text:
Linear Algebra/OLD/Vector Spaces

A vector space is a way of generalizing the concept of a set of vectors. For example, the complex number 2+3i can be considered a vector, since in some way it is the vector $$\begin{bmatrix} 2 \\ 3\end{bmatrix}$$.

The vector space is a "space" of such abstract objects, which we term "vectors".

Some familiar friends
Currently in our study of vectors we have looked at vectors with real entries: $$\mathbb{R}^2, \mathbb{R}^3, ..., \mathbb{R}^n$$, and so on. These are all vector spaces. The advantage we gain in abstracting to vector spaces is a way of talking about a space without any particular choice of objects (which define our vectors), operations (which act on our vectors), or coordinates (which identify our vectors in the space). Further results may be applied to more general spaces which may have infinite dimension, such as in Functional Analysis.

Notations and concepts
We write a vector, like we have before, bold, but you should write these on paper underlined or with an arrow on top. So we write $$\mathbf{v}=\begin{pmatrix} 2 \\ 3\end{pmatrix}$$ for that vector.

When we multiply a vector by a scalar number, we usually ascribe it a Greek letter, writing λv for the multiplication of v by a scalar λ. We write addition and subtraction of vectors as we have been doing before, x+y for the sum of vectors x and y.

With scalar multiplication and adding vectors, we can move to our definition of a vector space.

When we refer to an operation being 'closed' in a definition, we are saying that the result of the operation does not violate our definition. For example, if we are looking at the set of all integers, we can say that it is closed under addition, because adding any integers results in something inside the set of integers. However the set of integers is not closed under division, because dividing 3 by 2 (for example) doesn't result in a member of the set of integers.

Definition
A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called vector addition and scalar multiplication, respectively, such that, for $$x,y\in V$$ and α$$ \in F$$, where F is a field x+y and αx are well-defined elements of V with the following properties:
 * 1) commutativity of addition: x+y=y+x
 * 2) associativity of addition: x+(y+z)=(x+y)+z
 * 3) additive identity: there is a vector 0 such that 0+x=x for all x
 * 4) additive inverse: for each vector x, there exists another vector y such that x+y =0

 * 1) scalar associativity: α(βx) = (αβ)x
 * 2) scalar distributivity: (α + β)x=αx+βx
 * 3) vector distributivity: α(x+y)=αx+αy
 * 4) scalar identity: 1x=x

Alternative Definition
People who are familiar with group theory and field theory may find the following alternative definition more compact:

 * $$(V, +)$$ is an Abelian group.
 * $$\forall \alpha \in F, (\forall x, y \in V), \alpha (x+y) = \alpha x + \alpha y$$
 * $$\forall \alpha, \beta \in F, \forall x\in V, (\alpha + \beta)x = \alpha x + \beta x$$
 * $$\forall \alpha, \beta \in F, \forall x\in V, (\alpha \beta)x = \alpha (\beta x)$$
 * $$\forall x\in V, 1x = x$$

Some Basic Theorems

 * 1) The 0 vector is unique. Proof: Let 01 and 02 both be 0 vectors. Then 01=01+02=02.
 * 2) The additive inverse is unique. Proof: Suppose there is a x and y1 and a y2. are inverses of x Then y1=y1+(x+y2)=(y1+x)+y2=y2.
 * 3) 0x=0. Proof: Let y be the additive inverse of x. Then 0x= 0x + x + y = (0+1)x + y = x+y=0.
 * 4) (-1)x is the inverse of x. Proof: x+(-1)x=(1-1)x=0x=0.

Linear Spaces
The linear space is a very important vector space. Let n1, n2, n3, ..., nk be k elements of a field F. Then the ordered k-tuples (n1, n2, n3, ..., nk) form a vector space with addition being the sum of the corresponding numbers, and scalar multiplication by an element of F being the result of multiplying each one in the k-tuple. This would then be the k dimensional linear space.

Subspaces
A subspace is a vector space inside a vector space. When we look at various vector spaces, it is often useful to examine their subspaces.

The subspace S of a vector space V means that S is a subset of V and that it has the following key characteristics
 * S is closed under scalar multiplication: if λ∈R, v∈S, λv∈S
 * S is closed under addition: if u, v ∈ S, u+v∈S.

Any subset with these characteristics is a vector space.

The trivial subspace
The singleton set with the zero vector ({0}) is a subspace of every vector space.

Scalar multiplication closure: a 0=0 for all a in R

Addition closure: 0+0=0. Since 0 is the only member of the set we need to check only 0

Zero vector: 0 is the only member of the set and it is the zero vector.

Examples
Let us examine some subspaces of some familiar vector spaces, and see how we can prove that a certain subset of a vector space is in fact a subspace.

A slightly less trivial subspace
In R2, the set V of all vectors from R2 of the form (0,α) where α is in R is a subspace

Scalar multiplication closure: a (0,α) = (0,a α) and a α is in R

Addition closure: (0,α) +(0,β) =(0, α + β) and α + β is in R

Zero vector: taking α to be zero in our definition of (0, α) in V we get the zero vector (0,0)

A whole family of subspaces
Pick any number from R, say ρ. Then the set V of all vectors of the form (α, ρα) is a subspace of R2

Scalar multiplication closure: a (α, ρα) = (aα, ρaα) which is in V.

Addition closure: (α, ρα) +(β, ρβ) =(α + β, ρα + ρβ) = (α+β, ρ(α+β)) which is in V

Zero vector: taking α to be zero in our definition we get (0, ρ0) = (0,0) in V.

That means V2 = the set of all vectors of the form (α,2α) is a subspace of R2

and V3 = the set of all vectors of the form (α,3α) is a subspace of R2

and V4 = the set of all vectors of the form (α,4α) is a subspace of R2

and V5 = the set of all vectors of the form (α,5α) is a subspace of R2

and Vπ = the set of all vectors of the form (α,πα) is a subspace of R2

and V√2 = the set of all vectors of the form $$(\alpha,\sqrt{2}\alpha)$$ is a subspace of R2

As you can see, even a simple vector space like R2 can have many different subspaces.

Linear Combinations
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$ and $$S$$ is a nonempty subset of $$V$$. Then a vector $$x\in V$$ is said to be a linear combination of elements of $$S$$ if there exists a finite number of elements $$y_1, y_2, ..., y_n\in S$$ and $$a_1, a_2, ..., a_n\in F$$ such that $$ x = a_1y_1 + a_2y_2 + ... a_ny_n$$.

Spans
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$. The set of all linear combinations of $$y_1, y_2, ..., y_n\in V$$ is called the span of $$y_1, y_2, ..., y_n$$. This is sometimes denoted by $$Span(y_1, y_2, ..., y_n)$$.

Note that $$Span(y_1, y_2, ..., y_n\in V)$$ is a subspace of $$V$$.

Proof: Consider closure under addition and scalar multiplication for two vectors, x and y, in the span of the vectors $${v_1, v_2, ..., v_n}.$$

$$x = a_1v_1 + a_2v_2 + ... + a_nv_n$$

$$y = b_1v_1 + b_2v_2 + ... + b_nv_n$$

$$x+y = (a_1+b_1)*v_1 + (a_2+b_2)*v_2 + ... + (a_n+b_n)*v_n$$, which is also contained in the set.

$$k*x = (k*a_1)v_1 + (k*a_2)v_2 + ... + (k*a_n)v_n$$, which is also contained in the set.

Spanning Sets
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$ and $$y_1, y_2, ..., y_n$$ are vectors in such a vector space. The set $${y_1, y_2, ..., y_n}$$ is a spanning set for the vector space $$V$$ if and only if every vector in $$V$$ is a linear combination of $$y_1, y_2, ..., y_n$$. Alternately, $$\forall x\in V, (\exists a_1, a_2, ..., a_n\in F), x = a_1y_1 + a_2y_2 + ... +a_ny_n$$

Linear Independence
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$ and $$S = \{x_1, x_2, ..., x_n\}$$ is a finite subset of $$V$$. Then we say $$S$$ is linearly independent if $$a_1x_1+a_2x_2+...a_nx_n = 0$$ implies $$a_1 = a_2 = ... = a_n = 0$$.

Linear independence is a very important topic in Linear Algebra. The definition implies that linearly dependent vectors may form the nulvector as a non-trivial combination, from which we may conclude that one of the vectors can be expressed as a linear combination of the others.

If we have a vector space V spanned by 3 vectors $${v_1, v_2, v_3}$$ we say that v1, v2, and v3 are linearly dependent if there is a combination of one or two of them that can produce a third. For instance, if one of the following equations:

 * $$a_1v_1 + a_2v_2 = v_3$$
 * $$a_2v_2 + a_3v_3 = v_1$$
 * $$a_1v_1 + a_3v_3 = v_2$$

can be satisfied, then the vectors in V are said to be linearly dependant.

How can we test for linear independence? The definition sets it out to us: If V is a vector space spanned by 3 vectors of length N:

 * $$\tilde{V} = [v_1, v_2, v_3]$$

and we try to test whether these 3 vectors are linearly independent, we form the equations:

 * $$a_1v_1+a_2v_2+a_3v_3=0\,$$

and solve them. If the only solution is

 * $$a_1=a_2=a_3=0,\,$$

then the 3 vectors are linearly independent. If there is another solution they are linearly dependent.

?????? We can say that for V to be linearly independent it must satisfy this condition:

 * $$\tilde{V}\bar{a} = 0$$

Where we are using 0 to denote the null vector in V. If $$\tilde{V}$$ is square and invertable, we can solve this equation directly:

 * $$\tilde{V}^{-1}\tilde{V}\bar{a} = \bar{a} = \tilde{V}^{-1} \cdot 0$$
 * $$\bar{a} = 0$$

And if we know that $$\bar{a}$$ is zero, then we know that the system is linearly independent. If, however, $$\tilde{V}$$ is not square, or if it is not invertable, we can try the following technique:

Multiply through by the transpose matrix:
 * $$\tilde{V}^T\tilde{V}\bar{a} = 0$$

Find the inverse of $$[\tilde{V}^T\tilde{V}$$, and multiply through by the inverse:
 * $$[\tilde{V}^T\tilde{V}]^{-1}\tilde{V}^T\tilde{V}\bar{a} = [\tilde{V}^T\tilde{V}]^{-1} \cdot 0$$

Cancel the terms:
 * $$\bar{a} = \tilde{V}^T\tilde{V} \cdot 0$$

And our conclusion:
 * $$\bar{a} = 0$$

This again means that V is linearly independent.

Span
A span is the set of all possible vectors that are in a given vector space.

Basis
A basis for a vector space is the least amount of linearly independent vectors that can be used to describe the vector space completely. The most common basis vectors are the kronecker vectors, also called canonical basis:

 * $$i = \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}$$$$j = \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}$$$$k = \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}$$

In the cartesian graphing space, we say an ordered triple of coordinates is defined as:

 * $$v = \begin{bmatrix}x \\ y \\ z\end{bmatrix}$$

And we can make any point (x, y, z) by combining the kronecker basis vectors:

 * $$\begin{bmatrix}x \\ y \\ z\end{bmatrix} = xi + yj + zk$$

Some theorems:
 * A basis $$\mathcal{B}(v_1, \ldots, v_n)$$ of a vector space V has the maximal number of linearly independent vectors.
 * (Converse) A maximal number of linearly independent vectors in a vector space is a basis.

