Who do you think should control the economy? Do you think the economy should be left alone, and allowed it to reach a good point on its own? Or, do you think the economy should be influence by the government, though means of intervention?

During the 18th century, Adam Smith, brought about the idea of the 'invisible hand'. This idea was simply, the economy should be left alone and everything will be okay, or guided by the 'invisible hand'.

As humans, we are all guided by our own self-interest. Adam Smith thought that, this self-interest would allow sellers to compete with one and other to obtain more buyers.

This could lead us to equilibrium by giving buyers competitive options to choose from without the influence of outside forces, other than the 'invisible hand' or our own self-interest.

As humans, we are all guided by our own self-interest. Adam Smith thought that, this self-interest would allow sellers to compete with one and other to obtain more buyers.

This could lead us to equilibrium by giving buyers competitive options to choose from without the influence of outside forces, other than the 'invisible hand' or our own self-interest.

In *The Wealth of Nations, *Smith wrote,

| "But the annual revenue of every society is always precisely equal to the exchangeable value of the whole annual produce of its industry . . . As every individual, therefore, endeavours as much as he can both to employ his capital in the support of domestic industry, and so to direct that industry that its produce may be of the greatest value . . . He generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it . . . by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in many other cases, led by an By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it. I have never known much good done by those who affected to trade for the public good."invisible hand to promote an end which was no part of his intention . . . | |

He also wrote it in his book, *The Theory of Moral Sentiments*,

| "The produce of the soil maintains at all times nearly that number of inhabitants which it is capable of maintaining. The rich only select from the heap what is most precious and agreeable. They consume little more than the poor, and in spite of their natural selfishness and rapacity, though they mean only their own conveniency, though the sole end which they propose from the labours of all the thousands whom they employ, be the gratification of their own vain and insatiable desires, they divide with the poor the produce of all their improvements. They are led by an which would have been made, had the earth been divided into equal portions among all its inhabitants, and thus without intending it, without knowing it, advance the interest of the society, and afford means to the multiplication of the species."invisible hand to make nearly the same distribution of the necessaries of life, | |

Essentially, the 'invisible hand' is a market force, in which you can not see, pushes and pulls the demand and supply of commodities in a free market that could help achieve equilibrium.

]]>Below, I will show you where the elastic, inelastic, unit elastic, perfectly elastic, and perfectly inelastic points are located. Along with locating points, I will provide examples and brief explanations on each.

Point (A)

Point (B)

Point (C)

Point (D)

Point (E)

]]>- Perfectly Elastic
- Elasticity of Demand tends to be infinity
- At any specific price, quantity demanded is infinite
- Example: Foreign Currency Exchange, exhibits features of perfect competition when buying it

Point (B)

- Elastic
- Elasticity of Demand is greater than 1
- A change in price results in a larger change in quantity demanded
- Example: A seller increases the price of a good by a certain amount, the buyer's demand will decrease by a larger percentage as the price

Point (C)

- Unit Elastic
- Elasticity of Demand is equal to 1
- A change in price results in an equal change in quantity demanded
- Example: A seller increases the price of a good by a certain amount, the buyer's demand will decrease by the same percentage as the price

Point (D)

- Inelastic
- Elasticity of Demand is less than 1
- A change in price results in a smaller change in quantity demanded
- Example: A seller increases the price of a good by a certain amount, the buyer's demand will decrease by a smaller percentage as the price

Point (E)

- Perfectly Inelastic
- Elasticity of Demand tends to be 0
- At any price change, quantity demanded does not change
- Example: A medical drug that keeps you alive

Exchange is the act upon which an individual must give in order to receive. Natural and unnatural forms of exchange are thought of as needs and desires, respectively. Aristotle viewed barter, monetary, retail sale, and usury as forms of exchanges. This post will briefly go over these four forms of exchange, as distinguished by Aristotle. |

To begin, **barter** exchange is a non-monetary form of direct exchange of commodities. Barter is when a commodity is exchanged directly with another commodity.

He viewed this form as good, or natural, but found barter to be inconvenient. This leads to an absence of the coincidence of wants and inconsistency in the exchange ratio.

