mathematics development in europe

Mathematics Development in Europe

Overview

Overview

• Transmission period• Fibonacci and the 13th century• Cubic and quadratic equations• Euclidean postulate• Non Euclidean pioneers• Discovery of non Euclidean• Contribution: Gauss, Newton, Fermat and

Euler

Transmission period

• The collapse of Rome and the general chaos that followed has no great advancements in the mathematical community in it.

• The Dark ages and then the Middle Ages were upon the land and civilization let alone the science of mathematics was having trouble surviving the times.

Transmission period

Fibonacci and the 13th century

• Born-died 1170-1240• He helped introduce the Hindy – Arabic

numerals.

Fibonacci Series

• Also known as arithmetical sequence, invented in 1225 by Leonardo Fibonacci.

• Each number in the series which begins 1, 1, 2, 3, 5, 8, 13, 21, 34 is, after the first two figures, merely the sum of the previous two numbers.

• These numbers have occurred in nature and the Arts.

Fibonacci Series

• rabbits, bees, sunflowers,pinecones,...

• reasons for seed-arrangement (mathematical)

• connections to the Golden number

Cubic and quadratic equations

• Girolamo Cardano (1501 – 1576) who wrote cubic equation in Ars magna in 1545.

The Cubic Equations

• A particular equation• The general case

A particular equation

• The idea of Cardano about a particular equation can be written as

• Cardano’s idea is to introduce two new

variables and : = 2 ( a trick from Pythagorean triple)


• Rewrite equation as 3 = • here, Cardano’s solution method is a bag of

tricks. His idea is to observe, just by guessing.


• Take 3 = • verify + 3()()

[ - 3 + 3 - ] + (3 - 3 =

= • Thus, let , and determine . = 20 + • = 20 +


• As = 2, so (, then = • Let = , then equation = 20 + becomes = 20

+ • multiplying , = 20 + 8• Rearrange - 20 – 8 = 0 (this equation

becomes quadratic)


• Solve this quadratic equation - 20 – 8 = 0• = = = 10 =


• So = =

The general case

• Consider a cubic equation + a + b + c = 0• Make the change of variable = + a + b() + c = 0( - 3.. + 3. . - ) + a( - 2. .t + ) + b () + c = 0 + (-3. + ) - + b) + (- + - )= + (- + b) + ( - ) = 0

The general case

• Here a polynomial has no square term, and assume that - then make the change of variable =,

• [ …

The solution of a quadratic equation

• General quadratic equation: a + b + c = 0, and assume a • Divide out by a: [a + b + c ] = 0• this reduce to + + = 0• write as [+ ] + = 0


• This expression [+ ] can be turned into a perfect square: divide the coefficient of by two and square it, then add the result on as constant term,

• =


• The result: [ + ] + = or = - • It is convenient to put the right hand side over

a common denominator so that we have =


• Taking the square root of both sides, and as a positive real number has both a positive square root and negative square root, then

• = or =


• A little algebraic manipulation, = - • or =

Pierre de Fermat (1601- 1665)

• Fermat was a French mathematician who is best known for his work on number and theory

• One of his last theorem’s was proven by Andrew Wiles in 1994.


• Whilst in Bordeaux, Fermat produced work on maxima and minima, which was important. His methods of doing this were similar to ours, however as he has not a professional mathematician his work was very awkward.


• Fermat’s last theorem was that if you had the equation, xn + yn = zn

• This equation has no nonzero integer solutions x, y, and z when the integer exponent n can be no more than two.

• When n is more than two, the equation does not work.

Fermat’s Method• One of the ideas of calculus is to find the

tangent line to a given curve. the tangent line to a circle. in classical geometry, the tangent line to a

circle C at a point P is the line passes through P and is perpendicular to the radius P.

Fermat’s Method

• The tangent line to the curve is interpreted as the line passes through P and touches the curve at P.

• But, to Fermat, the tangent line has the special feature; it only intersect the curve at one point.

Fermat’s Method

• The idea of Fermat is illustrated by the concept of slope. If a line is given and two points ( ) and ( ) on that line, then the slope is

Example

• By using Fermat’s idea, find the tangent line to the curve at the point (2,4).

Solution

The intersection of the line with the curve can be calculated by

Equating the above two expressions, then

= In the form of quadratic equation,

- Using quadratic formula, then

Rewrite this as

Here, (m – 4) is a perfect square, then the solution becomes

Solution

• So, only one m is needed, but the equation looks two solutions. Thus, (m – 4) = 0 or m = 4.

• The equation for the line that passes through the point (2, 4) and intersects the curve is the line with slope m = 4;

The derivative

• Through Fermat, differential calculus emerged. This idea led to Descartes, Newton, Leibniz, and others develop mathematical knowledge for calculating tangents, finding maxima and minima functions, and performing operations in analysis and mechanics.

the tangent line• Given the graph of the function . Determine

the ‘slope’ of the graph at point

• Here, the tangent line (or the slope of tangent line of the graph) is the tangent line that approximates the graph at that point.

the tangent line

• However, the slope of the line at (c, f(c)) can be determined by the slopes of secant line (a line connecting two different points nearby on the curve – here (c, f(c)) and (c + h, f(c + h))

the tangent line

• Here, the term limit refers to the expression as tends to 0.

= - Newton quotient• This limit, the derivative of f at c or f’(c). When

the limit exists, then the function f is differentiable at c.

The theory of limit

• It is deep and subtle. Discovered more than 2000 years ago, and never got it right. Even Isaac Newton used limits (with trepidation), but never really understood them.

Example

• Calculate the slope of the tangent line to the graph of at 2

Solution • As the slope, m = , then = = = = = [ - 6h + 9] = 9

Fermat’s Lemma

• It is based on a geometric observation about differentiable functions.

• P is vertically higher than points nearby, and is called a local maximum, whereas Q P is vertically lower than points nearby, and is called a local minimum.

Fermat’s Lemma

• Here, the point P on the graph goes neither uphill nor downhill or in other words, the tangent line is horizontal (its slope is zero).

• Thus, the derivative of a function at a point of differentiability where the function assumes a local maximum is 0.

Fermat’s Lemma

• Here, the point Q on the graph goes neither uphill nor downhill or in other words, the tangent line is horizontal (its slope is zero).

• Thus, the derivative of a function at a point of differentiability where the function assumes a local minimum is 0.

Fermat’s Lemma

• The two rules are the content of Fermat’s Lemma.

• A point is called a critical point for the function if ’() = 0.

Example

• Sketch the graph of the function: - 3 + 4

Solution

= = = = (6 + 6 + 2 - 6 -3 - 12) = 6 - 6 - 12

Solution

as 0 0 = = 6 - 6 – 12 The quadratic factors: 0 = 6( + 1)( – 2)• Thus, = -1 , = 2

Solution

• Checking: for = -1, 3/2) = 21/2 1/2) = -15/2• Here, the graph is going uphill to the left of

= -1, and downhill to the right of = -1. thus, = -1 is the location of a local maximum.

Solution

• Checking: for = 2, 3/2) = -15/2 /2) = 21/2• Here, the graph is going downhill to the left of

= 2, and uphill to the right of = 2. thus, = 2 is the location of a local minimum.

Solution

• As and then the graph is shown as:

Euler (1707 –1783)

• important discoveries in calculus… graph theory.• introduced much of modern mathematical

terminology and notation, particularly for mathematical analysis,

• renowned for his work in mechanics, optics, and astronomy.

• Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time

Euler (1707 –1783)

• son of Protestant minister. Was minister but studied mathematics. Renowned for Algebra,Calculus

mathematics development in europe

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