Bases and Dimension
If a vector space V is such that: it contains a linearly independent set B of N vectors, and any set of N + 1 or more vectors in V is linearly dependent,

then V is said to have dimension N, and B is said to be a  basis of V.

TODO
Tell about what is a basis in a vector space and about coordinate transformations. (this article contains an abstract definition of a basis which is a generalization of a basis in vector space and can be used as the foundation to explain about bases and coordinate transformations.)

Discuss the geometry of subspaces (points, lines, planes, hypersurfaces) and connect them to the geometry of solutions of linear systems. Connect the algebra of subspaces and linear combinations of vectors to the algebra of linear systems.

title: Linear Algebra/OLD/Vector Spaces


text:
User:Vulture

See de:Benutzer:Vulture

There's not much though ...

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Vulture


text:
Talk:Japanese/Introduction

A question about contributing
Hi,

I have been talking with other Japan-related bloggers about how we can work together to improve the Japan-related content available online and this wikibooks project came up as one of the things that would be great to contribute to.

I was wondering how we could best get involved and also if anyone who has worked on the project would like to comment on it and I will post the comments on my blog dailyj.net

Please let me know. Thanks (Toritaiyo 17:33, 29 November 2007 (UTC))

 * Find a page you're interested in contributing to, be bold and have fun. Contact others on talk pages relevant to the page in question — Talk:Japanese for the whole book. --Swift (talk) 07:58, 12 June 2008 (UTC)

title: Talk:Japanese/Introduction


text:
PHP Programming

Introduction

 * /Introduction/

Setup and Installation

 * Setup and Installation
 * Installation on Windows
 * Easy Windows Setup Instructions
 * Installation on Linux
 * Installation on Mac OS X
 * Uses of PHP

Note: Before contributing, check out the |discussion page. How to write your examples.

The Basics

 * This section is about things that are important for any type of PHP development. Useful for a PHP programmer of any level.

 * 1) Beginning with "Hello World!"
 * 2) Nuts and Bolts
 * 3) Commenting and Style
 * 4) Comparison operators
 * 5) Arrays
 * 6) Control structures
 * 7) The if Structure
 * 8) The switch Structure
 * 9) The while Loop
 * 10) The do while Loop
 * 11) The for Loop
 * 12) The foreach Loop
 * 13) Functions
 * 14) /PHP Include Files/
 * 15) Files
 * 16) * Images
 * 17) Mailing
 * 18) Cookies
 * 19) Sessions
 * 20) Databases
 * 21) MySQL
 * 22) * PHP and MySQL
 * 23) PostgreSQL
 * 24) PHP Data Objects
 * 25) Neo4j
 * 26) /DBAL/
 * 27) Integration Methods (HTML Forms, etc.)
 * 28) Regular expressions

Advanced PHP

 * Advanced PHP includes high level programming and PHP techniques designed to make PHP even more useful and powerful.

Data Structures

 * 1) Data Structures

Object Oriented Programming (OOP)

 * 1) Classes
 * 2) Special Methods
 * 3) Overriding and Overloading
 * 4) /Inheritance/
 * 5) /SSH Class/

Caching

 * 1) Caching

Templating

 * 1) /Why Templating/
 * 2) /Templates/
 * 3) /Smarty templating system/
 * 4) * /Smarty templating system/Functions/
 * 5) * /Smarty templating system/Simple tutorial/
 * 6) /Flat Frog/
 * 7) /XML/
 * 8) /XSL/
 * 9) * /XSL/registerPHPFunctions/

Libraries

 * 1) PHP PEAR
 * 2) PHP Manual Function Reference
 * 3) PHP 5 Functions

Security

 * 1) /Configuration: Register Globals/
 * 2) /SQL Injection Attacks/
 * 3) /Building a secure user login system/
 * 4) /Cross Site Scripting Attacks/ (XSS)
 * 5) /Secure HTTP headers/
 * 6) /Encryption/

See also:
 * section on avoiding session fixation in the Sessions chapter.
 * (CSRF)

Command-Line Interface (CLI)

 * 1) /PHP CLI/
 * 2) /PHP-GTK/
 * 3) /Daemonization/

Appendices

 * /Alternative Hungarian Notation/
 * /Code Snippets/
 * /Coding Standards/
 * /formatting notes/
 * /Get Apache and PHP/
 * /headers and footers/
 * /html output/
 * /OOP5/Advanced Input validation/
 * /OOP5/Input validation/
 * /phpDocumentor/
 * /Reserved words/
 * /Contributors/

title: PHP Programming


text:
Physics Study Guide

<Cover and interwiki

Introduction

 * 1) /Purpose/
 * 2) /Laws/
 * 3) /Scientific Method/

Section One

 * 1) Units S.I. units
 * 2) Kinematics Displacement, velocity, acceleration
 * 3) /Force/ Newton's 3 Laws of Motion
 * 4) /Momentum/ Elastic and inelastic collisions
 * 5) Friction and the Normal Force
 * 6) /Work/ Force times distance
 * 7) /Energy/ The potential to do Work

Section Two

 * 1) /Gravity/ A universal force of attraction
 * 2) /Circular Motion/
 * 3) /Torque/
 * 4) /Periodic Motion/ the motion that keeps in regular intervals on time.
 * 5) /Waves/
 * 6) /Wave overtones/
 * 7) /Standing waves/
 * 8) /Sound/

Section Three

 * 1) /Fluids/
 * 2) /Fields/
 * 3) /Thermodynamics/
 * 4) /Electricity/ Charge, current, electric fields
 * 5) /Magnetism/ Magnetic dipoles, magnetic fields
 * 6) /Electronics/ Resistance, voltage, capacitance, inductance (incomplete)
 * 7) /Optics/ Light, lasers etc
 * 8) /Theories of Electricity/ Theories behind Electricity

Appendices

 * 1) Commonly Used Physical Constants
 * 2) Approximate Coefficients of Friction
 * 3) /Greek alphabet/
 * 4) /Logs/
 * 5) /Vectors and scalars/
 * 6) Symbols Used in This Book
 * 7) Exercises
 * 8) More topics
 * 9) /Style Guide/

title: Physics Study Guide


text: Physics Study Guide/Authors

title: Physics Study Guide/Authors


text:
Physics Study Guide/Linear motion

Kinematics is the description of motion. The motion of a point particle is fully described using three terms - position, velocity, and acceleration. For real objects (which are not mathematical points), translational kinematics describes the motion of an object's center of mass through space, while angular kinematics describes how an object rotates about its centre of mass. In this section, we focus only on translational kinematics. Position, displacement, velocity, and acceleration are defined as follows.

Position
"Position" is a relative term that describes the location of an object RELATIVE to some chosen stationary point that is usually described as the "origin".

A vector is a quantity that has both magnitude and direction, typically written as a column of scalars. That is, a number that has a direction assigned to it.

In physics, a vector often describes the motion of an object. For example, Warty the Woodchuck goes 10 meters towards a hole in the ground.

We can divide vectors into parts called "components", of which the vector is a sum. For example, a two-dimensional vector is divided into x and y components.

Displacement
Displacement answers the question, "Has the object moved?"

Note the $$\equiv$$ symbol. This symbol is a sort of "super equals" symbol, indicating that not only does $$\vec x_f- \vec x_i$$ EQUAL the displacement $$\Delta\vec{x}$$, but more importantly displacement is operationally defined by $$\vec x_f- \vec x_i$$.

We say that $$\vec x_f- \vec x_i$$ operationally defines displacement, because $$\vec x_f- \vec x_i$$ gives a step by step procedure for determining displacement.

Namely:

 * 1) Measure where the object is initially.
 * 2) Measure where the object is at some later time.
 * 3) Determine the difference between these two position values.

Be sure to note that displacement is not the same as distance travelled.

For example, imagine travelling one time along the circumference of a circle. If you end where you started, your displacement is zero, even though you have clearly travelled some distance. In fact, displacement is an average distance travelled. On your trip along the circle, your north and south motion averaged out, as did your east and west motion.

Clearly we are losing some important information. The key to regaining this information is to use smaller displacement intervals. For example, instead of calculating your displacement for your trip along the circle in one large step, consider dividing the circle into 16 equal segments. Calculate the distance you travelled along each of these segments, and then add all your results together. Now your total travelled distance is not zero, but something approximating the circumference of the circle. Is your approximation good enough? Ultimately, that depends on the level of accuracy you need in a particular application, but luckily you can always use finer resolution. For example, we could break your trip into 32 equal segments for a better approximation.

Returning to your trip around the circle, you know the true distance is simply the circumference of the circle. The problem is that we often face a practical limitation for determining the true distance travelled. (The travelled path may have too many twists and turns, for example.) Luckily, we can always determine displacement, and by carefully choosing small enough displacement steps, we can use displacement to obtain a pretty good approximation for the true distance travelled. (The mathematics of calculus provides a formal methodology for estimating a "true value" through the use of successively better approximations.) In the rest of this discussion, I will replace $$\Delta$$ with $$\delta$$ to indicate that small enough displacement steps have been used to provide a good enough approximation for the true distance travelled.

Velocity
[Δ, delta, upper-case Greek D, is a prefix conventionally used to denote a difference.] Velocity answers the question "Is the object moving now, and if so - how quickly?"

Once again we have an operational definition: we are told what steps to follow to calculate velocity.

Note that this is a definition for average velocity. The displacement Δx is the vector sum of the smaller displacements which it contains, and some of these may subtract out. By contrast, the distance travelled is the scalar sum of the smaller distances, all of which are non-negative (they are the magnitudes of the displacements). Thus the distance travelled can be larger than the magnitude of the displacement, as in the example of travel on a circle, above. Consequently, the average velocity may be small (or zero, or negative) while the speed is positive.

If we are careful to use very small displacement steps, so that they come pretty close to approximating the true distance travelled, then we can write the definition for instantaneous velocity as

[δ is the lower-case delta.] Or with the idea of limits from calculus, we have:

[d, like Δ and δ, is merely a prefix; however, its use definitely specifies that this is a sufficiently small difference so that the error--due to stepping (instead of smoothly changing) the quantity--becomes negligible.]

Acceleration
Acceleration answers the question "Is the object's velocity changing, and if so - how quickly?"

Once again we have an operational definition. We are told what steps to follow to calculate acceleration.

Again, also note that technically we have a definition for average acceleration. As for displacement, if we are careful to use a series of small velocity changes, then we can write the definition for instantaneous acceleration as:

Or with the help of calculus, we have:

Vectors
Notice that the definitions given above for displacement, velocity and acceleration included little arrows over many of the terms. The little arrow reminds us that direction is an important part of displacement, velocity, and acceleration. These quantities are vectors. By convention, the little arrow always points right when placed over a letter. So for example, $$\vec v$$ just reminds us that velocity is a vector, and does not imply that this particular velocity is rightward.