**Monetary** was also seen as good, and natural, in the eyes of Aristotle. This form of exchange is when a commodity is indirectly exchanged with another commodity.

It begins with a commodity, then this commodity is sold for money and the money is used to buy a desired commodity.

He viewed this form as good, or natural, but found barter to be inconvenient. This leads to an absence of the coincidence of wants and inconsistency in the exchange ratio.

It begins with a commodity, then this commodity is sold for money and the money is used to buy a desired commodity.

This could be thought of as modern day scalping, which is not usually looked down upon within the economic community. Since, it is voluntary and both sellers and buyers benefit from actions taken.

Lastly,

It is the use of money to gain more money without obtaining a good or service nor an end use value. Usury is also looked upon by Aristotle as unnatural.

The way a Pure Mathematician thinks is different from the way an Applied Mathematician thinks. A Pure Mathematician looks at life as something they want to enhance. Several will go their entire lives trying to construct an original theory, or one could consider it a work of art, to improve the mathematical foundation as a whole.

For example, Dr. John Nash wanted to come up with an original idea within mathematics. But instead he found a theory that was wrong in his opinion and improved the original theory so it will be true and won a Nobel Prize for his improvement in the subject.

An Applied Mathematician will look at life as a problem waiting to be solved. They will ask questions from an array of subjects and try to solve them using mathematics. These subjects can range from chemistry, physics, economics, and even music.

For example, people wanted to build a computer that could do a massive amount of computations in as little time as possible. This is called a Supercomputer.

They asked the question and then used Applied Mathematics to answer and solve the problem.

An Applied Mathematician will look at life as a problem waiting to be solved. They will ask questions from an array of subjects and try to solve them using mathematics. These subjects can range from chemistry, physics, economics, and even music.

For example, people wanted to build a computer that could do a massive amount of computations in as little time as possible. This is called a Supercomputer.

They asked the question and then used Applied Mathematics to answer and solve the problem.

An example of an Applied Mathematician could be Albert Einstein. He asked many questions on how different things work and solved them by using mathematics. Einstein is the originator of the Theory of Relativity, which states that from the point of view of an individual on Earth looking into space at an object traveling the speed of light it will appear to be moving slower than it really is. |

These two types of mathematics use different ways to solve problems. Pure Mathematics is really based off of techniques, and the way of solving a problem in Pure Mathematics is based on proofs.

Applied Mathematics, on the other hand, uses modeling and theories already proven by Pure Mathematicians to solve real world problems.

A Pure Mathematician would ask questions like: How could this theorem be related to mathematics? How could I improve this theorem? How could I prove this theory to be a contradiction?

An Applied Mathematician would ask questions like: Where could I apply this to? Could I use more theories to support my findings? Is this useful in answering the question?

Applied Mathematics, on the other hand, uses modeling and theories already proven by Pure Mathematicians to solve real world problems.

A Pure Mathematician would ask questions like: How could this theorem be related to mathematics? How could I improve this theorem? How could I prove this theory to be a contradiction?

An Applied Mathematician would ask questions like: Where could I apply this to? Could I use more theories to support my findings? Is this useful in answering the question?

Mathematics as a whole is fundamentally essential to everyday life. The history between Pure and Applied Mathematics goes hand in hand with one another.

All it took was individuals who asked questions, who went about solving them or proving a certain theory they create.

Ideally, they think the same way, but from a deeper perspective, one can see that Pure Mathematicians want to improve the understanding to further the reach of mathematics. Applied Mathematicians solve questions that are very well needed to improve life.

Pure Mathematics can be considered an art or a never ending musical symphony. It creates a beautiful masterpiece that is used by Applied Mathematicians. They feed off each other and make this world a better more understanding place.

]]>All it took was individuals who asked questions, who went about solving them or proving a certain theory they create.

Ideally, they think the same way, but from a deeper perspective, one can see that Pure Mathematicians want to improve the understanding to further the reach of mathematics. Applied Mathematicians solve questions that are very well needed to improve life.

Pure Mathematics can be considered an art or a never ending musical symphony. It creates a beautiful masterpiece that is used by Applied Mathematicians. They feed off each other and make this world a better more understanding place.

