  Elizabeth had forced Sara to marry Devereux’s best friend and supporter, the Duke of Nottingham.   Trapped in a marriage she never wanted, Sara still loved Devereux.   At their secret meeting, a duet between Sara and  Devereux supplies one of the opera’s high points.   Delightful flute mirrored the intense love they shared.  Telling him to flee, that they must never meet again, Sara gave Devereux, as a token of her love, her blue shawl.

By order of the Queen, Sir Walter Raleigh arrested Devereux.  Raleigh discovered Sara’s blue scarf.  The scarf proved Devereux loved a woman.  Blind with rage and jealousy, Elizabeth signed Devereux’s death warrant.

Recognizing his wife’s scarf, the Duke of Nottingham, drunk in his palace, assaulted Sara with a knife and threw her about.  Devereux’s unwise passion for Sara turned against him his best supporters – the Queen and his former friend the Duke of Nottingham.

Still in love with Devereux, too late, the Queen canceled his execution.   Moments before the executioner chopped off Devereux’s head, she pardoned Devereux.  A cannon shot signaled his death.   The Queen saw visions of Devereux’s headless ghost and a bloody crown.

Elizabeth could not order Devereux to love her.  Even the greatest power meets limits. In the background of the stage statues symbolized Time and Death. Renouncing her throne, she exclaimed “Let James be King!”  A blast of the orchestra’s brass marked her death.

Setting a fast and thrilling pace, the Queen’s transforming feelings supply the opera’s dramatic tension.  Her love transformed into fury, regret, sorrow, remorse, despair, and finally madness.  Donizetti called this work, “the opera of emotions.”

# Monday

Monday

A Poem by Omar Abdelhamid

It is Monday.

The sky is freshly rolled out

the air is soft and silent

A single crimson leaf

And the world is in front of my eyes

but my arms cannot reach out and grab it

I am a speck in the ground below

Some hear my whispers

Stoop down to listen.

Pick up the words I drop

And place them into my humble basket

1. Am.  Sinking.

My arms and neck  are stuck in the cement

I cannot reach out, nor do I have a soul to lift me

I am bound to something, something more than me

and it rushes through my bones

through my mind bouncing back and forth and breaking through every thought every memory

And I can feel the frozen bodies of those who fell behind me,  screaming, hoping,

leaving their legacies

to be plucked by the vulture time

who eats his meal with no haste

they leave it for me

for me  to drop

and for you to stoop down,

pick up, and place into my humble basket

They will find me here

if they care to look

frozen

a moment in time

an echo, a memory

And every echo is smaller

when there is no one there to hear

Send help

Give my words a way

To break me out

out of this cage

# Skyscrapers

Skyscrapers

A Short Story by Omar Abdelhamid

Exhausted from the walk and relieved to have found a seat, I braced myself for the roar of the train as it took off. Suddenly we were in the dark, and I could spot blotches of graffiti on the inside of the tunnel as we whizzed by. I could not think, because my thoughts,unable to keep up with the speed of the iron beast, were left to float in the emptiness of the tunnels, to be ignored, if even noticed, by later commuters. The screech of the friction of the rails as we shot through exhausted my mind, until I could see nothing, I could hear nothing, and I could know nothing. And West shot the beast towards home.

# The Achievements of the Han Dynasty

The Han dynasty lasted from 206 BC-220 AD and had many amazing achievements that changed the world for all and benefits our society in many ways. The Han dynasty had many achievements in science, and one of them was the seismograph. The Seismograph was an impressive instrument because it detected earthquakes from hundreds of miles away. Another scientific achievement was that they learned how to predict when the sun was going to have an eclipse. This discovery helped people because they were always ready when an eclipse came.  In medicine, doctors found new kinds of medications. This helped doctors treat more diseases and patients who were ill. Han craft workers also made an amazing invention when they learned how to create paper. They created paper by pounding the bark of Mulberry trees.  The invention of paper had a huge effect on the way people lived. Paper made it easier to record what was happening. With paper, students invented the first Chinese dictionary. Another idea that came up under the rule of emperor Wudi was Grand School. Grand School were schools that were created to help students get jobs in the local government. Grand School was the empire’s best school. They were set up in every province or state in the empire. Without Grand School and the Confucian emphasis on education, their society might not have had innovators to create these inventions. Overall, thanks to the Han dynasty’s hard work and achievements, society obtained tools and scientific advancements that benefit us to this day.

