Taming the Infinite

Chapter 1

 

Tokens, Tallies and Tablets

 

During all my entire life I hate math and every class of school or university that is related with math. I now that life is math and that in everything we know it is math but that is not what we have been thought in school.  In school in formed the idea that for approve math I have to know how to resolves some algebra or geometrical problems but the reality is that math is more that win a test. This book opens my eyes to really understand the origin of math and how all is connected and how interest all of it are and I have ignore for many years.

 

 

This first chapter explains how math began with numbers and how mathematics is universal. Numbers are the symbols of how math is represented.  “The history of mathematics begins with the invention of written symbols to denote numbers”.

 

Without numbers, the civilizations could not exist they are in everywhere.

 

History of numbers

 

-Numbers started 10,00 years ago with little clay tokens. Tokens where used as financial and tax purpose.

 

 

-Then the clay envelopes with symbols on it evolved. This method was explained by Ian Stewart as the possible beginning of inflations because the Bureaucrats of Mesopotamia didn’t need the contents that are in the envelopes.

 

-37,000 years ago were find in a cave the oldest tally marks//// this are the first numerals.

 

-The Babylonian system with BASE 60

 

- The Egyptians used drawings for whole numbers and an eye for the fractions.

 

-Now our culture uses sophisticated mathematics for science, technology and increasingly for commerce too.

 

“The main mathematics that does lie on the surface is arithmetic.”

 

 

 

 

 

Chapter 2

The logic of Shape

First steps in geometry

 

This chapter was about the beginning of Geometry.

 

Math has two types of reasoning own is symbolic the one that originated in numbers and the other is visual.

 

The first systematic use of diagrams, together with a limited use of symbols and a heavy dose of logic, occurs in the geometric writings of Euclid of Alexandria.

 

Euclid insisted that any mathematical statement must be given a logical proof before it could be assumed to be true.

 

Euclid combined the use of logical structure of proof with pictures.

 

Euclid started by listing a number of definitions, postulates and common notions (Logic).

 

Pythagoras

 

The most influential aspect of the Pythagorean cult´s philosophy is the belief that the universe is founded on numbers. Also they understood that mathematics is about abstract concepts not realities.

 

The Pythagoreans recognized the existence of nine heavenly bodies, Sun, Moon, Mercury, Venus, Earth, Mars , Jupiter and Saturn plus the Central Fire. Also the irrationals was invented by Eudoxus around 375 BC.

 

The Golde mean

 

 

-Numerical Value 1.618

-Irrational

-Regular pentagons are directly connected with the extreme and mean Ratio

 

Archimides

 

He specialize his works in spheres, circles and cylinders that now is associated with (pi) 3.14159. To estimate it he compared the circumference of a circle with the perimeters of two series of polygons inside the circle and surrounding it.

 

Chapter 3

Notations and Numbers

Where our number symbols come from

 

This chapter explains the different symbols that each civilization used and also explains how the numerical system and his application has evolutionated.

 

The different number systems are:

 

-Roman numerals: Letters

Example: 2012 = MMXII

 

-Babylonian sexagesimal system: base 60 fingers

 

-Egyptian number symbols

 

-Mayan: 20 numbers

 

-Greek Numerals: used letters.

 

-Indian: used ten symbols to denote decimal digits, also the positional notation was in use in India from 400 onwards.

 

Indian mathematicians: Aryabhata, Brahmagupta and Mahavira.

 

-Europe: Leonardo of pisa (Fibonacci) he introduced Hindu-Arabic number symbols to Europe.

 

-Negative Numbers

 

“Counting rods” instead of abacus by Chineses.

Hindu mathematicians found negative numbers useful to represents debts in financial calculations.

 

Chapter 4

Lure of the unknown

X marks the spot

 

Now in this chapter focuses on algebra, equations and also cubic equations. It explained how they began to be applied and how really worked in reality.

 

 

Babylonians had great influence on Algebra. They invented the linear equations (ax+b=0) 

 

And the quadratic: (ax2+bx+c=0).

 

+ and – also was useful in commerce and many other symbols were invented that we use today. Like x, =, <. >. ( ), √ .

