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Three Greatest Mathematicians - Archimedes, Gauss and Newton

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Danny Fischer

on 10 June 2014

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Transcript of Three Greatest Mathematicians - Archimedes, Gauss and Newton

Background of Newton
Sir Isaac Newton was an English physicist and mathematician who is widely regarded as one of the most influential scientists of all time and as a key figure in the scientific revolution.
Newton's Principia formulated the laws of motion and universal gravitation that dominated scientists' view of the physical universe for the next three centuries. It also demonstrated that the motion of objects on the Earth and that of celestial bodies could be described by the same principles. By deriving Kepler's laws of planetary motion from his mathematical description of gravity, Newton removed the last doubts about the validity of the heliocentric model of the cosmos.
Newton built the first practical reflecting telescope and developed a theory of color based on the observation that a prism decomposes white light into the many colors of the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound. In addition to his work on the calculus, as a mathematician Newton contributed to the study of power series, generalised the binomial theorem to non-integer exponents, and developed Newton's method for approximating the roots of a function.
Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge. He was a devout but unorthodox Christian and, unusually for a member of the Cambridge faculty, he refused to take holy orders in the Church of England, perhaps because he privately rejected the doctrine of the Trinity. In addition to his work on the mathematical sciences, Newton also dedicated much of his time to the study of alchemy and biblical chronology, but most of his work in those areas remained unpublished until long after his death. In his later life, Newton became president of the Royal Society. He also served the British government as Warden and Master of the Royal Mint.

Newton's Influence
He came up with the Binomial Theorem and was one of the two creators of calculus.
These discoveries represented a quantum leap in the fields of math and science allowing for calculations that more accurately modeled the behavior of the universe than ever before.
Without these advances in math, scientists could not design vehicles to carry us and other machines into space and also plot the best and safest course.
Calculus gave scientist the tools to set up a theoretical model of a situation and still account for varying factors.
This basic knowledge would help scientist such as Einstein to be able make even greater discoveries such as the Theory of Relativity and Nuclear Fission.


Binomial Therom
Background of Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor and astronomer.
Archimedes is regarded as one of the leading scientists in classical antiquity.
Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors
.
Background of Gauss
Johann Carl Friedrich Gauss was a German mathematician and physical scientist who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, and optics.
Sometimes referred to as the Princeps mathematicorum (Latin for "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.
Introduction
Throughout the ages of mathematics there have been many mathematicians that have influenced math in one way or another and brought a new way of solving problems. Archimedes of Syracuse, Carl Friedrich Gauss and Isaac Newton were three of the greatest mathematicians and helped influence maths in a major way.
Three Greatest Mathematicians - Archimedes, Gauss and Newton
Archimedes Influence
Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.
He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.
He also defined the spiral bearing his name, formulae for the volumes of solids of revolution, and an ingenious system for expressing very large numbers.
Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.
The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape.
Gauss's Influence
Gauss found that that a regular polygon with 17 sides could be drawn using just a compass and straight edge.
Gauss was able to come up with a method called "The method of least squares" which is helpful when analyzing data.
Gauss came up with the equation for finding costructible angles.
Gauss also invented 'The law of quadratic recipocity' which is also known as the 'Golden Theorem.'
Method to Expand Polynomials
The binomial theorem states a formula for expressing the powers of sums. The most succinct version of this formula is the shown immediately below.
Method of Exhaustion
Through proof by contradiction he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of pi. In Measurement of a Circle he did this by drawing a larger regular hexagon outside a circle and a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step.
After four such steps, when the polygons had 96 sides each, he was able to determine that the value of pi lay between 3(1\7) (approximately 3.1429) and 3(10\71) (approximately 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to pi multiplied by the square of the radius of the circle

Method of Exhaustion Graph
The Quadrature of the Parabola
In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is (4\3) times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio (1\4)
If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + .... which sums to (1\3)
Quadrature of the Parabola Example
Volume of an Object with an Irregular Shape
Volume is the space an object takes up
The process to find the volume of an irregular shape is called measuring. In order to find the volume of an irregular shape you can measure the displacement of a liquid. When you place an irregular object in water you measure the amount changed to determine the object's volume.
One way of solving the volume of an object with a irregular shape is using the displacement method.
Or you can use the regular volume formula which is 'volume = mass\density'
Another way is to draw it on graph paper and count the blocks or you can break the irregular objects down into regular shapes, solve their individual volume and then add them together.
Example of Volume of an Irregular Shape
Constructible Angles
There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle.
The angles that are constructible form an abelian group (In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers.) under addition modulo 2πpi(which corresponds to multiplication of the points on the unit circle viewed as complex numbers).
The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example the regular heptadecagon (the seventeen-sided regular polygon) is constructible because *see next circle for equation*
The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots).
The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes (is a positive integer of the form Fn = where n is a non-negative integer
In addition there is a dense set of constructible angles of infinite order.

Constructible Angle Equation and Example
Equation:

Example:

Method of Least Squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.
The most important application is in data fitting. The best fit in the least-squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the independent variable (the 'x' variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares.
Line of Best Fit (Least Sqaure Method)
A line of best fit is a straight line that is the best approximation of the given set of data.
It is used to study the nature of the relation between two variables.
It is used to study the nature of the relation between two variables.
A line of best fit can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal (and the line passes through as many points as possible).
A more accurate way of finding the line of best fit is the least square method .
Use the following steps to find the equation of line of best fit for a set of ordered pairs.
Step 1: Calculate the mean of the x-values and the mean of the y-values.
Step 2: Compute the sum of the squares of the x-values.
Step 3: Compute the sum of each x-value multiplied by its corresponding y-value.
Step 4: Calculate the slope of the line using the formula:
where n is the total number of data points.
Line of Best Fit (Least Square Method) Part II
Step 5: Compute the y-intercept of the line by using the formula:


Where are the mean of the x- and y-coordinates of the data points respectively.
Step 6: Use the slope and the y -intercept to form the equation of the line.
Example: Use the least square method to determine the equation of line of best fit for the data. Then plot the line.



Solution: Plot the points on a coordinate plane.

Line of Best Fit (Least Square Method) Part III
Calculate the means of the x-values and the y-values, the sum of squares of the x-values, and the sum of each x-value multiplied by its corresponding y-value.
Calculate the slope.
First, calculate the mean of the x-values and that of the y-values.
Use the formula to compute the y-intercept.
Line of Best Fit (Least Square Method) Part IV
Use the slope and y-intercept to form the equation of the line of best fit.
The slope of the line is –1.1 and the y -intercept is 14.0.
Therefore, the equation is y = –1.1 x + 14.0.
Draw the line on the scatter plot.
The easiest way to understand the binomial theorem is to first just look at the pattern of polynomial expansions below.
•(x+y)² =x²+2xy + y²
•(x+y)^3 = x^3 + 3x²Y+ 3xY2 + y3
•(x+y)^4 =x^4+ 4x^3Y +6x²Y² + 4XY^3 + Y^4
The generalized formula for the pattern above is known as the binomial theorem
By: Danny Fischer
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