Once upon a time, in the imaginary land of numbers… Yes, numbers! I bet that would’ve never come to mind. Which brings me to the question: Who thought of them and why?
In 50 A.D., Heron of Alexandria studied the volume of an impossible part of a pyramid. He had to find √(81-114) which, back then, was insolvable. Heron soon gave up. For a very long time, negative radicals were simply deemed “impossible”. In the 1500’s, some speculation began to arise again over the square root of negative numbers. Formulas for solving 3rd and 4th degree polynomial equations were discovered and people realized that some work with square roots of negative numbers would occasionally be required. Naturally, they didn’t want to work with that, so they usually didn’t. Finally, in 1545, the first major work with imaginary numbers occurred.

In 1545, Girolamo Carding wrote a book titled Ars Magna. He solved the equation x(10-x)=40, finding the answer to be 5 plus or minus √-15. Although he found that this was the answer, he greatly disliked imaginary numbers. He said that work with them would be, “as subtle as it would be useless”, and referred to working with them as “mental torture.” For a while, most people agreed with him. Later, in 1637, Rene Descartes came up with the standard form for complex numbers, which is a+bi. However, he didn’t like complex numbers either. He assumed that if they were involved, you couldn’t solve the problem. Lastly, he came up with the term “imaginary”, although he meant it to be negative. Issac Newton agreed with Descartes, and Albert Girad even went as far as to call these, “solutions impossible”. Although these people didn’t enjoy the thought of imaginary numbers, they couldn’t stop other mathematicians from believing that i might exist. (The History of Complex Numbers)

Rafael Bombelli was a firm believer in imaginary numbers. However, since he couldn’t do anything with them people doubted him. He did understand that i times i should...

...History of imaginarynumbers
I is an imaginarynumber, it is also the only imaginarynumber. But it wasn’t just created it took a long time to convince mathematicians to accept the new number. Over time I was created. This also includes complex numbers, which are numbers that have both real and imaginarynumbers and people now use I in everyday math.
I was created because everyone needed it. At first the square root of a negative number was thought to be impossible. However, mathematicians soon came up with the idea that a number to solve this equations could be created. Today the square root of negative one is known as I. I is very useful to the world engineers use it to study stresses on beams. Complex numbers help us study the flow of fluid around objects, such as water around a pipe. Last year in my electricity class we had to use imaginarynumbers all the time.They are also used in electric circuits, and help in transmitting radio waves. So if it weren’t for I, we might not be able to talk on our cell phones, or listen to the radio. It is definitely a good thing that I was created.
The very first mention of people trying to use imaginarynumbers dates all the way back to the 1st century....

...Introduction
The purpose of this research paper is to introduce the topic of “Complex and ImaginaryNumbers” and its applications. I chose the topic “Complex and ImaginaryNumbers” because I am interested in mathematics that is hard to be pictured in your mind, unlike geometry or equations.
An imaginarynumber is the square root of a negative number. That is why they are calledimaginary, what René Descartes called them, because he thought such a number could not exist. In this paper, I will discuss how complex numbers and imaginarynumbers were discovered, the interesting math of complex numbers, and how they are used in other areas of mathematics and science. Complex numbers are applied in engineering, control theory and improper integrals to take the place of certain imaginary values, as well as to simplify some explanations.
2. THe Concept
2.1 History of imaginarynumbers
Long ago in ancient Greece, there was a society of mathematicians called Pythagoreans who believed the only numbers were natural numbers and positive rational numbers (Rusczyk 357). Later, Hippasus discovered irrational numbers such as√2, then 0 and negative numbers were...

...
Abstract
A complex number is a number that can be written in the form of a+bi where a and b are real numbers and i is the value of the square root of negative one. In the form a + bi, a is considered the real part and the bi is considered the imaginary part. The goal of this project is show how the use of complex numbers originates in the history of mathematics.
Introduction
Complexnumbers are very important component of mathematics. They enable us to solve any polynomial equation of degree n. Simple equations like x3+1 would not have solutions if there were no complex numbers. The complex number has enriched other branches of mathematics such as calculus, linear algebra (matrices), trigonometry, and you can find its applications in applied sciences, and physics. In this project we will present the history of complex numbers and the long road to understanding the applications of this truly powerful number.
I) Ancient History
Russian Egyptologist V.S. Glenishchev traveled to Egypt in 1893 on routine business little did he know that what he would purchases would change how the world viewed Ancient Egyptian civilization, and shape mathematical landscape for years to come
Stolen from the valley of kings in 1878, at Deir el-Bahri, the Moscow Mathematical Papyrus which was sold...