Why do we need vectors? As a simple example, consider velocity. It is not enough to know how fast one is moving. We also need to know which direction we are moving. Less trivially, consider how many different ways an object could be experiencing an acceleration (a change in its velocity). Ultimately, there are three distinct ways an object could accelerate:

 * 1) The object could be speeding up.
 * 2) The object could be slowing down.
 * 3) The object could be traveling at constant speed, while changing its direction of motion.

More general accelerations are simply combinations of 1 and 3 or 2 and 3.

Importantly, a change in the direction of motion is just as much an acceleration as is speeding up or slowing down.

In classical mechanics, no direction is associated with time (you cannot point to next Tuesday). So the definition of $$\vec a_{av}$$ tells us that acceleration will point wherever the change in velocity $$\Delta\vec{v}$$ points.

Understanding that the direction of $$\Delta \vec{v}$$ determines the direction of $$\vec a$$ leads to three non-mathematical but very powerful rules of thumb:

 * 1) If the velocity and acceleration of an object point in the same direction, the object's speed is increasing.
 * 2) If the velocity and acceleration of an object point in opposite directions, the object's speed is decreasing.
 * 3) If the velocity and acceleration of an object are perpendicular to each other, the object's initial speed stays constant (in that initial direction), while the speed of the object in the direction of the acceleration increases. Think of a bullet fired horizontally in a vertical gravitational field. Since velocity in the one direction remains constant, and the velocity in the other direction increases, the overall velocity (absolute velocity) also increases.

Again, more general motion is simply a combination of 1 and 3 or 2 and 3.

Using these three simple rules will dramatically help your intuition of what is happening in a particular problem. In fact, much of the first semester of college physics is simply the application of these three rules in different formats.

= Equations of motion (constant acceleration) = A particle is said to move with constant acceleration if its velocity changes by equal amounts in equal intervals of time, no matter how small the intervals may be

Since acceleration is a vector, constant acceleration means that both direction and magnitude of this vector don't change during the motion. This means that average and instantaneous acceleration are equal. We can use that to derive an equation for velocity as a function of time by integrating the constant acceleration.

Giving the following equation for velocity as a function of time.

To derive the equation for position we simply integrate the equation for velocity.

Integrating again gives the equation for position.

The following are the equations of motion:

The following equations can be derived from the two equations above by combining them and eliminating variables.

Acceleration in One Dimension

Acceleration in Two Dimensions
(Needs content)

Acceleration in Three Dimensions
(Needs content)

What does force in motion mean?
Force means strength and power. Motion means movement. That’s why we need forces and motions in our life. We need calculation when we want to know how fast things go, travel and other things which have force and motion. ...

How do we calculate the speed?
If you want to calculate the average speed, distance travelled or time taken you need to use this formula and remember it: $$ {speed} = \frac{distance}{time\ taken} $$ This is an easy formula to use, you can find the distance travelled, time taken or average speed, you need at least 2 values to find the whole answer.

Is velocity the same thing as speed?
Velocity is a vector quantity that refers to "the rate at which an object changes its position", whereas speed is a scalar quantity, which cannot be negative. Imagine a kid moving rapidly, one step forward and one step back, always returning to the original starting position. While this might result in a frenzy activity, it would result in a zero velocity, because the kid always returns to the original position, the motion would never result in a change in position, in other words $$\Delta \vec x$$ would be zero.

Speed is measured in the same physical units of measurement as velocity, but does not contain an element of direction. Speed is thus the magnitude component of velocity. Velocity contains both the magnitude and direction components. You can think of velocity as the displacement/duration, whereas speed can be though as distance/duration.

Acceleration
When a car is speeding up we say that it is accelerating, when it slows down we say it is decelerating.

How do we calculate it?
When we want to calculate it, the method goes like that: A lorry driver brakes hard, and slows from 25 m/s to 5 m/s in 5 seconds. What was the vehicle's acceleration? $$ {acceleration} = \frac{change\ in\ velocity}{time\ taken} = \frac{5 - 25}{5} = \frac{-20}{5} = -4\ ms^{-2} $$ What is initial velocity and final velocity? Initial velocity is the beginning before motion starts or in the middle of the motion, final velocity is when the motion stops.

There is another way to calculate it and it is like that This equations which are written is the primary ones, which means that when you don’t have lets say final velocity, how will you calculate the equation? This is the way you are going to calculate.

Observing motion
When you want to know how fast an athletic person is running, what you need is a stopwatch in your hand, then when the person starts to run, you start the stopwatch and when the person who is sprinting stops at the end point, you stop the watch and see how fast he ran, and if you want to see if the athlete is wasting his energy, while he is running look at his movement, and you will know by that if he is wasting his energy or not.

This athletic person is running, and while he is running the scientist could know if he was wasting his energy if they want by the stop watch and looking at his momentum.

Measuring acceleration
Take a slope, a trolley, some tapes and a stop watch, then put the tapes on the slope and take the trolley on the slope, and the stopwatch in your hand, as soon as you release the trolley, start timing the trolley at how fast it will move, when the trolley stops at the end then stop the timing. After wards, after seeing the timing, record it, then you let the slope a little bit high, and you will see, how little by little it will decelerate.

= Newton =

Isaac Newton was an English physicist, mathematician (described in his own day as a "natural philosopher"), astronomer and alchemist. Newton is one of the most influential scientists of all time, and he is known, among other things, for contributing to development of classical mechanics and for inventing, independently from Gottfried Leibniz, calculus.

Newton's laws of motion
Newton is also known by his three laws of motion, which describe the relationship between a body and the forces acting upon it, and its motion in response to said forces.
 * 1) First Law (also known as the law of inertia) states that every body continues in its state of rest or state of uniform motion unless compelled to change that state by being subject to an external force. The moment of inertia is defined as the tendency of matter to resist any change in its state of motion or state of rest.
 * 2) Second Law The vector sum of the external forces $$F$$ on an object is equal to the mass $$m$$ of that object multiplied by the acceleration vector $$a$$ of the object, or algebraically $$F = ma$$.
 * 3) Third Law states that when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

=Symbols=

Some useful symbols seen and that we will see:

Force
A force is any interaction that tends to change the motion of an object. In other words, a force can cause an object with mass to change its velocity. Force can also be described by intuitive concepts such as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol $$F$$.

How to calculate the force?
When we want to calculate the force, and we have the mass and acceleration, we can simply use the simple formula stated in the Newton's second law above, that is $$F = ma$$, where $$m$$ is the mass (or the amount of matter in a body), and $$a$$ is the acceleration. Note that the Newton’s second law is defined as a numerical measure of inertia.

What is inertia?
Inertia is the tendency of a body to maintain its state of rest or uniform motion, unless acted upon by an external force.

Robert Hooke
Robert Hooke was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work.

Hooke's law
Hooke's law is a principle of physics that states that the force $$F$$ needed to extend or compress a spring by some distance $$X$$ is proportional to that distance, or algebraically $$F = -kX$$, where $$k$$ is a constant factor characteristic of the spring, its stiffness.

title: Physics Study Guide/Linear motion


text:
German/Lesson 4

<< Lesson 3 | Lektion 4 | Lektion 5 >>

Lesestück 4-1 ~ Eine Geschichte über Zürich

 * Zürich ist die größte Stadt der Schweiz. Sie liegt am Ausfluss des Zürichsees und ist die Hauptstadt des gleichnamigen Kantons, des Kantons Zürich. Zürich ist ausgesprochen schön gelegen, am nördlichen Ende des Zürichsees—bei klarem Wetter hat man eine gute Sicht auf die Glarner Alpen.

 * Zürich ist das Zentrum der schweizer Bankenwirtschaft. Neben den beiden Großbanken ('Credit Suisse' und 'UBS') haben auch etliche kleinere Bankinstitute ihren Sitz in der Stadt.

Although this short story contains quite a number of impressive German nouns and adjectives, with the aid of Vokabeln 4-1 following you should have no trouble reading and understanding it. The passage makes considerable use of the German genitive case (English possessive case), which you have not yet learned. However, a clue applicable here: translate des as "of the" or "of" and note there are other der-words that also mean "of the".

Vokabeln 4-1
die Alpen              Alps der Ausfluss           outlet, effluence           (of a lake) die Bankinstitute      banking institutes die Bankenwirtschaft   banking business das Ende               end die Großbanken        major banks die Hauptstadt         capital city das Haus               house der Kanton             canton                      (Swiss state) das Lesestück          reading passage die Schweiz            Switzerland die Sicht              view der Sitz               office das Wetter             weather das Zentrum            center (centre) das Zürich             Zurich                      (city and canton in Switzerland) der Zürichsee          Lake Zurich

d.h. (das heißt)       i.e. ("that is" in Latin) Glarner Alpen          Glarner Alps man hat...             one has...   nach Hause              (toward) home               (compare: zu Hause = "at home")

anrufen                call, telephone geben (gab, gegeben)   give kommen (kam, gekommen) come liegen (lag, gelegen)  lie (lay, lain)

am (an dem)            at the ausgesprochen          markedly bei                    in  beiden                  two etliche                a number of, quite a few, several gleichnamig            same named größte                 largest klar                   clear klein                  small neben                  besides nördlich               northern schweizer              of or pertaining to Swiss Pronunciation Guide >>

Grammatik 4-1 ~ Introduction to adjectives
An adjective is a part of speech which can be thought of as a "describing word"—typically, an adjective modifies a noun. In both English and German, adjectives come before the noun they describe or modify. In many other languages (such as French) they usually come after the noun. Here are some examples of adjectives (underlined) you have already encountered: Because nouns are capitalized in German, it is fairly obvious in these sentences where the adjectives occur: just before the nouns they modify. Note how the endings on German adjectives can change, depending upon the noun (keinen Käse; klarem Wetter; gute Sicht)—specifically, the gender and case of the noun they are modifying. Before explaining the basic rules governing adjective endings, you need to have a better understanding of person, gender, and case in German nouns—concepts that will be explored in the next few lessons.

Finally, realize that the ordinal numbers you learned in Lektion 3 are, in fact, adjectives—subject to the same rules governing word endings for adjectives.

Gespräch 4-1 ~ Das neue Mädchen

 * Markus und Helena sind Freunde.
 * Markus: Lena, wer ist das neue Mädchen? Die Brünette dort drüben.
 * Helena: Ich glaube, sie heißt „Karoline“.
 * Markus: Sie ist sehr schön.
 * Helena: Sie ist hübsch, wenn man kleine Mädchen mit langen dunklen Haaren mag.
 * Markus: Ja. Ihre Haare gefallen mir sehr.
 * Helena: Markus, du bist ein Ferkel!

This short conversational passage contains more examples of adjectives.

