Euclid of Alexandria

Euclid of Alexandria (Ancient Greek: Εὐκλείδης – Eukleídēs, lived c. 300 BCE, Alexandria, Egypt) systematized ancient Greek and Near Eastern mathematics and geometry. He wrote The Elements, the most widely used mathematics and geometry textbook in history. Older books sometimes confuse him with Euclid of Megara. Modern economics has been called “a series of footnotes to Adam Smith,” who was the author of The Wealth of Nations (1776 CE). Likewise, much of Western mathematics has been a series of footnotes to Euclid, either developing his ideas or challenging them.

Euclidean and Non-Euclidean geometry

Euclidean geometry is the geometry of space described by the system of axioms first stated systematically (though not sufficiently rigorous) in the Elements of Euclid.

The space of Euclidean geometry is usually described as a set of objects of three kinds, called “points,” “lines” and “planes”; the relations between them are incidence, order (“lying between”), congruence (or the concept of a motion), and continuity.

The parallel axiom (fifth postulate) occupies a special place in the axiomatics of Euclidean geometry. D. Hilbert gave the first sufficiently precise axiomatization of Euclidean geometry (see Hilbert system of axioms).

There are modifications of Hilbert’s axiom system as well as other versions of the axiomatics of Euclidean geometry. For example, in vector axiomatics, the concept of a vector is taken as one of the basic concepts. On the other hand, the relation of symmetry may be taken as a basis for the axiomatics of plane Euclidean geometry.

Non-Euclidean geometry means, in the literal sense — all geometric systems distinct from Euclidean geometry; usually, however, the term “non-Euclidean geometries” is reserved for geometric systems (distinct from Euclidean geometry) in which the motion of figures is defined, and this with the same degree of freedom as in Euclidean geometry.

The degree of freedom of motion of figures in the Euclidean plane is characterized by the condition that every figure can be moved, without changing the distances between its points, in such a way that any selected point of the figure can be made to occupy a previously-designated position; moreover, every figure can be rotated about any of its points.

In the Euclidean three-dimensional space, every figure can be moved in such a way that any selected point of the figure will occupy any prescribed position; besides, every figure can be rotated about any axis through any of its points.

Euclid’s postulates

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid’s fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates (“absolute geometry”) for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th.

In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent “non-Euclidean geometries” could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

References

  1. Euclidean geometry. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Euclidean_geometry&oldid=34034
  2. Non-Euclidean geometries. N.V. Efimov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Non-Euclidean_geometries&oldid=14875
  3. Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 88-92, 1989.

Displacement

The displacement of an object is defined as the vector distance from some initial point to a final point. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced.

It is therefore distinctly different from the distance traveled except in the case of straight-line motion in one direction. The distance traveled divided by the time is called the speed, while the displacement divided by the time defines the average velocity.

Displacement is defined as follow:

\[\Delta x=x_f−x_0\]

where \(\Delta x\) is displacement, \(x_f\) is the final position, and \(x_0\) is the initial position. Displacement has a direction as well as a magnitude!

Position

In order to describe the motion of an object, you must first be able to describe its position (where it is at any particular time). More precisely, you need to specify its position relative to a convenient reference frame.

So the position of a point \(P\) can be described by a pair or a set of coordinates, such as: \(P=(x, y)\) (in two dimensions) or \(P=(x, y, z)\) (in three dimensions).

Position vector

The position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point \(P\) in space in relation to an arbitrary reference origin \(O\). Usually denoted \(\vec{r}\), or \(\vec{s}\), it corresponds to the straight-line from \(O\) to \(P\).

Position 1

In other words, it is the displacement or translation that maps the origin to \(P\).

In one dimension \(x(t)\) is used to represent position as a function of time. In two dimensions, either cartesian or polar coordinates may be used, and the use of unit vectors is common. A position vector \(r\) may be expressed in terms of the unit vectors as follow:

\[\vec{r}(t)=x\vec{i}+y\vec{j}\]

In three dimensions, cartesian or spherical polar coordinates are used, as well as other coordinate systems for specific geometries.

\[\vec{r}(t)=x\vec{i}+y\vec{j}+z\vec{k}\]

The vector change in position associated with a motion is called the displacement.