 

 

Chapter 5

Eternal Triangles

Trigonometry and Logarithms

 

I love trianglesm this before doing the proposition 1 of Euclid´s Elements. Euclidean geometry is based on trianles, and most other interesting shapes, such as circles ans ellipses, can be approximated by polygons.

 

Trigonometry measure triangles, and also is usefull ofr all mathematics.

 

Improtant functions:

 

 

The logarithm (Log x, logxy =logx + logy)

Base 10 logarithms

         Number e

                                               

 

Chapter 6

Curves and Coordinates

Geometry is Algebra is Geometry

 

                      Fermants was the first person that describes coordinates, he discovers also the connection between algebra and geometry.

 

Descartes modern notion of coordinates.

 

The coordinates helps to represent the functions in a graphical way.

 

Chapter 7

Patterns in Numbers

 

                            The numbers that can be expressed as the product of two smaller numbers are said to be composite.   Those who cannot be expressed are primes.

 

First few numbers that are primes: 2 3 5 7 11 13 17 19 23 29 31 37 41

 

Euclid: First number theory

Diaphanous: Triangles

Gauss: Related number theory to geometry, Euclid’s polygons.                   

 

Chapter 8

The system of the world       

The invention of Calculus

 

“The most significant single advance in the history of mathematics was calculus, invented independently around 1680 by Isaac Newton and Gottfried Leibinz. The row soured relations between English mathematicians and those of continental Europe for a century, and the English were the main losers. “

 

Calculus

Is the mathematics of instantaneous rates of change.

 

Differential Calculus: provides methods for calculating rates of change, and it has many geometric applications, in particular finding tangents to curves.

 

Derivative of a function: the rate at which  f(x) is changing compared to how x is changing (the rate of change of f (x) with respect to x. 

 

 F( x + h ) – f (x) / h

 

 

 

Integral calculus does the opposite of differential, it given the rate of change of some quantity it specifies the quantity itself. Geometric applications of integral calculus include the computation of areas and volumes.

 

-Reverse process of derivation. 

-∫ g (x) dx

 

 

Calculus is about functions: procedures that take some general number and calculate an associated number.

 

 

Chapter 9

Patterns in Nature

Formulating laws of physics

 

“The main message in Newton´s Principia was not the specific laws of nature that he discovered and used, but the idea that such laws exist- together with evidence that the way to model nature´s laws mathematically is with differential equations.”

 

Newton role  was essential to science     the laws that he proposed were useful to this date.             

 

Types of different equations:

 

Ordinary : (ODE) refers to an unknown function y of a single variable x, and relates various derivates of y, such as dy/dx and dy/dx.

 

Partial: (PDE) refers to an unknown function Y or two or more variables, where X  and Y are coordinates in the plane and T is time.      

 

Wave equation: become one of the most impotant equations in mathematical physics, because waves arise in many different circumstances.

 

Applications of PDE:

 

-Electromagnetism

-Music (Drums and chord)

-Heat Flow

-Fluid dynamics.

 

ODE

-Energy

 

 

Chapter 10

Impossible Quantities

Can negative numbers have square roots?

 

                                                                  

Natural numbers 1,2,3 the integers, also include zero and negative whole numbers.

 

Rational numbers: composed of fractions p/q where p and q are integers and q is not zero.

 

Real numbers: Generally introduced as decimals that can go on forever. Whatever that means- and represent both the rational numbers, as repeating decimals and irrational numbers like Pi.

 

 

Square root of -1 = imaginary number

 

 

 

 

 

 

Chapter 11

Firm Foundations

Making calculus make sense

 

“Calculus has been developed to be an indispensable tool for the study of the natural world and problems that arose from this connection led to a  wealth of new concepts and methods”

 

in the 19th century Mathematiciasn started doubting the use of a function so they separate concepts in:

 

Meaning of the term, function.

Representing a function by a formula, a power serias

Fourier series or whatever.

Properties the function possessed.

Guaranteed which properties.

 

Example: Single polynomial defines a continuous function. A single Fourier seris might not.

 

 

Chapter 12

Impossible Triangles

Is Euclid´s geometry the only one?

 

Desargues: discover the first non-trivial theorem in projective geometry. 

