Theorem:
If A, B and C are collinear points, then the real numbers x, y, z not all zero can be found such that
x + y + z = 0 xA + yB + zC = 0.
and also the inverse of this thorem is true: If three such numbers not all zero can be ound then the points are collinear.
Conversely
Theorem:
If A, B, C are three given points which are not collinear and we can find three real numbers x, y, z such that x+y+z = 0 and xA + yB + zC = 0,
then we must have x=y=z=0.
Lonicera ”Dropmore Scarlet”
Clematis”Ville de Lyon”
A clever way to express the equation of a line is to use the parameter form.
If C is any point on the line determined by two points a and B then we may always write
C = (1-t) A + t B
where the ratio of the real numbers t/(t-1) = AC / CB. t is the parameter with values from 0 to 1.
I give a proof for this statement:
Let AC = t AB.
This equation can be translated into
C-A = t (B-A) → C = (1 – t) A + t B.
It is also a fact that
AB = AC + CB
AC = t (AB) → AC = t(AC + CB) → (1 -t)AC and t/(t-1) = AC/CB
Q.E.D.
Venus transit June 6th 2012
Picture of Venus transit photographed with an Hα- filter.
λ = 6566 A. Resolution 1 A.
In this picture it is possible to cpmåare the size of the Earth to that of the Sun since Venus is of the same size as the Earth.
The vectors P1, P2, P3, … are said to be linearly dependent if the real numbers k1, k2, k3, .. not all zero can be found so that
k1P1 + k2P2 + ….. + knPn = 0.
Since it is possible to solve for e.g. P1= -k2P2/k1-k3P3/k1-….
This means that all the vectors lie on the same line through the origin.
Conversely, if two vectors lie along the same line they are linearly dependent vectors.
A set of vectors which are not dependent are said to be
linearly independent.
An example of linearly independent vectors are E1 = (1,0) and E2 = (0,1).
They form the basis for the two dimensional vector-space. In favt any two linearly-independent vectors can form the basis of a two dimensional vector space. The dimension is identical to the number of vectors necessary.
Two geometric formulae:
Menelaos theorem: A line cuts the sides BC, CA and AB of a triangel in the points L, M and N respectively. If L = xB + x’C + y’A, M = yC + y’A and
N = zA + z’B, where x +x’ = y + y’ = z +z’ = 1,
then xyz = -x’y’z’.
Rhododendron catawbiense
A line has length between two points but no width.
A point has no components i.e. it can’t be divided into parts.
A straight line traces the closest distance between two points.
A curved line has no straight segments.
A surface or a plane is a completeky flat surface between its border lines. Lines constitutes the borders of planes.
Two straight lines are said to intersect each other in a point where they meet. The inclination between those lines is called the angle between the lines.
The point of intersection is called the vertex of the angle.
A circle is a planar figure limited by a curved line called the perimeter or periphery.
The distance from the center to the perimeter is called the radius and is everywhere on the circle the same.
The diameter is a straight line from periphery to peiriphery through the center of the circle. It divides the circle into two equal parts-half-circles.
A triangle is a figure made up of three straight lines. These are called its sides.
A quadrilateral figure is made up of four straight lines.
A polygonal figure is borders by more than four straight lines.
A triangle composed of three equal sides is called equilateral.
A triangle having two equal sides is termed isosceles triangle.
A right triangle has a perpendicular angle.
Paeony ‘Karl Rosenfield’
There are two types of proof:
Direct proofs: Where it is proved (deduced or induced) from the basic axioms, definiyions or earlier proved theorems that the statement is correct.
Indirect proofs: The principle for this type of proofs is that it is proved that the opposite of the stated proposition cannot be true since this would lead to a contradiction or to something impossible.
In order to solve a geometrical problem it is often necessary to perform geometrical construction. This means e.g that the solution may be facilitated by
connecting given dots
extrapolate given lines
Bisect given lines or angles in halves,
Construct a line parallell to a given line
Constuct a line perpendicular to a given line.
construct circles etc.
Euclid then proceeds to define the equal-sign (2+3 =5), inequality sign ( A > B) and what is meant by addition (the sum of parts) and multiplication )repeated adition of the same number (A +A +A = 3A) and the difference between two quantitites A – B which eguals the quantity you must add to B to get A. .
Last Wednesday the planet Venus passed between the Sun and the earth. A rare astronomical event that will not happen again until 2117.
Venus transit
A photograph of the Venus – Jupiter conjunction 12/3 2012
I have started to read Euclid’s ‘Elements’. This is perhaps the most influential book in mathematics and was used for more than two-thousand years in schools all over the western world. My edition is written by Christian Fredrik Lindman lecturer of mathematics at the upper secondary school in Strängnäs, Sweden and it was printed in 1867.
In the first chapter many central terms and mathematical entities are defined.
Mathematics is the science dealing with quantities.
A quantity is an entity that can be increased or decreased by the addition or subtraction of more of the same quantity.
Geometry is the science dealing with quantities that have an extension in space.
Definition is a list of characteristics typical of that object
Postulat is a theorem that cannot be proved.
An axiom is an obvious statement.
Problem is a theorem where you must show how the problem shall be solved and then prove that it is solved.
Theorem Is a statement of a mathematical truth that must be proved.
Corollary is a theorem that is a direct consequence of another theorem,
Rosa ‘Flora Danica’
Since the divergence of a vectorfield provides us with the number of field lines radiating outward from the source of the vectorfield it can be intuitively understood that the volume integral of the divergenbde of F equals the surface integral of F over the closed surface A:
∫ ∇⋅F dv = ∫F⋅dS
Another important theorem in vaectoranalysis is Stokes Theorem
∫∇×F ds = ∫F⋅dl
De första talen människan använde sig av var förmodligen positiva heltalen de sk. naturliga talen. De användes för att ange kvantiteter av olika ting: fem tomater, 10 persikor etc..
Då människan började med handel kunde man bli skuld satt och då var de negativa talen användbara för att beskriva detta.
Hur skulle man fördela jaktbyten och dylikt? Då behövdes bråktal sk rationella tal.
De vetenskapliga framstegen och de indisk-arabiska siffrorna krävde större noggrannhet varvid decimaltalen infördes. Alla tal på tallinjen kallas de reella talen.
Alla cirklar är likformiga med förhållandet pi mellan omkrets och diameter. Detta tal är inte möjligt att uttrycka vare sig som decimaltal eller bråktal. pi är ett irrationellt tal.
The prime-numbers can be regarded as the building blocks of the number system.
Prime numbers are numbers that can only be divided by one and itself.
Every number cn be factorized into primenumbers.
Christian Goldbach proposed that every even number can be formed as the sum of two prime numbers. Goldbach lived in Königsberg during the 17th century.
Gauss himself devoted great attention to priimenumbers and proposed the folloeing theorem: The number of primes smaller than N equals N/ln(N).
This was proven byu Charles Hermite in the 19th century.
Prime numbers smaller than a cetain number can be generated with a method from antiquity: the sieve of Erathostenes. another possibility is to generate Mersenne primenumbers 2n – 1.
Exercise 1 in Sjunnesson Ma 1c:
Är 2 * 3* 5* 7 + 3
Ett primtal?
Lösning: bryter man ut tre fås talet 3(2*5*7 + 1). Detta tal är delbart med tre eftersom tre är en faktor i talet. 3(2*5*7+1)/3 = 2*5*7 + 1. alltså är talet inget primtal då det ju är delbart med tre.
Save the Tiger!
Miniräknare/calculator
Kristians Kunskapsbank 