Distance

Distance is defined to be the magnitude or size of displacement between two positions.

Note that the distance between the two positions is not the same as the distance traveled between them. Distance traveled is the total length of the path traveled between two positions.

Although displacement is described in terms of direction, distance is not. Distance has no direction and, thus, no sign.

It is important to note that the distance traveled, however, can be greater than the magnitude of the displacement (by magnitude, we mean just the size of the displacement without regard to its direction; that is, just a number with a unit).

The distance formula

Given endpoints \(P_1=(x_1,y_1)\) and \(P_2=(x_2,y_2)\), the distance between two points is given by:

\[d(P_1,P_2)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

Similarly, the distance between two points \(P_1=(x_1,y_1,z_1)\) and \(P_2=(x_2,y_2,z_2)\) in xyz-space is given by the following generalization of the distance formula:

\[d(P_1,P_2)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]

Volume

Volume is the measure of space occupied by a three-dimensional body.

The volume of a three-dimensional body is a numerical characteristic of the body; in the simplest case, when the body can be decomposed into a finite set of unit cubes (i.e., cubes with edges of unit length), it is equal to the number of these cubes. Volumes of three-dimensional bodies (i.e., sets in three-dimensional Euclidean space) for which volume can be defined have properties analogous to those of areas of plane figures:

  1. volume is non-negative;
  2. volume is additive (the volume of the union of two bodies is the sum of their volumes);
  3. volume is invariant for displacements;
  4. the volume of the unit cube is equal to one.

Specific volume (chemistry)

The specific volume, \(\nu\), of a substance is the ratio of the substance’s volume to its mass. It is the reciprocal of density and an intrinsic property of matter as well.

\[\nu =\dfrac{V}{m}=\dfrac{1}{\rho}=\left[\dfrac{\textrm{m}^3}{\textrm{kg}}\right]\]

Coordinate systems

A coordinate system is a system that uses one or more numbers, called coordinates, to uniquely determine the position of a point or other geometric elements on 1D, 2D, and 3D dimensions.

Each of these numbers indicates the distance between the point and some fixed reference point, called the origin. The first number, known as the \(x\) value, indicates how far left or right the point is from the origin. The second number, known as the \(y\) value, indicates how far above or below the point is from the origin. The origin has a coordinate of \((0, 0)\).

Longitude and latitude are a special kind of coordinate system, called a spherical coordinate system since they identify points on a sphere or globe. However, there are hundreds of other coordinate systems used in different places around the world to identify locations on the earth. All of these coordinate systems place a grid of vertical and horizontal lines over a flat map of a portion of the earth.

A complete definition of a coordinate system requires the following:

  • the projection in 1, 2 or 3 dimensions;
  • the location of the origin;
  • the units that are used to measure the distance from the origin.

Common coordinate systems

  • Cartesian coordinate system
  • Polar coordinate system
  • Cylindrical coordinate systems
  • Spherical coordinate systems

Cartesian coordinate system

The term “cartesian coordinates” (also called rectangular coordinates) is used to specify the location of a point in the plane (two-dimensional), or in three-dimensional space.

Coordinate systems 2

In such a coordinate system you can calculate the distance between two points and perform operations like axis rotations without altering this value. The distance between any two points in rectangular coordinates can be found from the distance relationship.

The most common coordinate system for representing positions in space is one based on three perpendicular spatial axes generally designated \(x\), \(y\), and \(z\). The three axes intersect at the point called the origin \(O=(0,0,0)\).

Any point \(P\) may be represented by three signed numbers, usually written \((x, y, z)\) where the coordinate is the perpendicular distance from the plane formed by the other two axes.