Proves: Triangles ABC and A ´ B ´C are in perspective, which means that the three lines AA´, BB´AND CC´all pas through the same point O.

 

Euclid´s axioms

 

Legendre:

Adrien-Marie was the one that discovered the existence of Similar Triangles.

Similar Triangles: are the ones that having the same angles, but with edges of different sizes.

 

Saccheri

 

Without using the Fifth Postulate, Saccheri proved that angles C and D are equal. So that left two distinct possibilities:

 

-Hypothesis of the obtuse angle: Both C and D are greater than a right angle.

 

-Hypothesis of the acute angle: both C and D are less than a right angle.

 

Similar triangles: having the same angles but different size.

 

Non Euclidean geometry: natural geometry of a curved surface.  

 

Chapter 13

The Rise of Symmetry

How not to solve an equation

 

 

Group theory started with the Babylonians and is the framework for studying symmetry. Algebra has to do with symmetry. In this chapter Abel proved that trying to prove the algebraic solutions for equations of the 5th degree that it is impossible but Galois is the one that that look that the solutions for algebraic equations is related with symmetry.

 

Chapter 14

Algebra Comes of Age

Numbers give way to structures

 

Number Theory: By understand reciprocity laws and Fermat´s last theorem. Not change distances or angles.

 

Theory of Invariants: algebraic expressions that do not change when certain changes of variable are performed.

 

Euclidean Group:

Do not change distances and angles. Quantities not change when transformation from the group is applied.

 

 

-Translation 

-Rotation

-Reflection

-Glide reflection

 

Kinds of geometry:

-Elliptic Geometry

-Hyperbolic Geometry

-Projective Geometry

 

LIE group: most significant symmetries (satisfies algebraic identities and topological manifolds

 

.

Other types of algebraic systems:

• Ring

• Field 

Algebra

 

Chapter 15

Rubber Sheet Geometry

Qualitative beat quantitative

 

Rubber sheet geometry more properly known as topology.

Topology is the geometry of shapes that can be deformed or distorted in extremely convoluted ways.

 

Lines can bend, shrink or stretch; circles can be squashed so that they turn into triangles or squares.

 

The most important aspect of this geometry is continuity.

3 dimensions and 2 dimensions

 

 

 

Chapter 16

The Fourth Dimension

Geometry out of his world

 

In the four dimensions three planes of Space are three and the fourth is the time.

 

Calculus intention, which was intended to be more comprehensible but it wasn’t.

 

Hyper numbers: l, I, and k quaternions by any number of units. Ç

 

Commutative law of multiplication, ab= ba. 

 

All of these led to the same description of a vector (x, y, z).

 

Chapter 17

The Shape of Logic

 

Putting Mathematics on fairly firm foundations

 

Numbers are just conceptual concept they are just symbols that represent something what is important is what that symbols represent.

 

Axioms for Whole Numbers:

• Exists a number 0

• Every number has a successor s(n) which we think of as  (n + 1)

• If P(n) is a property of numbers such that P(0) is true, and whenever P(n) is true than P(s(n)) is true, then P(n) is true for every n. (Principle of Mathematical Induction).

 

 

Cantor

 

Transfinite numbers : different sizes of infinity. Infinity, invarious guises, seems unavoidable in mathematics.

 

Gödel: Proved that if mathematics is logically consistent, then it is impossible to prove that.

 

Chapter 18

How likely is that?

The rational approach to chance

 

Probability is one of the most used mathematical techniques in this time.

 

Example:

 

All probabilities are between 1 and 0

1 certain and 0 impossible.

 

 

 

Chapter 19

Number Crunching

Calculating Machines and computational mathematics

 

This chapter focuses on the use of numbers in the actual time. All digital computers, the calculators etc. worked with binary notations represented as strings of 0 and 1.

 

Mathematics in like a never ending thing because one solve of one problem open another and another, it is infinite.

 

Chapter 20

Chaos and Complexity

Irregularities have patterns to

 

Chaos: formless disorder.

Newton is deterministic in the most solutions to a differential equation given initial conditions.

 

Theoretical monsters helped mathematics breakaway traditional shapes likes rivers clouds etc.

 

-Snowflake curve

-Hilbert´s space filling curve

-Sierpinski Gasket