Vokabeln 4-2
die Brünette            brunette die Haare               hair(s) das Mädchen             girl das Ferkel              piglet

gefallen               appeal to glauben                  believe heißen                  name, call mag                     like, desire, wish

dort                    there (dort) drüben           over there dunkel                  dark ihr                     her hübsch                  cute klein                   short lang                    long neue                    new wenn                    if wer? who?

Pronunciation Guide >>

Grammatik 4-2 ~ Nouns and pronouns in the accusative and dative
As was noted previously when the concept of case was introduced for pronouns (Grammatik 2-2), there are four cases used in German. Recall that the nominative case in German corresponds to the subjective case in English and applies to nouns and pronouns used in a sentence as the subject of a verb. Nouns (and pronouns) that are used as objects of transitive (action) verbs are in the English objective case. If these are direct objects (recipients of the action of a verb), then these nouns are in the accusative case in German. If indirect objects, then these nouns are in the dative case in German. Essentially, the English objective case is divided, in German, into an accusative case used for direct objects and a dative case used for indirect objects.

Pronouns
For comparison with English, recall that the singular personal pronouns (nominative case) are "I", "you", and "he/she/it" (1st, 2nd, and 3rd persons). The objective case, personal pronouns in English are "me", "you", and "him/her/it"—and are used for both direct and indirect objects of verbs. For example:

 * He gives it [the Direct Object] to me [the Indirect Object].

The German accusative case, personal pronouns (singular) are: mich, dich, ihn/sie/es. The German dative case, personal pronouns (singular) are: mir, dir, ihm/ihr/ihm. Thus, the above English example sentence becomes, in German:

 * Er gibt es  [the Direct Object]  mir  [the Indirect Object].

Because mir is a dative pronoun, there is no need in German to use a modifier as in English, where "to" is used as a signal of an indirect object. The following table summarizes the German pronouns in three cases for both singular and plural number:

 * * Polite form

Recall from Gespräch 2-1 the "incomplete" sentence Und Ihnen? ('And you?'). Note that the pronoun agrees in case (here, dative) with the implied sentence — Und wie geht es Ihnen? The same rule is evident in Gespräch 1-1 (Und dir?). Such agreement is important to convey the correct meaning. Tables giving the German personal pronouns in all cases can be found in an appendix: Pronoun Tables.

Nouns
Nouns do not change their form (spelling) relative to case in German; instead, a preceding article indicates case. You have learned the nominative case definite and indefinite articles (Grammatik 3-3: der, die, das and ein, eine. ein) for each of the three noun genders. Now we will learn the accusative (used to signal a direct object) and dative (used to signal an indirect object) articles. First, the definite articles:

This table might seem a bit overwhelming (and there is yet one more case in German: the genitive!), but some points to note can make memorizing much easier. First, as you can see from the table, gender does not really exist for plural nouns. No matter what the noun gender in its singular number, its plural always has the same set of definite articles: die, die, den for nominative, accusative, and dative cases. The plural der-words are similar to the feminine singular der-words, differing only in the dative case. Another point: the dative for both masculine and neuter nouns is the same: dem. Finally, for feminine, neuter, and plural nouns, there is no change between nominative and accusative cases. Thus, only for masculine nouns is there a definite article change in the accusative compared with the nominative.

The following examples demonstrate the use of the definite article in various parts of speech:

In the last example, you need to know that in both English and German, the noun (or pronoun) that follows the verb 'to be' is a predicate noun, for which the correct case is the nominative. That is why, in English, 'It is I' is grammatically correct and 'It is me' is simply incorrect.

The indefinite articles are as follows:

Of course, there are no plural indefinite articles in German or English (ein means "a". "an", or "one"). It is important to see that there is a pattern in the case endings added to ein related to the der-words in the definite articles table above. For example, the dative definite article for masculine nouns is dem—the indefinite article is formed by adding -em onto ein to get einem. The dative definite article for feminine nouns is der—the indefinite is ein plus -er or einer. These ending changes will be covered in greater detail in a future lesson. You will see that there are a number of words (adjectives, for example) whose form relative changes by addition of these endings to signal the case of the noun they modify. Finally, we can see a pattern relationship between these "endings" and the 3rd person pronouns as well:

We could construct a similar table to compare the definite articles to the 3rd person pronouns. And in that case, we would also see how the plural definite articles (die, die, den) compare with the third person plural pronouns (sie, sie, ihnen).

Grammatik 4-3 ~ Interrogatives
You have encountered nearly all of the interrogatives commonly used in German (review Grammatik 1-2): wann                 when warum                why             Warum sind Sie müde? was                  what            Was ist das? wer                  who             Wer ist das Mädchen? wie                  how             Wie geht es dir? wieviel              how much        Wieviel Uhr ist es? wo                   where           Wo ist das Buch? wohin                where (to)      Wohin gehst du? In a question, interrogatives replace the unknown object and establish the class of answer expected. Note that the English construction for some of the questions differs from the German in that the former uses the progressive form of "do".

Übersetzung 4-1
Translate the following sentences into German:

title: German/Lesson 4


text:
Talk:Esperanto/Introducing yourself

Comments

 * So far, I like it. One suggestion: you instroduce the concept of direct and indirect objects, and you give a rule for forming direct object, but you do not yet give a rule for forming indirect objects.  Those of us who have studied languages before are at the edges of our seats with anticipation...

Has anyone any comments on this lesson? Especially those with no knowledge of Esperanto, is this first lesson clear enough to follow (especially the grammar)? --Gabriel Beecham 12:31, 31 Oct 2003 (UTC)

 * Looks like you are structuring things along the lines I have been using for German, and I started out there by copying the Spanish language textbook well set up by Thomas. So I would say "good start"! What to actually present in each lesson is a difficult decision, and my feeling is, you may shift things around as you progress. I do believe you should start simple, but make lots of ties (as you are doing) to English grammar as an English speaker can relate to these connections and some learners (like myself) need to see rules to understand why things are said as they are in the foreign language. I'd say you have a good handle on these concepts. - Marsh 20:36, 13 Nov 2003 (UTC)

--:Interesting, indeed fairly facsinating, and good choices of words introductory words (Or perhaps it's just that I like reading and cities). Fairly useful, for a very basic introduction, but as an English speaker learning Spanish and hoping to learn Esparanto, I've found that I've had to go to several other sites to find much what I'm looking for. It might be helpful if you gave certain common or useful words such as "no" "on" "in" "as" (or all ases) "yes" "who" "what" "when" "why" "how" and "how many/much", perhaps as your next lesson. Thanks though, nice introduction, and useful lesson. 6:24, 9 December 2004 (EST)

Does anyone have ideas for the review section? -travisg 23:42, 27 Sep 2004 (UTC)

'It is being "verb'ed", so to speak.'
I am familiar with nouns being "verb'ed" in an entirely different sense, idiomatically speaking. e.g. Google is a company that built a search engine; "Google" was "verb'ed" when people started using the word to mean "search using Google's engine." This usage is quite different - we mean to say that the verb is being applied to the direct object.

Disclaimer: My first language is (USA) English. Ingcake (discuss • contribs) 03:40, 3 June 2015 (UTC)

Nice introduction. I would just like to pick up on one point. At the beginning of the section headed "Grammar", there is an indication that the rules are without exception. However by the time we reach the bit on adverbs there is clear mention of exceptions being present. This seems incongruous. I have a suggestion to remedy this. Instead of saying that most adverbs end in "e" and there are exceptions to this, why not say that adverbs are of two groups - those that remain root-based (or you could use some other technical description here if you like) and don't need modification e.g. morgaŭ, and those that are derived, e.g rapide derived from the adjectival root rapid. In that way you don't even need to refer to exception! I personally don't view adverbs as having exceptions but simply of two main classes - unmodified root (or you could say non-derived)(the aŭ class) and the derived type (the -e class). Thank you for considering this. Stelalumo (discuss • contribs) 07:52, 27 July 2017 (UTC)

Errors
I'm new to Esperanto, and I am finding this book fascinating! Keep it up. When you say "Please note that not all adverbs use this rule, but the overwhelming majority of them do." do you mean that not all adverbs end -e? A brief explanation would be beneficial here. I thought that since this is a sythetic language, there are no exceptions to the rules! :) --Ralph walden 02:22, 12 August 2006 (UTC)

 * There are no exceptions; the rule is that to turn a root into an adverb, you add and -e. That's always true, but not all adverbs are formed from roots. There are some words that are purely adverbs, like ("almost"). There is no "" root that this comes from (what would "" or "" mean?).

 * But you're right - this really needs a bit more discussion. There is just too much crammed into this one lesson, and nothing is getting treated well. --Indiana (talk) 23:03, 10 February 2010 (UTC)

It seems a bit odd to say "there is no verb conjugation in Esperanto." Vidi, vidas, vidis, vidos, vidus -- that's conjugation, buddy. If you are going to use grammatical terms please use them properly. Rikat (talk) 03:51, 16 May 2009 (UTC)

 * Perhaps it would have been more constructive to edit the phrase in question to "there is no verb conjugation in Esperanto for number or gender", rather than to criticize and do nothing else. --Indiana (talk) 23:03, 10 February 2010 (UTC)

The pronouns section seems to be incorrect. I think mi can only mean "I" not "me". I'm pretty sure min is "me". --Yair rand (talk) 20:36, 14 December 2009 (UTC)

 * can mean "me". If you point at a picture of yourself and say "that's me", that would be "". This is not an inconsistency in Esperanto, but due to a quirk in English. In English you can use either "I" or "me" as the predicate for the copula in certain conditions: for example, when announcing yourself you can say "it is I" or "it is me". --Indiana (talk) 23:03, 10 February 2010 (UTC)

Restructuring
This first "lesson" is way too grammar heavy. How about starting with a sentence like "My name is (Bob). I'm from (USA)." rather than starting with a long grammatical reference? Amazingly people can learn languages without knowing anything explicitly about grammar. Introduce the nouns, verbs, etc later, or introduce them more gradually. Pengo (talk) 05:47, 1 June 2008 (UTC)

 * I agree - I looked at what was being introduced in the first lesson ("Nouns, Adjectives, Adverbs, Personal and possessive pronouns, Verbs - Present Tense, Objects") and went cross-eyed. This really needs to be scaled down and separated substantially. --Indiana (talk) 23:03, 10 February 2010 (UTC)

title: Talk:Esperanto/Introducing yourself


text:
Physics Study Guide/Greek alphabet

=About the Common uses in Physics= While these are indeed common usages, it should be pointed out that there are many other usages and that other letters are used for the same purpose. The reason is quite simple: there are only so many symbols in the Greek and Latin alphabets, and scientists and mathematicians generally do not use symbols from other languages. It is a common trap to associate a symbol exclusively with some particular meaning, rather than learning and understanding the physics and relations behind it.

=See Also= Greek alphabet on Wikipedia

Greek letters used in mathematics, science, and engineering

title: Physics Study Guide/Greek alphabet


text:
Cascading Style Sheets

This book is a guide to Cascading Style Sheets (CSS), a technique widely used in web pages including Wikipedia to describe their visual style and appearance. CSS can take HTML to new places creatively and functionally. Once you learn how to style mark-up, you can additionally learn JavaScript functions that make dynamic web pages.