Often positions are specified by a position vector \(\vec{r}\) which can be expressed in terms of the coordinate values and associated unit vectors:

\[\vec{r}=x\vec{i}+y\vec{j}+z\vec{k}\]

With above definitions of the positive x, y, and z-axis, the resulting coordinate system is called right-handed; if you curl the fingers of your right hand from the positive x-axis to the positive y-axis, the thumb of your right-hand points in the direction of the positive z-axis. Switching the locations of the positive x-axis and positive y-axis creates a left-handed coordinate system. The right-handed and left-handed coordinate systems represent two equally valid mathematical universes. The problem is that switching universes will change the sign on some formulas.

In addition to the three coordinate axes, we often refer to three coordinate planes. The xy-plane is the horizontal plane spanned by the x and y-axes. It is identical to the two-dimensional coordinate plane and contains the floor in the room analogy. Similarly, the xz-plane is the vertical plane spanned by the x and z-axes and contains the left wall in the room analogy. Lastly, the yz-plane is the vertical plane spanned by the y and the z-axis and contains the right wall in the room analogy.

Polar coordinate system

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

Coordinate systems 3

Where \(P=(r,\theta)\). The reference point (analogous to the origin of a cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth.

The polar coordinates \(r\) and \(\theta\) can be converted to the cartesian coordinates \(x\) and \(y\) by using the trigonometric functions sine and cosine:

\(x=\sin\theta\)

\(y=\cos\theta\)

The cartesian coordinates \(x\) and \(y\) can be converted to polar coordinates \(r\) and \(\theta\) with \(r\geq 0\) and \(\theta\) in the interval \((-\pi, \pi ]\) by: \(r=\sqrt{x^2+y^2}\)

Number

A number is a primitive mathematical entity, whose concept arises from the need to count, as an abstraction of the concept of quantity, or to assign the position in a list of elements, or to identify the relationship between quantities of the same type.

Odd and even numbers

An odd number is an integer that is not divisible by 2 (without remainder). Odd numbers are: \(\pm 1, \pm 3, \pm 5, \pm 7, \pm 9, \pm 11, …\)

An integer that is not odd is called even number.

An even number is an integer divisible (without remainder) by 2. Even numbers are: \(0, \pm 2, \pm 4, \pm 6, \pm 8, \pm 10, …\)

An integer \(k\) is even if and only if the last digit of \(k\) is even. For example 2139856 is even, while 2146903 is not even (but odd). An integer not divisible by 2 (i.e. leaving remainder 1) is called an odd number.

Numerical sets (number system)

Numerical sets (number system)

Numbers can be classified into sets, called number systems:

  • Natural numbers (counting numbers) \(\mathbb{N}\)
  • Whole numbers
  • Integers \(\mathbb{Z}\)
  • Rational numbers \(\mathbb{Q}\)
  • Real numbers \(\mathbb{R}\)
  • Complex numbers \(\mathbb{C}\)

Natural number (counting number)

The most basic numbers used in algebra are those we use to count objects: \(1, 2, 3, 4, 5, …\) and so on. These are called the counting numbers also called natural numbers.

The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly.

Whole number

Including zero with the counting numbers gives a new set of numbers called the whole numbers.

\[0, 1, 2, 3, 4, 5 …\]

The discovery of the number zero was a big step in the history of mathematics.

Integer number

The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign).

Rational number

A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator.

Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator.

Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them.

Real number

A real number is a value of a continuous quantity that can represent a distance along a line, they include all the measuring numbers. Every real number corresponds to a point on the number line.

Complex number

A complex number is a number that can be expressed in the form \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\) is a solution of the equation \(x^2 = −1\).

Because no real number satisfies this equation, \(i\) is called an imaginary number. For the complex number \(a + bi\), \(a\) is called the real part, and \(b\) is called the imaginary part.

Probability

Probability is a mathematical language used to discuss uncertain events, and probability plays a crucial role in statistics.

Any measurement or data collection effort is subject to several sources of variation. By this, we mean that if the same measurement were repeated, then the answer would likely change.

Probability distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in a statistical experiment.