Applied Mathematics is a hit or miss, meaning either it works or it does not work, and in order to be improved it depends on the works of Pure Mathematics.

What we know today was developed by Pure Mathematicians. They did not have technology we have today to apply their work to practical use.

For example, Number Theory was thought to be one of the most nonessential subjects within Algebra. Number Theory is the study of numbers and their properties. It is considered to be a part of Pure Mathematics.

What we know today was developed by Pure Mathematicians. They did not have technology we have today to apply their work to practical use.

For example, Number Theory was thought to be one of the most nonessential subjects within Algebra. Number Theory is the study of numbers and their properties. It is considered to be a part of Pure Mathematics.

It was not until recently Number Theory became a part of the applied world, since we can apply the theories learned in Number Theory to breaking and making codes.

In today’s society, the subject is used greatly by NSA and any intelligence agency. When one wants to buy items online, Number Theorists make it where it is safe to do so. They do this by making a code for when one enters their card information.

The information is then encrypted so it will not be readable if it was obtained by an individual wanting to steal from another individual. Without the beauty of Number Theory a lot of crime will be going on today.

In today’s society, the subject is used greatly by NSA and any intelligence agency. When one wants to buy items online, Number Theorists make it where it is safe to do so. They do this by making a code for when one enters their card information.

The information is then encrypted so it will not be readable if it was obtained by an individual wanting to steal from another individual. Without the beauty of Number Theory a lot of crime will be going on today.

Applied Mathematics, however, has essentially been around for a very long time. This type of math can be as straightforward as a person counting money, or as complicated as creating an algorithm from scratch to shut down the nation’s defense system.

For example, Quantum Theory is surely amongst the most difficult subjects within Applied Mathematics. Quantum Theory is basically principles within energy and matter and how they behave in a very small, atomic, states.

Quantum Theory is used in physics and is a key theory to understanding Quantum Mechanics. Without Quantum Theory there would be no laptops, one would not be able to instant message anybody, and there would not be any lasers.

Applied and Pure Mathematics both play a huge role in our everyday life, as seen through our history.

]]>For example, Quantum Theory is surely amongst the most difficult subjects within Applied Mathematics. Quantum Theory is basically principles within energy and matter and how they behave in a very small, atomic, states.

Quantum Theory is used in physics and is a key theory to understanding Quantum Mechanics. Without Quantum Theory there would be no laptops, one would not be able to instant message anybody, and there would not be any lasers.

Applied and Pure Mathematics both play a huge role in our everyday life, as seen through our history.

For example, mathematics was used in making the cars that individuals drive around, the phone you use, the clothes you wear, and many more things.

Even though Pure and Applied Mathematics seem to both mean the same thing, if one was to look deeper into the aspects of each they will find that they are in fact different in many various ways.

]]> Today we are going to go over an area within complex analysis known as Stereographic Projection. According to an online dictionary stereographic projection is “a one-to-one correspondence between the points on a sphere and the extended complex plane where the north pole on the sphere corresponds to the point at infinity of the plane” (Dictionary). |

To begin, complex analysis is used in just about everything we do in life. Wolfram Alpha says complex analysis is “the study of complex numbers together with their derivatives, manipulation, and other properties” (Weisstein). The site goes on to say “Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems”.

Stereographic projection is used in many areas of the physical life. One area is physics where they use the metric tensor. A metric tensor “is a function which tells how to compute the distance between any two points in a given space” (Stover).

In this way, “the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be ‘arithmetized’ by way of introducing a sort of generalized coordinate system” (Borisenko and Tarapov 1979).

It is said in this problem done by Hobson, that to find “the metric tensor in a new coordinate system, consider the stereographic projection of a sphere of radius r” (Hobson 2006). It is also said that “the projection is done by drawing lines from the south pole through every point on the sphere and extending each line until it intersects the plane tangent to the north pole” (Hobson 2006).

This shows that within physics you can see that complex analysis is used from using stereographic projection on metric tensor.