Contents

 * 1) /Introduction/
 * 2) /Applying CSS to HTML and XHTML/ — linking, embedding and inlining
 * 3) /Applying CSS to XML/
 * 4) CSS Construction
 * 5) /Syntax/
 * 6) /Defining Style Rules/
 * 7) /Lengths and Units/
 * 8) /Selectors/
 * 9) /Inheritance Rules/
 * 10) The Important Keyword
 * 11) CSS Presentation
 * 12) /Color/
 * 13) /Fonts and Text/
 * 14) /Hyperlinks/
 * 15) /Pseudo Elements and Pseudo Classes/
 * 16) /Clipping/
 * 17) /Lists/
 * 18) /Box Model/ — setting the size and shape of elements
 * 19) /Borders/
 * 20) /Background/
 * 21) /Shorthand Properties/
 * 22) /Cursor/
 * 23) /Gradients/
 * 24) /Sprites/
 * 25) CSS Layout
 * 26) /Centering/
 * 27) /Positioning/
 * 28) /Tables/
 * 29) /Floating Elements/
 * 30) /Typography/
 * 31) /Media Types/
 * 32) /Multi-Column Layout/
 * 33) /z-index/
 * 34) CSS URI Schemes
 * 35) /MHTML URIs/
 * 36) /Data URIs/
 * 37) CSS Applications
 * 38) /CSS Menus/
 * 39) /Breadcrumbs/
 * 40) /User Style Sheets/
 * 41) Troubleshooting
 * 42) Standards Mode and Quirks Mode
 * 43) Browser Compatibility
 * 44) CSS Source Order (Troubleshooting the cascade of repeated selectors)
 * 45) Appendices
 * 46) /Hacks and Filters/
 * 47) /Browser Extensions/
 * 48) /Links/
 * 49) /Index of Properties/
 * 50) /Index/
 * 51) /Glossary/
 * 52) /Detailed TOC/
 * 53) /Contributors/
 * 54) /To Do/

Related Wikibooks

 * JavaScript
 * XHTML
 * The CSS Cascade
 * Style Presentation Modes in Opera 12.16

de:Websiteentwicklung: CSS CSS Le langage CSS CSS Pemrograman CSS CSS CSS Lembaran Gaya Lata Cascading Style Sheets CSS CSS CSS CSS CSS CSS

title: Cascading Style Sheets


text:
German/Level III/Der Engländer in Österreich

Lektion Fünf

Rathaus von St. Pölten

Gespräch 5-2 ~ Der Engländer in Österreich
Republik Österreich Wenn er auf den Kontinent fährt, wandert Herr Standish gern. Heute früh fährt er in die Stadt St. Pölten in Niederösterreich. Er spricht mit einer fremden Frau:
 * Herr Standish: Entschuldigen Sie bitte. Wo ist hier ein Hotel?
 * Die Frau: Gleich dort drüben. Das ist das Hotel "Zur Post".
 * Herr Standish: Gibt es ein Restaurant darin?
 * Die Frau: Ja gewiss! Ein Restaurant mit einfacher Küche, besonders zum Abendessen. Aber ich könnte Ihnen ein anderes Restaurant empfehlen. Es heißt 'Alt-Wien', und es gibt dort das beste Frühstück. Das Restaurant ist links neben dem Hotel, um die Ecke.
 * Herr Standish: Danke sehr. Und können Sie mir sagen, wo das Rathaus von St. Pölten ist?
 * Die Frau: Wie bitte?
 * Herr Standish: Wie komme ich zum Rathaus?
 * Die Frau: Rechts um die Ecke und dann immer geradeaus – ungefähr ein Kilometer.
 * Herr Standish: Danke sehr.
 * Die Frau: Bitte sehr. Wiedersehen.
 * Herr Standish: Auf Wiedersehen.

Vokabeln 5A
das Abendessen            supper (evening meal) [das] Österreich          Austria die Ecke                  corner das Frühstück             breakfast das Hotel                 hotel der Kilometer             kilometer die Küche                 cooking, cuisine der Kontinent             continent (Europe) [das] Niederösterreich    (federal state of) Lower Austria das Rathaus               city hall das Restaurant            restaurant die Stadt                 city

Bitte sehr                You're welcome Entschuldigen Sie         Pardon me, excuse me Es gibt dort...            There is there... Gibt es...? Is there..? Guten Tag                 good day (parting) immer geradeaus           straight on ahead können Sie                could you (polite form) Wie bitte? Pardon me? (polite "come again?")

empfehlen                 recommend fahren                    travel kommen                    come, go, get wandern                   wander sagen                     say, tell sprechen                  speak

anderer, andere, anderes  other besonders                 especially bitte                     please das                       that dann                      then darin                     therein ein                       a (indefinite article) eins                      one (cardinal number) fremd                     unknown gern                      gladly gleich                    just, right (correct), right here, same heute früh                this morning hier                      here (in this place) ich                       I (personal pronoun) links                     left (direction) neben                     next to rechts                     right (direction) ungefähr                  approximately von                       of (Rathaus von St. Pölten = St. Polten City Hall) wie                       how (interrogative) wo                        where (interrogative) zu                        to (zum = contraction of zu dem)

Andere Wörter 4A
der Bahnhof               train station der Flughafen             airport die Polizeiwache          police station die Post                  post office

genau                     exact(ly) heute                     today

Lesestück 5-1 ~ Eine Geschichte über St. Pölten
Karte: St. Pölten in Österreich

Niederösterreich ist sowohl flächenmäßig als auch nach Einwohnern das größte der neun österreichischen Bundesländer. Sankt Pölten ist die Landeshauptstadt von Niederösterreich. Der Name St. Pölten geht auf den heiligen Hippolytos zurück, nach dem die Stadt benannt wurde.

Die Altstadt befindet sich dort, wo vom 2. bis zum 4. Jahrhundert die Römerstadt Aelium Cetium stand. 799 wurde der Ort als "Treisma" erwähnt. Das Marktrecht erhielt St. Pölten um 1050, zur Stadt erhoben wurde es 1159. Bis 1494 stand St. Pölten im Besitz des Bistums Passau, dann wurde es landesfürstliches Eigentum. Bereits 771 findet sich ein Benediktinerkloster, ab 1081 gab es Augustiner-Chorherren, 1784 wurde deren Kollegiatsstift aufgehoben, das Gebäude dient seit 1785 als Bischofssitz. Zur Landeshauptstadt von Niederösterreich wurde St. Pölten mit Landtagsbeschluss vom 10. Juli 1986, seit 1997 ist es Sitz der Niederösterreichischen Landesregierung. Luftbild von St. Pölten

Vokabeln 5B
Die Altstadt                  old town Der Augustiner                Augustinian Der Besitz                    possession, holding Das Bistum                    diocese Der Bischofssitz              bishop's see (a seat of a bishop's authority) Die Bundesländer              federal states Die Chorherren                men's choir Das Eigentum                  proprietorship Die Einwohner                 inhabitants Das Gebäude                   premises Die Geschichte                history Das Jahrhundert               century Das Kloster                   monastery, friary Das Kollegiatsstift           monastery college Die Landeshauptstadt          regional or state capital city Die Landesregierung           provincial (state) government Der Landtagsbeschluss         day of jurisdictional reorganization Das Marktrecht                right to hold markets Der Name                      name Der Ort                       place, spot, city Die Römerstadt                Roman town Der Sitz                      official place

Bistum Passau                 a dioecian region in Bavaria sowohl... als auch            both... and zurück auf                    goes back to

aufheben (hob auf, aufgehoben) merged in (or turned into?) befinden sich                 situated, located (befand sich, haben sich befunden) finden sich*                  found (located) benennen (benannte, benannt)  call (as to label) erhalten (erhielt, erhalten)  receive erheben (erhob, erhoben)      arise, raise erwähnen (erwähnte, erwähnt)  mention stehen (stand, gestanden)     stand (stood, stood) werden (wurde, [ist]geworden) become

ab                            from auf                           up bereits                        already bis                           until, by, up to flächenmäßig                   (no direct translation) ~ when measured in surface heilig                        holy landesfürstlich               baronial or princely (holdings) nach                          in terms of  um                             around

(* one short form of anfinden: findet sich (an); in colloquial language you can cut the "an"; but in THIS special case it is the short form of "(be)findet sich (dort)")

Pronunciation Guide >>

 * Read more about St. Pölten at the German Wikipedia (source of article above).


text:
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High School Mathematics Extensions/Mathematical Proofs

 * "It is by logic that we prove, but by intuition that we discover."

Introduction
Mathematicians have been, for the past five hundred years or so, obsessed with proofs. They want to prove everything, and in the process proved that they can't prove everything (see this). This chapter will introduce the axiomatic approach to mathematics, and several types of proofs.

Direct proof
The direct proof is relatively simple — by logically applying previous knowledge, we directly prove what is required.

Example 1

Prove that the sum of any two even integers $$x$$ and $$y$$ is even.

Solution 1

We know that since $$x$$ and $$y$$ are even, they must have 2 as a factor. Then, we can write the following:
 * Let $$x=2a$$, $$y=2b$$ , for some integers $$a,b$$

Then:
 * $$\begin{matrix}x+y&=&2a+2b\\&=&2(a+b)\end{matrix}$$

by the distributive property of integers

The number $$2(a+b)$$ clearly has 2 as a factor, which implies it is even. Therefore, $$x+y$$ is even.

Example 2

Prove the following statement for non-zero integers $$a,b,c$$:

If $$a$$ divides $$b$$ and $$b$$ divides $$c$$, then $$a$$ divides $$c$$.

Solution 2

If an integer $$x$$ divides an integer $$y$$, then we can write $$y=qx$$ , for some non-zero integer $$q$$. So let's say that $$b=qa$$ and $$c=rb$$, for some non-zero integers $$q$$ and $$r$$. Then:
 * $$\begin{matrix}c&=&rb\\&=&r(qa)\\&=&(rq)a\end{matrix}$$

by the associative property of integer multiplication.

But since $$q$$ and $$r$$ are integers, their product $$qr$$ must also be an integer. Therefore, $$c$$ is the product of some integer multiplied by $$a$$, so we get that $$a$$ divides $$c$$.

Mathematical induction
Deductive reasoning is the process of reaching a conclusion that is guaranteed to follow. For example, if we know then we can conclude:
 * All ravens are black birds, and
 * For every action, there is an equal and opposite reaction
 * This bird is a raven, therefore it is black.
 * This billiard ball will move when struck with a cue.

Induction is the opposite of deduction. To induce, we observe how things behave in specific cases and from that we draw conclusions as to how things behave in the general case.

Suppose we want to show that a statement (let us call it $$S$$ for easier notation) is true for all natural numbers. This is how induction a proof by induction works:
 * 1) First, we show that $$S$$ is true for the natural number 1. This is usually called the basis or the base case.
 * 2) Then, we show that $$S$$ is true for natural number $$n+1$$ whenever it is true for natural number $$n$$.
 * 3) By mathematical induction, $$S$$ is true for all natural numbers.