# Annie and Communism

The cold war was a period of tension between the two major world powers, the U.S. and The Soviet Union, and the two major world political structures, Communism and Capitalism during the 20th century. It was, of course, much more complex and interesting than quickly outlined here. But this simple description gives us a bit of context for what we are about to discuss.

The Cold War was all about getting people on either side of the tension. Both sides used all kinds of propaganda to get their own citizens more strongly associated with the beliefs of their government.

This propaganda was perpetuated through various different mediums, including cartoons, posters, and (as made possible by the popularity of advancing technology) movies.

# Anthropology, Placebos, and Magic Voodoo Doctors

When the anthropology students of Horace Miner harshly judged and mocked the cultures of the people they studied and read about, Miner showed his students the humanity of these cultures and put them in a better light in a very clever way. He wrote an article  about the Nacirema, a Tribe with very strange customs and traditions, such as a mouth-rite ritual done by sticking horse hairs in the mouth. How strange indeed. A culture with medicine men and women and a charm-box in the washing room.

# whispering night

Whispering night

A whispering night

Silences itself to life

Until normality

Having long been the rhythm of the sun

Sets and is but a whisper

And the cold trickles up the veins

When the buttress of the fragile soul

Breaks and drops it’s  child

And not the dark

But the dimming light is seen

With squinting

” let it live ” is cried

And the light does push so gallantly for life

Till the final desperate flicker and the  end

Puts it to rest at last

And sight excludes it

And when the mind is pressed and the night is dark and the music is soft but deadly

A child quickly falls and falls and falls

Asleep

-Omar Abdelhamid

# On the Elements of Love

by Omar Abdelhamid

To encompass the benefit and the reason for the survival of love in humans, one can describe all forms of “Love” as the basis of all motivation. The reason for doing anything that has ever been done is love.

We can further break down love into 3 different forms.

One kind of love is loving something for what it has done before. Another is loving for what it is doing. And the last is loving for what it can do.

All these forms are similar, despite being listed in the dictionary as three entirely different definitions of the word love. Because all three types of love is giving very passionately because there is something that can, is, or has been given to you before. You love because you were given or will be given and you are grateful. So love is in a sense gratitude as well.

These categories seem vague at first glance, so it would help to provide examples for each kind.

The Winner of the Haiku Day Haiku Contest is……. Aaron Wang!

Here is Aaron’s poem:

Look, it’s New Year’s Day,

Parties, food, and wild laughter,

What a beginning.

Thank you Aaron, and thanks to all who entered.
Happy Holidays!
Uncategorized

By Alwin Peng

The quadratic formula is used in order to find the value of x in an equation such as ax^2 + bx + c = 0.

It is x = (-b + (b^2 – 4ac)^(1/2))/2a.

It simplifies a more complicated way to solve the expression. Below, I will show you why the formula works by explaining how it was derived.Deriving the quadratic formula goes as follows:

ax^2 + bx + c = 0

ax^2 + bx = -c .We subtract both sides by C

x^2 + bx/a = -c/a .We divide both sides by A

x^2 + bx/a + (b^2)/(4a^2) = -c/a + (b^2)/(4a^2) .We add (b^2)/(4a^2) to both sides   so that the left side of the equation is factorable

(x + b/2a)^2 = -c/a + (b^2)/(4a^2) We factor the left side

(x + b/2a)^2 = -4ac/(4a^2) + (b^2)/(4a^2) We multiply the top and bottom of the fraction -c/a by 4a so that it can be added with (b^2)/(4a^2), achieving the common denominator of 4a^2 in both fractions.

(x + b/2a)^2 = (-4ac + (b^2))/(4a^2) We now add the like terms

x + b/2a = (((b^2) – 4ac)^(1/2))/2a We square root both sides

At last, we subtract both sides by b/2a, isolating x and getting the quadratic formula:

x = (-b + (b^2 – 4ac)^(1/2))/2a

The method  that we just used to obtain the quadratic formula is known as “completing the square”, but just plugging values into the quadratic formula is a faster and simplified way to complete the square.

# Fibonacci Sequence

Fibonacci Numbers

by Eugene Toth

Fibonacci numbers are an amazing sequence of numbers which appear all throughout human history and throughout nature.  One may see fibonacci numbers in the great pyramid or a nautilus’ shell.  This amazing sequence of numbers have a simple pattern but a stunningly complex role in the world around us.