Stereographic projection is used in many areas of the physical life. One area is physics where they use the metric tensor. A metric tensor “is a function which tells how to compute the distance between any two points in a given space” (Stover).

In this way, “the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be ‘arithmetized’ by way of introducing a sort of generalized coordinate system” (Borisenko and Tarapov 1979).

It is said in this problem done by Hobson, that to find “the metric tensor in a new coordinate system, consider the stereographic projection of a sphere of radius r” (Hobson 2006). It is also said that “the projection is done by drawing lines from the south pole through every point on the sphere and extending each line until it intersects the plane tangent to the north pole” (Hobson 2006).

This shows that within physics you can see that complex analysis is used from using stereographic projection on metric tensor.

Additionally, area is within engineering. On this site called the civil engineering notes we can see many cases of examples of how it is used exactly.

It says that “stereographic projection is one of the convenient methods of projecting the linear and planar features” (Fosson, “Structural Geology”). Along with this information it goes on to say that “it allows the three-dimensional orientation data to be represented and analyzed in two dimensions”.

This site actually gives all areas it is used in from landslide hazard failures, earthquake studies, structural geological analysis, and even mining industry using fossil fuels.

It says that “stereographic projection is one of the convenient methods of projecting the linear and planar features” (Fosson, “Structural Geology”). Along with this information it goes on to say that “it allows the three-dimensional orientation data to be represented and analyzed in two dimensions”.

This site actually gives all areas it is used in from landslide hazard failures, earthquake studies, structural geological analysis, and even mining industry using fossil fuels.

We can see that the application of complex analysis goes very far and wide. This application known as stereographical projection is just one of the many fields within complex analysis that ventures into the worlds of physics and engineering.

Since these fields are so applied this gives way for us to improve our life as a whole using this type of projection could indeed make even more contributions to the world of fields out there and to all the unknown we have yet to discover.

]]>Since these fields are so applied this gives way for us to improve our life as a whole using this type of projection could indeed make even more contributions to the world of fields out there and to all the unknown we have yet to discover.

Pierre de Fermat was born in Beaumont-de-Lomagne, France on August 17, 1601. Written in an article by Boyer, Fermat was “called the founder of the modern theory of numbers” (Fermat, 2016). He was considered one of the most profound mathematicians during the 17th century. It is said he studied law, foreign languages, literature, science, and even mathematics (2016). From his studies, we know that he was primarily a lawyer and did “mathematics in his spare time” according to wolfram. |

Fermat was a very influential man, as you may can tell. In an article on the story of mathematics site it is said that he was able to “discover several new patterns in numbers which had defeated mathematicians for centuries” (Article).

It also goes on to say that he is credited “for early developments that led to modern calculus, and for early progress in probability theory” (Article). Fermat was actually so amazing at everything he did that he was fluent in four different languages like Italian and even Greek.

Over time Fermat discovered many different theorems within mathematics. For example, the Two Square Theorem which shows any prime number that is “divided by 4 leaves a remainder of 1 [and] can always be rewritten as the sum of two squares” (Article).

The one theorem I like best is called Fermat’s Principle of Least Time. He used the methods that were influenced by Heron. The principle says that a path between two points by a ray or a light is a path that can be traversed in the least possible time.

In an article on MAA, it says that “Fermat became interested in finding a theoretical derivation of Snell’s Law” (Sanchis, 2014).

Snell’s Law was discovered by Willebrord Snell, it dealt with “the change in direction that occurs when a beam of light crosses a boundary from one medium into another” (2014). The proof of this principle could be solved using algebraic optimization.

If we were looking at a coordinate plane, we are going to find C(x,0) to minimize the travel time of a light beam alone path A to C to B.

We will assume the velocity above the x-axis is \(v_{1}\) and below is \(v_{2}\). Since \( Time=\dfrac{Distance}{Velocity} \), the time of travel is \( T(x) = \dfrac{\sqrt{(x-a)^{2}+b^{2}}}{v_1} + \dfrac{\sqrt{(x-c)^{2} + d^{2}}}{v_2} \).