To understand how the last step works, notice the following
 * $$S$$ is true for 1 (due to step 1)
 * $$S$$ is true for 2 because it is true for 1 (due to step 2)
 * $$S$$ is true for 3 because it is true for 2 (due to previous)
 * $$S$$ is true for 4 because it is true for 3 (due to previous)
 * $$S$$ is true for 5 because it is true for 4 (due to previous)
 * and so on...

Example 1 Show that the identity
 * $$1+2+\dots+n=\frac{n(n+1)}{2}$$

holds for all positive integers.

Solution Firstly, we show that it holds for 1
 * $$1 = \frac{(1 + 1)1}{2} = \frac{2}{2} = 1 $$

Suppose the identity holds for some natural number k:
 * $$1 + 2 + 3 + ... + k = \frac{1}{2}(k + 1)k $$

This supposition is known as the induction hypothesis. We assume it is true, and aim to show that,
 * $$1 + 2 + 3 + ... + k + (k + 1) = \frac{1}{2}(k + 2)(k + 1) $$

is also true.

We proceed

\begin{matrix} 1 + 2 + 3 + ... + k & & =& \frac{1}{2}(k + 1)k\\ \\ 1 + 2 + 3 + ... + k &+ (k + 1) &=& \frac{1}{2}(k + 1)k + (k + 1)\\ \\ & & = & (k + 1)(\frac{k}{2} + 1)\\ \\ & & = & \frac{1}{2}(k + 1)(k + 2) \end{matrix} $$ which is what we have set out to show. Since the identity holds for 3, it also holds for 4, and since it holds for 4 it also holds for 5, and 6, and 7, and so on.

There are two types of mathematical induction: strong and weak. In weak induction, you assume the identity holds for certain value k, and prove it for k+1. In strong induction, the identity must be true for any value lesser or equal to k, and then prove it for k+1.

Example 2 Show that n! > 2n for n ≥ 4.

Solution The claim is true for n = 4. As 4! > 24, i.e. 24 > 16. Now suppose it's true for n = k, k ≥ 4, i.e.
 * k! > 2k

it follows that
 * (k+1)k! > (k+1)2k > 2k+1
 * (k+1)! > 2k+1

We have shown that if for n = k then it's also true for n = k + 1. Since it's true for n = 4, it's true for n = 5, 6, 7, 8 and so on for all n.

Example 3 Show that
 * $$1^3 + 2^3 + ...+ n^3 = \frac {(n+1)^2n^2}{4} $$

Solution Suppose it's true for n = k, i.e.
 * $$1^3 + 2^3 + ...+ k^3 = \frac {(k+1)^2k^2}{4} $$

it follows that

\begin{matrix} 1^3 + 2^3 + ...+ k^3 + (k+1)^3 & = &\frac {(k+1)^2k^2}{4} + (k+1)^3\\ & = & (k+1)^2 (\frac{k^2}{4} + (k+1))\\ & = & \frac {1}{4}(k+1)^2 (k^2 + 4k + 4)\\ & = & \frac {1}{4}(k+1)^2 (k + 2)^2 \end{matrix} $$ We have shown that if it's true for n = k then it's also true for n = k + 1. Now it's true for n = 1 (clear). Therefore it's true for all integers.

Exercises
1. Prove that $$1^2 + 2^2 + ... + n^2 = \frac{ n(2n^2 + 3n +1)}{6} $$

2. Prove that for n ≥ 1,
 * $$ (1 + \sqrt{5})^n = x_n + y_n\sqrt{5}$$

where xn and yn are integers.

3. Note that
 * $$\sum_{i=1}^n[i^k - (i-1)^k] = n^k$$

Prove that there exists an explicit formula for
 * $$\sum_{i=1}^ni^m$$ for all integer m. E.g.

$$1^3 + 2^3 + ... + n^3 = \frac{n^2(n+1)^2}{4}$$

4. The sum of all of the interior angles of a triangle is $$180^\circ$$; the sum of all the angles of a rectangle is $$360^\circ$$. Prove that the sum of all the angles of a polygon with n sides, is $$(n - 2)\cdot 180^\circ$$.

Proof by contradiction

 * "When you have eliminated the impossible, what ever remains, however improbable must be the truth." Sir Arthur Conan Doyle

The idea of a proof by contradiction is to:
 * 1) First, we assume that the opposite of what we wish to prove is true.
 * 2) Then, we show that the logical consequences of the assumption include a contradiction.
 * 3) Finally, we conclude that the assumption must have been false.

√2 is irrational
As an example, we shall prove that $$\sqrt{2}$$ is not a rational number. Recall that a rational number is a number which can be expressed in the form of p/q, where p and q are integers and q does not equal 0 (see the 'categorizing numbers' section here).

First, assume that $$\sqrt{2}$$ is rational:

\sqrt{2} = \frac{a}{b} $$ where a and b are coprime (i.e. integers with no common factors, with greatest common divisor 1). If a and b are not coprime, we remove all common factors. In other words, a/b is in simplest form. Now, continuing:

\begin{matrix} \sqrt{2} &=& a/b \\ 2 &=& a^2/b^2 \\ 2b^2 &=& a^2 \end{matrix} $$

We have now found that a2 is some integer multiplied by 2. Therefore, a2 must be divisible by two. If a2 is even, then a must also be even, for an odd number squared yields an odd number. Therefore we can write a = 2c, where c is another integer.

\begin{matrix} 2b^2 &=& a^2 \\ 2b^2 &=& (2c)^2 \\ 2b^2 &=& 4c^2 \\ b^2 &=& 2c^2 \end{matrix} $$

We have discovered that b2 is also an integer multiplied by two. It follows that b must be even. We have a contradiction! Both a and b are even integers. In other words, both have the common factor of 2. But we already said that a/b is in simplest form, with no common factors. Since such a contradiction has been established, we must conclude that our original assumption was false. Therefore, √2 is irrational.

Contrapositive
Some propositions that take the form of if xxx then yyy can be hard to prove. It is sometimes useful to consider the contrapositive of the statement. Before I explain what contrapositive is let us see an example
 * "If x2 is odd then x is also odd"

is harder to prove than
 * "if x is even then x2 is also even"

although they mean the same thing. So instead of proving the first proposition directly, we prove the second proposition instead.

If A and B are two propositions, and we aim to prove
 * If A is true then B is true

we may prove the equivalent statement
 * If B is false then A is false

instead. This technique is called proof by contrapositive.

To see why those two statements are equivalent, we show the following boolean algebra expressions is true (see Logic)
 * $$p \Rightarrow q \equiv q' \Rightarrow p' \!$$

(to be done by the reader).

Exercises
1. Prove that there is no perfect square number for 11,111,1111,11111......

2. Prove that there are infinitely number of k's such that, 4k + 3, is prime. (Hint: consider N = p1p2...pm + 3)

Reading higher mathematics
This is some basic information to help with reading other higher mathematical literature. ... to be expanded

Quantifiers
Sometimes we need propositions that involve some description of rough quantity, e.g. "For all odd integers x, x2 is also odd". The word all is a description of quantity. The word "some" is also used to describe quantity.

Two special symbols are used to describe the quanties "all" and "some"
 * $$\forall$$ means "for all" or "for any"
 * $$\exists$$ means "there are some" or "there exists"

Example 1 The proposition:
 * For all even integers x, x2 is also even.

can be expressed symbolically as:
 * $$(\forall x)(x\mbox{ is even} \Rightarrow x^2\mbox{ is even}) $$

Example 2 The proposition:
 * There are some odd integers x, such that x2 is even.

can be expressed symbolically as:
 * $$(\exist x)(x\mbox{ is odd} \Rightarrow x^2\mbox{ is even}) $$

This proposition is false.

Example 3 Consider the proposition concerning (z = x'y' + xy):
 * For any value of x, there exist a value for y, such that z = 1.

can be expressed symbolically as:
 * $$(\forall x)(\exist y)(z = 1) $$

This proposition is true. Note that the order of the quantifiers is important. While the above statement is true, the statement
 * $$(\exist y)(\forall x)(z = 1) $$

is false. It asserts that there is one value of y which is the same for all x for which z=1. The first statement only asserts that there is a y for each x, but different values of x may have different values of y.

Negation
Negation is just a fancy word for the opposite, e.g. The negation of "All named Britney can sing" is "Some named Britney can't sing". What this says is that to disprove that all people named Britney can sing, we only need to find one named Britney who can't sing. To express symbolically:
 * Let p represent a person named Britney
 * $$[(\forall p)(p\mbox{ can sing})]' = (\exists p)(p\mbox{ cannot sing})$$

Similarly, to disprove
 * $$(\forall x)(x\mbox{ is odd} \Rightarrow x^2\mbox{ is even}) $$

we only need to find one odd number that doesn't satisfy the condition. Three is odd, but 3×3 = 9 is also odd, therefore the proposition is FALSE and
 * $$(\exists x)(x\mbox{ is odd} \Rightarrow x^2\mbox{ is odd}) $$

is TRUE

In summary, to obtain the negation of a proposition involving a quantifier, you replace the quantifier by its opposite (e.g. $$\forall$$ with $$\exist$$) and the quantified proposition (e.g. "x is even") by its negation (e.g. "x is odd").

Example 1
 * $$(\forall x)(\exists y)(x(x + 1)(x + 2)(x + 3) + 1 = y^2) $$

is a true statement. Its negation is
 * $$(\exists x)(\forall y)(x(x + 1)(x + 2)(x + 3) + 1 \ne y^2) $$

Axioms and Inference
If today's mathematicians were to describe the greatest achievement in mathematics in the 20th century in one word, that word will be abstraction. True to its name, abstraction is a very abstract concept (see Abstraction).

In this chapter we shall discuss the essence of some of the number systems we are familiar with. For example, the real numbers and the rational numbers. We look at the most fundamental properties that, in some sense, define those number systems.

We begin our discussion by looking at some of the more obscure results we were told to be true Most people simply accept that they are true (and they are), but the two results above are simple consequences of what we believe to be true in a number system like the real numbers!
 * 0 times any number gives you 0
 * a negative number multiplied by a negative number gives you a positive number

To understand this we introduce the idea of axiomatic mathematics (mathematics with simple assumptions). An axiom is a statement about a number system that we assume to be true. Each number system has a few axioms, from these axioms we can draw conclusions (inferences).

Let's consider the Real numbers, it has axioms Let a, b and c be real numbers
 * For a, b, and c taken from the real numbers
 * A1: a+b is a real number also (closure)
 * A2: There exist 0, such that 0 + a = a for all a (existence of zero - an identity)
 * A3: For every a, there exist b (written -a), such that a + b = 0 (existence of an additive inverse)
 * A4: (a + b) + c = a + (b + c) (associativity of addition)
 * A5: a + b = b + a (commutativity of addition)

 * For a, b, and c taken from the real numbers excluding zero
 * M1: ab is a real number also (closure)
 * M2: There exist an element, 1, such that 1a = a for all a (existence of one - an identity)
 * M3: For every a there exists a b such that ab = 1
 * M4: (ab)c = a(bc) (associativity of multiplication)
 * M5: ab = ba (commutativity of multiplication)

 * D1: a(b + c) = ab + ac (distributivity)

These are the minimums we assume to be true in this system. These are minimum in the sense that everything else that is true about this number system can be derived from those axioms!