# Pascal’s Triangle

Pascal’s Triangle

by Eugene Toth

Mathematicians named Pascal’s triangle after the French mathematician Blaise Pascal.    Pascal’s triangle is a triangular graph.

In Pascal’s triangle each number is the sum of the two directly above it.  The numbers in each row are numbered beginning with 0 for the first row. Each number is positioned either to the left or to the right of the numbers in the rows above. The sum of the elements of a single row is twice the sum of the row preceding it.  A Pascal’s triangle can expand infinitely.

The Core staff of Infinite Initiative has decided on a change in the direction of the purpose of our organization.We are going to attempt to allow children to be able to express themselves through writing, so that they can get their ideas out eloquently and in a way that adults will be able to understand. The program will provide a writing and blogging skills program, and will also provide authorship on our websites to people who go through the program, with hopes that it will not only provide skills, but an audience for our children’s ideas.The project is in early development stages, but we would like as much support and publicity of our website as possible. Thank you in advance for any support given. Email omar@infiniteinitiative.org if you are interested in supporting or want any more information.

We will still continue to be posting, so scroll down to see our latest posts.

-Omar

# Newton’s Laws of Motion

Newton’s Laws of Motion

By Eugene Toth

Aristotle believed that everything had a natural state.  Water would stay in the hydrosphere.  It always flowed to the hydrosphere or water bodies.  Rocks fell to the geosphere or Earth.  Air entered the atmosphere or the rest of the air around us.  Fire rose in the form of smoke to a place above the atmosphere. People believed Aristotle’s theory until Galileo started studying physics.  Galileo’s studies of falling objects proved Aristotle wrong.  Then, in 1687, Sir Isaac Newton published his book Philosophiae Naturalis Principia Mathematica, or ,in English, Mathematical Principles of Natural Philosophy. People more commonly know Newton’s book as “The Principia.”

Sir Issac Newton Painted by Godfrey Kneller

# Living on Water

By Eugene Toth

“Call me Ishmael. Some years ago–never mind how long precisely–having little or no money in my purse, and nothing particular to interest me on shore, I thought I would sail about a little and see the watery part of the world. It is a way I have of driving off the spleen and regulating the circulation.”

– Moby Dick, Herman Melville

In almost all places on Earth, we have been living in houses of all shapes and sizes.  We have built mobile homes and houseboats.  Many have ridiculed the idea of houseboats.  We have not developed houseboats as much as other forms of housing.

# How to make Green Fire!

By Eugene Toth

WARNING!

Only persons 18 years old and older should perform this experiment. If you do not fit this criteria, you need adult supervision when performing this experiment. Perform this experiment outdoors in a fire safe area. Infinite Initiative disclaims responsibility for explosions, fire or any other damages.

Ever wanted to resemble a mad scientist?  Have you read the story of Dr. Frankenstein?  Have you ever wanted to produce green fire?  The spectacle of green fire will amaze and thrill.  In fact, green fire is no more dangerous than normal fire.

# Peanuts and Child Philosophy

Charles M. Schulz, famous for being the creator of the Peanuts cartoons, once remarked about his profession:

” If I were a better artist, I’d be a painter, and if I were a better writer, I’d write books- but I’m not, so I draw cartoons.”

The key element of his skills description is his lack of proficiency in writing. He may have not been good at writing, but he was still able to express very powerful messages. He did this by taking on the view of children, who have ideas like he did, but are not able to organize them into eloquent writing,and instead were more abstract in their mindset. He used his persona as a child to justify expressing his ideas in abstract and unclear ways.

# Vibratory Motion

Vibratory Motion

By Eugene Toth

Vibratory motion, or vibration differs from rotational motion and linear motion.  In vibration, motion progresses alternately changing direction at fixed intervals.

In the 6th century, the Greeks first studied vibration when they plucked the strings of instruments.   Pythagoras of Samos in Greece studied the relationship of vibrations to music.  People know Pythagoras of Samos better for discovering the Pythagorean Theorem (a2+b2=c2) in a triangle. The word vibration comes from the Latin word “vibrationis” for shaking.

# Materialism in America- The Repercussions of Materialism

Materialism In America

Above is the link to a study about the repercussions of materialism on Individuals and society as a whole.

This research paper was written by Omar Abdelhamid with the help of Mr. Terrance Brown and Ms. Liza Tarbell.