Now we will need to derive this and set it equal to 0, \( \dfrac{\dfrac{1}{2}[(x-a)^{2}+b^{2}]^{\dfrac{-1}{2}}*2(x-a)}{ v_{1}}+\dfrac{\dfrac{1}{2}[(x-c)^{2}+d^{2}]^{\dfrac{-1}{2}}*2(x-c)}{v_{2}}=0 \).

This simplifies to \( \dfrac{x-a}{v_{1}\sqrt{(x-a)^{2}+b^{2}}}+\dfrac{c-x}{ v_{2}\sqrt{(x-c)^{2}+d^{2}}}=0 \).

Since, \( \sin{\theta_{1}}=\dfrac{x-a}{\sqrt{(x-a)^{2}+b^{2}}}\) and \(\sin{\theta_{2}}=\dfrac{c-x}{ \sqrt{(x-c)^{2}+d^{2}}} \).

We now have, \( \dfrac{\sin{\theta_{1}}}{v_{1}}-\dfrac{\sin{\theta_{2}}}{v_{2}}=0 \).

This simplifies to \( \dfrac{\sin{\theta_{1}}}{\sin{\theta_{2}}}=\dfrac{ v_{1}}{v_{2}} \) which is Snell’s Law.

Thus giving us the answer of, \( v_{2} \sin{\theta_{1}} = v_{1}\sin{\theta_{2}} \) (2014).

]]>It also goes on to say that he is credited “for early developments that led to modern calculus, and for early progress in probability theory” (Article). Fermat was actually so amazing at everything he did that he was fluent in four different languages like Italian and even Greek.

Over time Fermat discovered many different theorems within mathematics. For example, the Two Square Theorem which shows any prime number that is “divided by 4 leaves a remainder of 1 [and] can always be rewritten as the sum of two squares” (Article).

The one theorem I like best is called Fermat’s Principle of Least Time. He used the methods that were influenced by Heron. The principle says that a path between two points by a ray or a light is a path that can be traversed in the least possible time.

In an article on MAA, it says that “Fermat became interested in finding a theoretical derivation of Snell’s Law” (Sanchis, 2014).

Snell’s Law was discovered by Willebrord Snell, it dealt with “the change in direction that occurs when a beam of light crosses a boundary from one medium into another” (2014). The proof of this principle could be solved using algebraic optimization.

If we were looking at a coordinate plane, we are going to find C(x,0) to minimize the travel time of a light beam alone path A to C to B.

We will assume the velocity above the x-axis is \(v_{1}\) and below is \(v_{2}\). Since \( Time=\dfrac{Distance}{Velocity} \), the time of travel is \( T(x) = \dfrac{\sqrt{(x-a)^{2}+b^{2}}}{v_1} + \dfrac{\sqrt{(x-c)^{2} + d^{2}}}{v_2} \).

Now we will need to derive this and set it equal to 0, \( \dfrac{\dfrac{1}{2}[(x-a)^{2}+b^{2}]^{\dfrac{-1}{2}}*2(x-a)}{ v_{1}}+\dfrac{\dfrac{1}{2}[(x-c)^{2}+d^{2}]^{\dfrac{-1}{2}}*2(x-c)}{v_{2}}=0 \).

This simplifies to \( \dfrac{x-a}{v_{1}\sqrt{(x-a)^{2}+b^{2}}}+\dfrac{c-x}{ v_{2}\sqrt{(x-c)^{2}+d^{2}}}=0 \).

Since, \( \sin{\theta_{1}}=\dfrac{x-a}{\sqrt{(x-a)^{2}+b^{2}}}\) and \(\sin{\theta_{2}}=\dfrac{c-x}{ \sqrt{(x-c)^{2}+d^{2}}} \).

We now have, \( \dfrac{\sin{\theta_{1}}}{v_{1}}-\dfrac{\sin{\theta_{2}}}{v_{2}}=0 \).

This simplifies to \( \dfrac{\sin{\theta_{1}}}{\sin{\theta_{2}}}=\dfrac{ v_{1}}{v_{2}} \) which is Snell’s Law.

Thus giving us the answer of, \( v_{2} \sin{\theta_{1}} = v_{1}\sin{\theta_{2}} \) (2014).