Let's consider the following true identity
 * (x + y)z = xz + yz

which is not included in the axioms, but we can prove it using the axioms. We proceed:

\begin{matrix} (x + y)z & = & z(x + y) \ \mbox{by M5}\\ & = & zx + zy \ \mbox{by D1}\\ & = & xz + yz \ \mbox{by M5}\\ \end{matrix} $$

Before we proceed any further, you will have notice that the real numbers are not the only numbers that satisfies those axioms! For example the rational numbers also satisfy all the axioms. This leads to the abstract concept of a field. In simple terms, a field is a number system that satisfies all those axiom. Let's define a field more carefully:

 A number system, F, is a field if it supports + and × operations such that:
 * For a, b, and c taken from F
 * A1: a + b is in F also (closure)
 * A2: There exist 0, such that 0 + a = a for all a (existence of zero - an identity)
 * A3: For every a, there exist b (written -a), such that a + b = 0 (existence of an additive inverse)
 * A4: (a + b) + c = a + (b + c) (associativity of addition)
 * A5: a + b = b + a (commutativity of addition)

 * For a, b, and c taken from F with the zero removed (sometimes written F*)
 * M1: ab is in F (closure)
 * M2: There exist an element, 1, such that 1a = a for all a (existence of one - the identity)
 * M3: For every a there exists a b such that ab = 1 (inverses)
 * M4: (ab)c = a(bc) (associativity of multiplication)
 * M5: ab = ba (commutativity of multiplication)

 * D1: a(b + c) = ab + ac (distributivity)

Now, for M3, we do not let b be zero, since 1/0 has no meaning. However for the M axioms, we have excluded zero anyway.

For interested students, the requirements of closure, identity, having inverses and associativity on an operation and a set are known as a group. If F is a group with addition and F* is a group with multiplication, plus the distributivity requirement, F is a field. The above axioms merely state this fact in full.

Note that the natural numbers are not a field, as M3 is generally not satisfied, i.e. not every natural number has an inverse that is also a natural number.

Please note also that (-a) denotes the additive inverse of a, it doesn't say that (-a) = (-1)(a), although we can prove that they are equivalent.

Example 1

Prove using only the axioms that 0 = -0, where -0 is the additive inverse of 0.

Solution 1
 * 0 = 0 + (-0) by A3: existence of inverse
 * 0 = (-0) by A2: 0 + a = a

Example 2

Let F be a field and a an element of F. Prove using nothing more than the axioms that 0a = 0 for all a.

Solution
 * 0 = 0a + (-0a) by A3 existence of inverse
 * 0 = (0 + 0)a + (-0a) by Example 1
 * 0 = (0a + 0a) + (-0a) by distributivity and commutativity of multiplication
 * 0 = 0a + (0a + (-0a)) by associativity of addition
 * 0 = 0a + 0 by A3
 * 0 = 0a by A2.

Example 3

Prove that (-a) = (-1)a.

Solution 3
 * (-a) = (-a) + 0
 * (-a) = (-a) + 0a by Example 2
 * (-a) = (-a) + (1 + (-1))a
 * (-a) = (-a) + (1a + (-1)a)
 * (-a) = (-a) + (a + (-1)a)
 * (-a) = ((-a) + a) + (-1)a
 * (-a) = 0 + (-1)a
 * (-a) = (-1)a

One wonders why we need to prove such obvious things (obvious since primary school). But the idea is not to prove that they are true, but to practise inferencing, how to logically join up arguments to prove a point. That is a vital skill in mathematics.

Exercises
1. Describe a field in which 1 = 0

2. Prove using only the axioms if u + v = u + w then v = w (subtracting u from both sides is not accepted as a solution)

3. Prove that if xy = 0 then either x = 0 or y = 0

4. In F-, the operation + is defined to be the difference of two numbers and the × operation is defined to be the ratio of two numbers. E.g. 1 + 2 = -1, 5 + 3 = 2 and 9×3 = 3, 5×2; = 2.5. Is F- a field?

5. Explain why Z6 (modular arithmetic modular 6) is not a field.

Problem Set
1. Prove

\frac{1}{\sqrt 1} + \frac{1}{\sqrt 2} + ... + \frac{1}{\sqrt n }\ge \sqrt n $$ for $$n\ge 1$$

2. Prove by induction that $$2n^3 - 3n^2 + n + 31 \ge 0 $$

3. Prove by induction
 * $$ {n \choose 0} + {n\choose 1} + {n\choose 2} + ... + {n\choose n} = 2^n$$

where
 * $${n \choose m} = \frac{n!}{m!(n-m)!} $$ and $$n! = n\cdot (n-1)\cdot (n-2)\cdot ... \cdot 2\cdot 1$$
 * and 0! = 1 by definition.

4. Prove by induction $$ {n \choose 0} + 2{n\choose 1} + 2^2{n\choose 2} + ... + 2^n{n\choose n} = 3^n$$

5. Prove that if x and y are integers and n an odd integer then $$\frac{x^n + y^n}{x + y} $$ is an integer.

6. Prove that (n~m) = n!/((n-m)!m!) is an integer. Where n! = n(n-1)(n-2)...1. E.g. 3! = 3×2×1 = 6, and (5~3) = (5!/3!)/2! = 10.

''Many questions in other chapters require you to prove things. Be sure to try the techniques discussed in this chapter.''


text:
Talk:German/Lesson 2

Untitled
We could use a photo illustrating the conversation in Gespräch 2-2 ~ Das neues Mädchen. One of a young couple talking in a social setting; even better, with a short, long-haired brunette in the background - Marsh 03:56, 16 Jan 2004 (UTC)

Would the translation for spielen be play, as is on the page, or the infinitive to play? Lord Emsworth 02:35, 1 Nov 2003 (UTC)
 * I think spielen is the infinitive, I can play should be Ich kann spiele. Maybe Thomas can clear this up. Etothex 04:48, 1 Nov 2003 (UTC)

I think it is a matter of how you want to present verbs. I originally had them as "to play" (which is correct), but it is also correct I believe to simply list the english verb that spielen represents: play. The vocabulary is not a direct translation list, but a list of words in German from the excercise and what they mean in that context) in English. I have two text books and a large Dictionary in front of me and one text gives the verbs thus: spielen - to play; whereas another textbook and the dictionary (New Cassell's) list them thus: spielen - play. - Marsh 05:11, 1 Nov 2003 (UTC)
 * Hmm, ok. I looked up spielen on LEO and I got the infinitive 'to play', with(present tense)

ich 	spiele 	 AKLL OF THESE ARE "play" IN ENGLISH du 	spielst er/sie 	spielt wir 	spielen ihr 	spielt sie 	spielen but then babelfish/altavista translated 'I can play' as Ich kann spielen, so I don't know what's correct. I suppose since you know more German than I do that you're right. :) Etothex 08:39, 1 Nov 2003 (UTC)

Yes, kann spielen is correct: ich      kann spielen du       kannst spielen (formal: Sie können spielen) er/sie/es kann spielen wir      können spielen ihr      könnt spielen (formal: Sie können spielen) sie      können spielen The formal "Sie" is captialized.

Some other forms with the infinitive: ich       muss   / darf   / soll   / will du        musst  / darfst / sollst / willst er/sie/es muss   / darf   / soll   / will wir       müssen / dürfen / sollen / wollen ihr       müsst  / dürft  / sollt  / wollt sie       müssen / dürfen / sollen / wollen

And some things (negation of the above) which are different: ich muss       I must ich darf       I am allowed to ich muss nicht  I don't need to ich darf nicht  I must not ich soll       I should ich soll nicht I should not ich kann       I can (I am able to, I am allowed to) ich kann nicht I cannot (like above) ich will       I want to ich will nicht  I dont't want to --Vulture 09:37, 1 Nov 2003 (UTC)

 * Well, I'm saying either way of presentation is correct. Not giving the English "to play", but instead using "play" is just a matter of personal preference. My very big German dictionary does not present English verbs as "to xxxxx" in the German to English translations. - Marsh 09:44, 1 Nov 2003 (UTC)

 * Yes, the "to" is just a confirmation (clarification) that it is the inifinitive. The dictionary usually only lists irregular forms, so it's somewhat reduntant (although my dictionary says "to" ;). --Vulture 09:53, 1 Nov 2003 (UTC)

Fräulein was used instead of Frau to refer to an unmarried woman. It's no longer used since the 60s or 70s. It's rather infrequently used to address a waitress. Otherwise it has a humoristic or ironic meaning. --Vulture 07:53, 4 Nov 2003 (UTC)

 * Hey, that's when I learned German! What is "Miss" now? Anyway, certainly go in and change anything you see to modernize the German. Really appreciated! - Marsh 18:41, 4 Nov 2003 (UTC)

 * Fräulein was replaced by Frau, there is no special "Miss". You might use it you translate something english and "Miss" (instead of Mrs) is important. But you might also simply use "Miss" in this case. --Vulture 09:19, 5 Nov 2003 (UTC)

 * Hard to believe a culture would dump a word implying youthful beauty (Fräulein) for a word that has a more (IMHO) frumpy connotation (Frau). Did the young people simply start calling "chicks" --> "Fraus"?  Or was there always a different set of colloquial expressions, and Fräuline went out with the Miss, Mrs., Ms. trends as in the US? - Marsh 17:34, 5 Nov 2003 (UTC) Along the same lines, would changing the sentence in Gesprach 2-2 >> Helena: Sie ist hübsch, wenn man kleine Mädchen mit langen schwarzen Haaren mag. to >>Helena: Sie ist hübsch, wenn man kleine Fräulein mit langen schwarzen Haaren mag. put more of an edge on her jealose reply to Karl?  That is, could one translate Fräulein in this case as "girl (tart)"? Or is there a word that would be more suitable (or is Mädchen just fine)?  - Marsh

 * It was considered inappropriate to see if somebody is married in the name (the "title"), especially if it's only used for females (i.e. no "Herrlein"). "Mädels", "Mädchen" is more like "chicks". "Schlampe" might do for Helena (*). My dictionary gives "slut" as translation.