...Paper - I
1. Sources: Archaeological sources:Exploration, excavation, epigraphy, numismatics, monuments Literary sources: Indigenous: Primary and secondary; poetry, scientific literature, literature, literature in regional languages, religious literature. Foreign accounts: Greek, Chinese and Arab writers.
2. Pre-history and Proto-history: Geographical factors; hunting and gathering (paleolithic and mesolithic); Beginning of agriculture (neolithic and chalcolithic).
3. Indus Valley Civilization: Origin, date, extent, characteristics, decline, survival and significance, art and architecture.
4. Megalithic Cultures: Distribution of pastoral and farming cultures outside the Indus, Development of community life, Settlements, Development of agriculture, Crafts, Pottery, and Iron industry.
5. Aryans and Vedic Period: Expansions of Aryans in India. Vedic Period: Religious and philosophic literature; Transformation from Rig Vedic period to the later Vedic period; Political, social and economical life; Significance of the Vedic Age; Evolution of Monarchy and Varna system.
6. Period of Mahajanapadas: Formation of States (Mahajanapada): Republics and monarchies; Rise of urban centres; Trade routes; Economic growth; Introduction of coinage; Spread of Jainism and Buddhism; Rise of Magadha and Nandas. Iranian and Macedonian invasions and their impact.
7. Mauryan Empire: Foundation of the Mauryan Empire, Chandragupta, Kautilya and Arthashastra; Ashoka;...

...Complex Numbers
All complex numbers consist of a real and imaginary part.
The imaginary part is a multiple of i (where i =[pic] ).
We often use the letter ‘z’ to represent a complex number eg. z = 3 +5i
The conjugate of z is written as z* or [pic]
If z1 = a + bi then the conjugate of z (z* ) = a – bi
Similarly if z2 = x – yi then the conjugate z2* = x + yi
z z* will always be real (as i2 = -1)
For two expressions containing complex numbers to be equal, both the real parts must be equal and the imaginary parts must also be equal.
If z1 = a + bi , z2 = x + yi and 2z1 = z2 + 3 then
2( a + bi) = x + yi + 3
hence 2a + 2bi = x + 3 + yi
so 2a = x + 3 (real parts are equal)
and 2b = y (imaginary parts are equal)
When adding/subtracting complex numbers deal with the real parts and the imaginary parts separately
eg. z1 + z2 = a + bi + x + yi
= a + x + (b + y)i
When multiplying just treat as an algebraic expression in brackets
eg. z1 z2 = (a + bi)(x + yi)
= ax + ayi + bxi + byi2
= ax - by + (ay + bx)i (as i2 = -1)
Division by a...

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...
Today one of the most cherished ideologies of America is the fact that everyone is and should be created equal. With this cherished ideology bringing a sense of pride and diversity to America we must keep in mind that this cherished ideology did not always exist. Since 1865 various individuals and groups have not been able to receive and express their rights to full equal status in the United States. These different individuals and groups have seemingly fought for their rights in equality and have become pioneers in the fight for evolution for equality.
In 1865 African Americans in the United States under the 13th amendment were freed from the terrible burden of slavery. Through the 14th amendment they were given the right to citizenship and the right to equal protection. The 15th amendment gave them the right to vote regardless of their skin color race or any other type of servitude. These amendments were meant to be enforced and make a serious change in the everyday life of the average American.
With these amendments passing in 1865 they were meant to make a serious change towards the evolution of equality. These changes did not seem to happen right away and African Americans were still not being treated with equality. The average African American at this time were being denied there newly given rights every day making life extremely hard to stay...

...Name:
Date:
Graded Assignment
Final Exam Part 2
I. Map
On this world map, indicate the following features:
Amazon Rainforest
Panama Canal
The Himalayas
The Ring of Fire
The Mississippi River
The Gobi Desert
(10 points)
II. Graphic Organizer
Fill in the table below about these five major world religions. Do not fill in the shaded boxes.
(10 points)
Religion
Name at least
one Holy Text
How do you achieve
enlightment?
Describe their view about the afterlife.
Hinduism
Bhagvada Gata
Do good deeds to get good karma until you break the samsara or cycle of reincarnation and reach enlightenment
Buddhism
Believe the Four Truths are true and real, follow the Eightfold Path, meditation is one of the major steps to reach enlightenment
Judaism
Old Testament
God promised the Jews, people of Israel, paradise and those who hate the Jews and mistreat them are going to go to Hell
Christianity
New Testament
Islam
Quran
People who believe in all the five pillars and do them and do righteous deeds go to heaven while the disbelievers and those who sin are punished and go to Hell
III. Short Answer
1. Explain the role of river valleys in the development of civilizations. Name at
least two river valleys as examples. (10 points)
Rivier valleys first and foremost provided water, a basic necessity for humans. It also provided fertile soil for agriculture, which led to settlements and brought hunting and gathering to an end....

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