During the 7th century, in India, an extraordinary mathematician came to existent. His name is Brahmagupta. It is said on an online article for famous mathematicians he was “born in 598 AD in Bhinmal, India” (2012). |

Bhinmal is a city in the northwest part of India. According to another online article about famous scientist he was the son of Jisnugupta, who was an astrologer (2014). Little is known about other members of his family, since this was also a long time ago.

Brahmagupta could be considered to have many titles, even though he considered himself an astronomer, according to that famous scientist article he is “mainly remembered for his contributions to mathematics”.

He rose up in the ranks within the scientist community and became the director of the astronomical observatory of Ujjain. According to a biography, Brahmagupta wrote a couple outstanding books, one being*Brahmasphutasiddhanta* which covered topics in “mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation…” and much more (O'Connor & Robertson).

The other book he wrote was called*Khandakhadyaka*, which covers similar things as the *Brahmasphutasiddhanta* did.

Although he wrote books mainly about astronomical, he had some controversial and excellent mathematical ideas. In the article found on the famous scientist site, it is said that Brahmagupta was the “first person in history to see zero as a number with its own properties”.

It is also said that he saw zero as a number you would get after subtracting a number by itself, and that “zero divided by any other number is zero”. Brahmagupta had also thought that zero divided by zero equals zero, which is wrong because this is actually undefined.

Brahmagupta may have had some wrong thoughts about math, but one of the formulas he discovered was right on. It deals with “determining the area of a cyclic quadrilateral given only the four sides lengths” (“Brahmagupta’s Formula”). According to that article, a nice proof for this formula is,

Given:

A cyclic quadrilateral with sides a, b, c, d, and the area of K can be found as:

\(K=\sqrt{(s-a)(s-b)(s-c)(s-d)}\)

where \(s=\dfrac{a+b+c+d}{2}\) is the semi-perimeter of the quadrilateral.

If we draw AC, we find that \([ABCD]=\dfrac{ab\sin{B}}{2}+\dfrac{cd\sin{D}}{2}=\dfrac{ab\sin{B}+cd\sin{D}}{2}\). Since \(B+D=180^{\circ}\), \(\sin{B}:\sin{D}\). Hence, \([ABCD]=\dfrac{(ab+cd)\sin{B}}{2}\). Multiplying by 2 and squaring, we get:

\(4[ABCD]^{2}=\sin^{2}{B}(ab+cd)^{2}\).

Substituting \(\sin^{2}{B}=1-\cos^{2}{B}\) results in \(4[ABCD]^{2}=(1-\cos^{2}{B})(ab+cd)^{2}=(ab+cd)^{2}-\cos^{2}{B}(ab+cd)^{2}\).

By the Law of Cosines, \(a^{2}+b^{2}-2ab\cos{B}= c^{2}+d^{2}-2cd\cos{D}$. $\cos{B}=-\cos{D}\), so a little rearranging gives \(2\cos{B}(ab+cd)=a^{2}+b^{2}-c^{2}-d^{2}\).

Now, to put everything together:

\(4[ABCD]^{2}=(ab+cd)^{2}-\dfrac{1}{4}(a^{2}+b^{2}-c^{2}-d^{2})^{2}\)

\(16[ABCD]^{2}=4(ab+cd)^{2}-(a^{2}+b^{2}-c^{2}-d^{2})^{2}\)

\(16[ABCD]^{2}=(2(ab+cd)+(a^{2}+b^{2}-c^{2}-d^{2}))(2(ab+cd)-(a^{2}+b^{2}-c^{2}-d^{2}))\)

\(16[ABCD]^{2}=(a^{2}+2ab+c^{2}+2cd-d^{2})(-a^{2}+2ab-b^{2}+c^{2}+2cd+d^{2})\)

\(16[ABCD]^{2}=((a+b)^{2}-(c-d)^{2})((c+d)^{2}-)a-b)^{2}\)

\(16[ABCD]^{2}=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-b+a)\)

\(16[ABCD]^{2}=16(s-a)(s-b)(s-c)(s-d)\)

\([ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}\) ("Brahmagupta's Formula").

