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Project in MATH
Transcript of Project in MATH
Thomas Carlyle's geometric solution
In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. The idea of using such a circle to solve a quadratic equation is attributed to Thomas Carlyle (1795–1881). Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
Given the quadratic equation
x2 − sx + p = 0
the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation.
Methods used by the Babylonians
Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC), the other mainly Seleucid from the last three or four centuries BC. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia.
Pythagorean geometric solution
The Pythagorean theorem takes its name from ancient greek mathematician phytagoras who was perhaps the first to offer a proof of the theorem.
The Pythagorean theorem is pythagoras most famous mathematical contribution. according to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. the later discovered that square root of 2 is irrational and therefore cannot be expressed as a ratio of two integers. greatly troubled Pythagoras and his followers
Earliest methods used to solve quadratic equations
Even though Pythagoras is one of the most famous ancient philosophers and mathematicians he is only known through the work of his disciples.
He was born on the island Samos, and was educated by Thales, Anaximander and Anaximenes. Because of his aversion to Polycrates' tyranny, he left the island to settle in Crotoa in South Italy. There, he founded the Pythagoreanismian movement: a society with aims in the fields of religion, politics and philosophy.
*they were devout in their belief that any two lenghts were integral multiplies of some unit length
The pythagorean theorem states that "the area of square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides"
The Egyptians applied the equation which was to become Pythagoras' Theorem long before Pythagoras
was born. They used it mainly for building and measuring out land for agricultural use.
it was the Greeks who formulated these into theorems. That is they raised the level of mathematics to a rigid discipline where generalizations were formulated into truths. Pythagoras was one such person and the famous equation
x^2 + y^2 = z^2
Pythagorean geometric solution 4
Pythagorean geometric solution 2
Pythagorean geometric solution 3
A carpenter needs to add 2 braces to a barn door. if the door measures 12x16ft how much wood will he need for both braces?
12^2 + 16^2=c^2
get the square root of 400 and c^2
2c = 40
Methods used by the Babylonians 2
In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, and baked hard in an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600 BCE, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation to accurate to three sexagesimal places (seven significant digits).
Lasheras, Marc Gabriel
Methods used by the Babylonians 3
The Babylonian system of mathematics was sexagesimal (base 60) numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a superior highly composite number, having factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 (including those that are themselves composite), facilitating calculations with fractions. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values (much as in our base ten system: 734 = 7×100 + 3×10 + 4×1). The Sumerians and Babylonians were pioneers in this respect.
Methods used by the Babylonians 4
As well as arithmetical calculations, Babylonian mathematicians also developed algebraic methods of solving equations. Once again, these were based on pre-calculated tables.
To solve a quadratic equation, the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form
x^2 + bx = c
where here b and c were not necessarily integers, but c was always positive. They knew that a solution to this form of equation is
x = -b/2 + squareroot of (b/2)^2 + c
Methods used by the Babylonians 5
and they would use their tables of squares in reverse to find square roots. They always used the positive root because this made sense when solving "real" problems. Problems of this type included finding the dimensions of a rectangle given its area and the amount by which the length exceeds the width.
Tables of values of n3 + n2 were used to solve certain cubic equations. For example, consider the equation
Multiplying the equation by a2 and dividing by b3 gives
Substituting y = ax/b gives
which could now be solved by looking up the n3 + n2 table to find the value closest to the right hand side. The Babylonians accomplished this without algebraic notation, showing a remarkable depth of understanding. However, they did not have a method for solving the general cubic equation.
The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is
x(x − s) + (y − 1)(y − p) = 0.
The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle)
x2 − sx + p = 0.
Thomas Carlyle's geometric solution 2
Thank you for listening!
Comparison of Babylonian methods to present
Their methods are almost the same as the
ones we use today. The only difference is that
it was made more simple today.
Comparison of Carlyle's circle method to present
The Carlyle's circle method is more complex and it needs a too much detailing in it because you can only find its answer by graphing a perfect circle which is difficult unless you have an instrument like a compass.
Comparison of Pythagorean geometric solution to present
It is one of the ways we solve our problems in the present. It is another way to get the answer of a quadratic equation easier than the others.