(*) to say, not to be - "Zicke" (nanny goat ;), cow, bitch, prude, silly thing) might better characterize Helena. I don't think those translations are right though. --Vulture 08:42, 6 Nov 2003 (UTC)

 * For now I can leave it at "mädchen" I don't want her attitude to be too obvious: A more subtle "oh yeh, if you like that kind of girl" is what I'm aiming for - Marsh 16:27, 6 Nov 2003 (UTC)

In German: Lesson 2 I've changed one "Fremder" to "Fremde" and another one to "Fremden". Both are nominative plural, and I don't know exactly why they differ, but I'm quite sure of this because I'm a native German speaker. It probably has something to do with "fremd" being an adjective. "Der Fremde" is a noun, but the endings are the same as those of the adjective: "fremde Leute" (="foreign people") but "die fremden Leute" (="the foreign people").--El 15:57, 7 Nov 2003 (UTC)

 * Ok, Marsh. I'm not sure if it's necessary to introduce reflexive verbs in the first lesson. You could avoid it by saying: "Heinrich trifft Karl. Sie sind Freunde." -- Hansm 18:38, 16 Dec 2003 (UTC)
 * Of course, it's (almost) right to say "Haben Sie Herr Standish schon angetroffen?". But I feel, the word "antreffen, angetroffen" is pretty difficult for the first lesson. You could say "Haben Sie Herrn Standish schon getroffen?" Anyway, this is more common.
 * Or how about "Haben Sie Herrn Standish schon gesehen?" - Note: Actualy, "Herr Standish" stands in the Dativ, here. Nevertheless, many Germans would also say "Herr Standish" in colloquial language.
 * Since you want to give a lesson about pronouns and the 2 forms for the 2nd person, I would try to keep the rest as easy as possible. The sentence "Aber verstehe ich richtig, dass er zu Besuch ist?" has a relativ sentence. Wouldn't it easyer just to say: "Ist er zu Besuch?" -- Hansm 18:51, 16 Dec 2003 (UTC)
 * Sounds good to me. Simplifying the early lessons is always a better idea. I will implement changes when I get the chance. Danke! - Marsh 05:06, 19 Dec 2003 (UTC)
 * Well, I have done some simplifications and adapted the list of vocabularies. I hope, it's not too much against your initial idaes. -- Hansm 00:27, 22 Dec 2003 (UTC)
 * Looks good to me. Danke sehr - Marsh 01:58, 25 Dec 2003 (UTC)

I think there's a little problem with "heißen" in section "Grammatik 2-3": It's only used in a passive meaning like it's said there: "Thus heißen can best be translated as 'to be named' or 'to be called'." But the 4 examples demonstrate an active meaning of "call/name". This would be translated as "nennen" in German, so the examples really may make it clearer, but they are wrong... or at least have a quite archaic sound ;) - Monsi 17:59, 3 Mar 2004 (UTC)
 * I believe there are a number of German nouns that make sense in the present tense in German, but are not so easily or singularly translated into English. English would never say "I name Eric" for "my name is..." - But if you are saying German would not any more really say "Ich heiße Erick" but....what (?) "Ich nenne Erick?" Still the same non-correspondence between German present tense and English. Please clarify; is it that heißen is really only used in the passive? - Marsh 05:44, 4 Mar 2004 (UTC)
 * Yes, heißen is only used in the sense of "to be called". It used to be used like "jdn etw. heißen"(to call sb. sth.), but that usage is archaic, dating back to the 1920s or so I'd say. "nennen" requires the reflexive pronoun if used to refer to oneself: "Ich nenne mich Judith", but that's not really common when introducing oneself, it's like saying "I call myself Judith". "nennen" is what we use to say "to call sb. sth." nowadays though, e. g. "er nannte ihn einen Idiot" = "he called him an idiot".
 * I was not happy with the way the lesson was going using heißen—too complicated for so early in the text, and even the examples come from sentences that have since moved deeper into the book. I will rewrite and drop heißen here; but it or nennen will show up sooner than latter. Are some of the examples with heißen "modern", or all "archaic" - Marsh 17:10, 5 Mar 2004 (UTC)

There is a lot more going on here: It's even worse for the male form: 'Der Herr' without anything else indicating the use as title/honorific would actually refer to god.
 * 1) The line about 'to be' refering to the use of present continous makes no sense at all. 'Heißen/Nennen' (Heißen is wrong as we know, substitute with 'Nennen'and it works). But nevertheless, 'Nennen' needs a passive construction similar to its closest English equivalent 'to be called xyz'. Continous does not! come into it. "Die Firma wird Trans-Global genannt." -> "The company is called Trans-Global" Both passive voice.
 * 2) Contrary to the commentary there is a different way to express 'is called'. It takes - surprise - the verb 'heißen' only with a change of subjects: Forget the 'They' and a passive voice. It is rather "Die Firma heißt Trans-Global." Nice, clean and easy.
 * 3) What's the deal with 'Um!-Global'? Proper names, especially of Latin origin, only get the worse for translation. Trans-Global sounds like a good German import-export firm. There probably is one for real just around the corner!
 * 4) Concerning Company = 'Gemeinde'. 'Gemeinde' means either 'parish' (religious group) or 'district' (local bureaucratic entity). I would go for 'Firma'.
 * 5) 'Die Geschäftsmänner'. Thought long and hard about it and the conclusion is, that maybe with pc and the rest of it, the plural of 'Geschäftsmann' is now 'Geschäftsleute'. We just don't use pure masuline plurals anymore, even if they happen to be all male.
 * 6) I will quickly nip over and correct 'das! Vereinigtes! Königreich'
 * 7) Somewhere I changed 'friends' = 'Brüder' into 'Freunde'. To call a mate your sibbling is theoretically possible in a colloquial setting (a bit like Hey Bro'). For the beginner, it could only be confusing.
 * 8) There is also a problem with 'die Frau' = 'Ms.' That is just confusing vocab practice because 'die Frau' = 'the woman'. That does not convey the meaning of honorific adress. You have to cut out the defnite article. Better then to learn it as a phrase than as an ambivlent single word.

That's it for the moment.
 * Floflei6

OK. Fix what you regard as archaic (a lot of these words come from LEO which has no systyem for indicating what is currently acceptable and what is archaic). I'll change the Herr und Frau as you suggest. - Marsh 17:42, 7 Apr 2004 (UTC)

According to LEO, it is "Vereinigtes Königreich" - marsh 18:49, 6 Jan 2005 (UTC)

"Vereinigtes Königreich" without articel, but "Das Vereinigte Königreich ...

"Das ist richtig." rather unusual
In Germany "Das ist richtig" is mostly used as "That's correct" as a teacher says to his pupil ("Did I make a mistake?" - "No, that's correct"). But here it is used twice in unusual situtations:
 * "Ist er zu Besuch? - Ja. Das ist richtig." should be simply "Ja." or "Ja, er wohnt zur Zeit bei mir." or "Ja" with another additional information.
 * "Zu viel ist zu viel. - Das ist richtig." should be "So ist es.", "Da haben Sie Recht." or something like that.
 * Obviously such subtleties reflect (among other things) changes in the use of words/phrases over time. I have no problem with using your suggestions - marsh 18:51, 14 November 2005 (UTC)

studieren
"Today we study German." would rather be "Heute lernen wir Deutsch." To study means "studieren" or "lernen" - but in German the two words are not the same thing. "Studieren" is what you do at a university - usually for a couple of years (even though we tend to drop out or change our subjects quite often // studieren can as well mean to examine thoroughly) jacs --Jacs 01:08, 31 January 2006 (UTC)
 * Considering that the lesson here is going from English to German (from a story about University studies), it complicates things to substitute the "more correct" lernen - so the implication is simply that "today we study German (at the university)". I do not see the need for a change? - marsh 05:05, 16 February 2006 (UTC)

title: Talk:German/Lesson 2


text:
German/Appendices/Grammar II

<< Beginning German | Basic German | Intermediate German

Conjugating 'to be'
Ich bin groß. I am tall. Du bist sehr groß. You are very tall. Sie ist klein. She is short. Sie sind groß. They are tall.

In these cases, we use the correct form of sein for each situation. Please notice the final two sentences both use 'Sie', and we must look at the verb to determine the difference between 'she' and 'they'.

In German, the English infinitive 'to be' is translated as sein.

This is the table of the forms of 'sein', with rough English translations. Note that in English, there are only three forms (am, is, are) while German has five (bin, bist, ist, sind, seid).

Also, the verb conjugation of the two you-formals are always the exact same.

German

sein

English

to be

Conjugating Normal Verbs
Er spielt Volleyball. He plays volleyball. Ich mache Hausaufgaben. I do my homework. Wir kommen aus München. We come from Munich. Was machst du? What are you doing?

In these sentences, different verbs and endings are used. Note that the verb is always in second position.

When conjugating normal verbs, use the endings shown below (a memory hook is the "best ten" endings). Note that in normal verbs, such as spielen and machen, ihr-form and er/sie/es-form are the same and the wir-form, sie (pl)-form and the formal are all the same as the infinitive.

-en

spielen - to play

machen - to make/do

Conjugating Irregular Verbs
Ich habe keine Zeit. I have no time. Gib mir das Buch! Give me the book. Sie wandert gern. She likes to hike. Er liest einen Roman. He is reading a novel.

In each of these sentences, we use an irregular verb. Irregularity occurs in the ich-form or the du-form and er/sie/es-forms. There are three types of irregularity.

E in the first syllable
One form of irregularity occurs sometimes when the verb contains an 'e' in the first syllable. The change is simple: the du-form and er/sie/es forms both change the 'e' to an 'i.e.' or an 'i'. Two common examples are shown. Note that the er/sie/es-form and ihr-form are no longer the same.

sehen - to see

geben - to give

Haben
A similar, yet different, change occurs in the verb "haben". As in the irregularity above, the du-form and er/sie/es-form change.

haben - to have

Verbs ending in Consonant-N
Some verbs change the ich-form for obvious reasons. "Wandern" and "basteln"  are two examples. Both drop the first e in the ich-form.

wandern - to hike

basteln - to build

Conjugating Modals
Ich will ins Kino gehen. I want to go to the movies. Dürfen wir hier essen? May we eat here? Was kann ich für dich tun? What can I do for you? Er mag Romane lesen. He likes to read books.

Modals are a new kind of verb. They are the equivalent to helping verbs in English. There are seven basic modals: können (can), mögen (like), dürfen (may), wollen (want), sollen (should), müssen (must), and möchten (would like). Möchten isn't technically a modal, but it acts like one in most aspects.

Modals are conjugated very differently. The ich-form and er/sie/es-form are always alike and singular has a different verb in the first syllable (except in sollen and möchten). Below are the conjugations of the six basic modals and möchten.

können - can

mögen - like

dürfen - may

wollen - want

sollen - should

müssen - must

möchten - would like

Separable Verbs
Du siehst schön aus! You look good! Ich muss mein Zimmer aufräumen. I have to clean my room. Komm mit! Come with! Probier diese Jeans an! Try these jeans on.

Some verbs in German are separable: they have a prefix that can be separated from the base. When the verb is used with a modal, it regains the prefix at the end of the sentence. When it is the main verb of the sentence, the prefix is moved to the end of the sentence.

An "example" in English would be the word "intake". When it is used as a verb, it becomes "take ... in". When it is used as an adjective or a noun, it becomes "intake" again.

Two easy examples of separable verbs are aussehen and mitkommen. Note that aussehen is also irregular.

aussehen - to appear

mitkommen - to come along/with

title: German/Appendices/Grammar II