]]>He rose up in the ranks within the scientist community and became the director of the astronomical observatory of Ujjain. According to a biography, Brahmagupta wrote a couple outstanding books, one being

The other book he wrote was called

Although he wrote books mainly about astronomical, he had some controversial and excellent mathematical ideas. In the article found on the famous scientist site, it is said that Brahmagupta was the “first person in history to see zero as a number with its own properties”.

It is also said that he saw zero as a number you would get after subtracting a number by itself, and that “zero divided by any other number is zero”. Brahmagupta had also thought that zero divided by zero equals zero, which is wrong because this is actually undefined.

Brahmagupta may have had some wrong thoughts about math, but one of the formulas he discovered was right on. It deals with “determining the area of a cyclic quadrilateral given only the four sides lengths” (“Brahmagupta’s Formula”). According to that article, a nice proof for this formula is,

Given:

A cyclic quadrilateral with sides a, b, c, d, and the area of K can be found as:

\(K=\sqrt{(s-a)(s-b)(s-c)(s-d)}\)

where \(s=\dfrac{a+b+c+d}{2}\) is the semi-perimeter of the quadrilateral.

If we draw AC, we find that \([ABCD]=\dfrac{ab\sin{B}}{2}+\dfrac{cd\sin{D}}{2}=\dfrac{ab\sin{B}+cd\sin{D}}{2}\). Since \(B+D=180^{\circ}\), \(\sin{B}:\sin{D}\). Hence, \([ABCD]=\dfrac{(ab+cd)\sin{B}}{2}\). Multiplying by 2 and squaring, we get:

\(4[ABCD]^{2}=\sin^{2}{B}(ab+cd)^{2}\).

Substituting \(\sin^{2}{B}=1-\cos^{2}{B}\) results in \(4[ABCD]^{2}=(1-\cos^{2}{B})(ab+cd)^{2}=(ab+cd)^{2}-\cos^{2}{B}(ab+cd)^{2}\).

By the Law of Cosines, \(a^{2}+b^{2}-2ab\cos{B}= c^{2}+d^{2}-2cd\cos{D}$. $\cos{B}=-\cos{D}\), so a little rearranging gives \(2\cos{B}(ab+cd)=a^{2}+b^{2}-c^{2}-d^{2}\).

Now, to put everything together:

\(4[ABCD]^{2}=(ab+cd)^{2}-\dfrac{1}{4}(a^{2}+b^{2}-c^{2}-d^{2})^{2}\)

\(16[ABCD]^{2}=4(ab+cd)^{2}-(a^{2}+b^{2}-c^{2}-d^{2})^{2}\)

\(16[ABCD]^{2}=(2(ab+cd)+(a^{2}+b^{2}-c^{2}-d^{2}))(2(ab+cd)-(a^{2}+b^{2}-c^{2}-d^{2}))\)

\(16[ABCD]^{2}=(a^{2}+2ab+c^{2}+2cd-d^{2})(-a^{2}+2ab-b^{2}+c^{2}+2cd+d^{2})\)

\(16[ABCD]^{2}=((a+b)^{2}-(c-d)^{2})((c+d)^{2}-)a-b)^{2}\)

\(16[ABCD]^{2}=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-b+a)\)

\(16[ABCD]^{2}=16(s-a)(s-b)(s-c)(s-d)\)

\([ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}\) ("Brahmagupta's Formula").

Pythagoras of Samos was a brilliant mathematician. He was born on the island of Samos around the year 569 BC, so it is said (Douglas, 2005). His parents were both apart of his life. His father was a merchant and Pythagoras was always with him, so he got to travel to many different places. |

Note, since Pythagoras lived so long ago and there is little to no account of his early life. It was, however, believed that he was heavily influenced by Pherekydes, Thales, and Anaximander.

Pythagoras became influential himself and was able to gather together a group of individuals who had the same beliefs as him. According to the encyclopedia he established “philosophical, political, and religious society whose members believed that the world could be explained using mathematics” (Atkins & Koth, 2002).

They had a huge belief of having only whole numbers and rational numbers. This group of individuals called themselves Pythagoreans and the school was called The School of Pythagoras.