# World Cup and Increasing Nationalism

The FIFA 2014 World Cup has become a principal aspect of many people’s lives. Doubtless, players and coaches are greatly affected by the outcomes of these games; however, fans and countries as a whole are impacted by these games. Some of these impacts are beneficial to people and nations, but some are proving to be dangerous and may get us dangerously close to a global conflict.

What is the meaning of Nationalism?

na·tion·al·ism
ˈnaSHənəˌlizəm/
noun
1. patriotic feeling, principles, or efforts.
• an extreme form of this, especially marked by a feeling of superiority over other countries.
plural noun: nationalisms
• advocacy of political independence for a particular country.

Lets take a closer look at this definition. It has two main parts; a nation is better than other nations and a nation deserves freedom from another nation.

# Angular Momentum

By Eugene Toth

Rotation Inertia

Bodies that have rotation have rotational inertia.  They will continue to rotate around axis unless stopped by an outside force. For example, the Earth rotates around the Sun. Because of rotational inertia, astronauts are exceedingly careful to avoid getting into a spin.

Just as there are linear and rotational velocities, there are also linear and rotational accelerations.  As we use the Roman letter α for rotational velocity, we also use it for rotational acceleration.  Just as linear velocity is equal to angular velocity times the distance from the center of rotation. Therefore linear acceleration is equal to angular acceleration.

By the second law of motion, we know that force = mass x linear acceleration.  Let a represent angular velocity.  Let r represent rotation.  “τ” is the greek letter “tau.”  In our case, we will use it to symbolize “torque.” This formula gives us:

As you exert force on the pedal, the petal turns and the pedal and chain gains angular momentum

F = M r a

Force = Mass x Rotation x angular velocity

Force = mass x radius x alpha

# Velocity

By Eugene Toth
What is Velocity?  Many think velocity is just a scientific name for speed.  They are partially right. Scientist define velocity as distance covered in unit time as in feet per second or miles per hour.

The curve of Force and Velocity. As the force exerted on an object is decreased, the velocity of the object is decreased.

If a ball rolls at 2 feet per second, it’s average velocity is two feet per second, and its average velocity is 2 feet per second. At any given moment, however, the object is going faster or slower. This can be expressed as a fraction

2 Feet       OR     2 feet/1 second
1 Second

It is important to recognize that the numerator and and denominator are units and not numbers.

The average velocity is 4 feet per second after it has traveled two feet.

In the third second, it covers 18 feet with an average velocity of 6 feet per second.

# Chuck E. Cheese and a Rise in Materialism

I was recently at a Chuck E. Cheese restaurant / arcade. I picked up on a few details about the place that you may not have noticed, but are very negatively influential to children who play there.

For those not familiar with the place, it is an arcade of different games and you get tickets from every game that you are able to exchange for receipts to buy prizes. Each game is worth a token that you can buy with real money.

Prizes of Chuck E. Cheese

See the problem yet?

Cynicism

There are two types of games, games for fun, and games that are chance based that get you tickets. The ticket games are often more prominent; however, they are usually rigged, making it hard or impossible to get a real prize. The number of fun games are decreasing and decreasing as the point of the arcade becomes more centered around getting enough tickets to get a prize. Again, these games are not fun at all, and only involve pulling a lever or sliding a coin down with accurate timing. Even worse, jackpot prizes reach only 25 or 50 tickets. The prizes are very expensive and often poorly made or easy to lose. They are often worthless. You can buy a small pack of candy for 50 tickets,and a good prize would be 4,000 tickets, an almost impossible task. These are usually supposed to be saved up for, but the store encourages children to cease to invest and instead buy small and cheap toys.

How is modern day economy similar to places like Chuck E. Cheese?

# Ode To Binary

0100111101000100010001010010000001

0101000100111100100000010000100100100101

0011100100000101010010010110010000110100001

01000001101000010100100001001101001011011100

110000101110010011110010010000001101001011100

# France Advancing into the Future

By Eugene Toth

The New Yorker provided an excellent article about a hydrogen power plant that the government is building that will give almost infinite energy to anyone who uses power coming from it.  The power plant will be a vacuum filled with trillions of hydrogen atoms.  Then, the inside of the vacuum will heat up to enormous temperatures without melting the metal.  The extreme heat will create the hydrogen atoms to move.  This will make the hydrogen atoms fuse creating huge amounts of nuclear energy with only one millionth of normal nuclear energy’s nuclear waste.  This spare nuclear waste will be drained of its resources until it will basically not exist making the power plant, in some sense, have no energy waste at all.