In the encyclopedia it goes on to say that their beliefs were secrecy, vegetarianism, refusal to eat beans, immortality, and even reincarnation.

There are many other beliefs they had, but the main direction of Pythagoreans was “philosophy, mathematics, music, and gymnastics” it is also said that “reality is mathematical; philosophy is used for spiritual purification” and much more.

The thing that sets this group apart from most, is that men and women were allowed to join, which was unheard of during this time period.

One of the most famous theorem that came from Pythagoras and his followers was known as the Pythagorean Theorem. This theorem states that for any right triangle with legs of length a and b and hypotenuse if length c, then \(a^{2}+b^{2}=c^{2}\). A simple proof of this theorem could be constructed.

Given a right triangle ABC, let angle \(ABC = 90^{\circ}\).

Then, \(\sin(C) = \dfrac{AB}{AC}\) and \(\cos(C) = \dfrac{BC}{AC}\).

Using the identity \(\sin^{2}(C)+\cos^{2}(C)=1\), then \((\dfrac{AB}{AC})^{2}+(\dfrac{BC}{AC})^{2}=1\).

Therefore, after simplifying you will get \((AB)^{2}+(BC)^{2}=(AC)^{2}\) (Molokach).

It is believed that one of Pythagoras’ followers, Hippasus of Metapontum, had discovered something that went against their beliefs. In the legend it is said that this person was also killed by the group because of what he discovered. In an article written by Brian Clegg, he shows the actual problem that Hippasus was working on (2009).

According to stories Hippasus asked the question of how long is the diagonal of a 1x1 square. Using the Pythagoras Theorem, we can see that the answer to this problem is \(\sqrt{2}\). This is because \(1^{2}+1^{2}=c^{2}\), which implies \(c=\sqrt{1^{2}+1^{2}}=\sqrt{2}\).

Since, this could not be simplified to a rational number it seemed to fail their beliefs. This in turn, probably, caused the Pythagoreans to kill Hippasus.

]]>They had a huge belief of having only whole numbers and rational numbers. This group of individuals called themselves Pythagoreans and the school was called The School of Pythagoras.

In the encyclopedia it goes on to say that their beliefs were secrecy, vegetarianism, refusal to eat beans, immortality, and even reincarnation.

There are many other beliefs they had, but the main direction of Pythagoreans was “philosophy, mathematics, music, and gymnastics” it is also said that “reality is mathematical; philosophy is used for spiritual purification” and much more.

The thing that sets this group apart from most, is that men and women were allowed to join, which was unheard of during this time period.

One of the most famous theorem that came from Pythagoras and his followers was known as the Pythagorean Theorem. This theorem states that for any right triangle with legs of length a and b and hypotenuse if length c, then \(a^{2}+b^{2}=c^{2}\). A simple proof of this theorem could be constructed.

Given a right triangle ABC, let angle \(ABC = 90^{\circ}\).

Then, \(\sin(C) = \dfrac{AB}{AC}\) and \(\cos(C) = \dfrac{BC}{AC}\).

Using the identity \(\sin^{2}(C)+\cos^{2}(C)=1\), then \((\dfrac{AB}{AC})^{2}+(\dfrac{BC}{AC})^{2}=1\).

Therefore, after simplifying you will get \((AB)^{2}+(BC)^{2}=(AC)^{2}\) (Molokach).

It is believed that one of Pythagoras’ followers, Hippasus of Metapontum, had discovered something that went against their beliefs. In the legend it is said that this person was also killed by the group because of what he discovered. In an article written by Brian Clegg, he shows the actual problem that Hippasus was working on (2009).

According to stories Hippasus asked the question of how long is the diagonal of a 1x1 square. Using the Pythagoras Theorem, we can see that the answer to this problem is \(\sqrt{2}\). This is because \(1^{2}+1^{2}=c^{2}\), which implies \(c=\sqrt{1^{2}+1^{2}}=\sqrt{2}\).

Since, this could not be simplified to a rational number it seemed to fail their beliefs. This in turn, probably, caused the Pythagoreans to kill Hippasus.