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Talk:Blender 3D: Noob to Pro/Penguins from spheres

Possibly moving the module
Hi!

While this is a really neat tutorial on creating a cute little penguin, this module doesn't introduce any new functions or tools of blender. Translation, rotation, scaling, extrusion, merging etc. are all tools from previous modules. Maybe this tutorial could be moved to the 'Miscellaneous Tutorials' section?

( I actually get this feeling from a lot of this books modules - that it's just a collection of miscellaneous tutorials. When learning about a new program shouldn't the main focus be on explaining the tools? Of course a little bit of repetition never hurt anyone, and to learn new tools, old tools will have to be used in the process of explaining what the new ones do. Sure now we know how to make a hat, a person and a penguin, but what about a toaster, a parrot, a moon, or the greek god Zeus when he's in a bad mood - are there going to be tutorials about these things too? )

Oh, and another thing! The term "noob notes" seems a bit weird here. Why not just "Observe" or "Hint" or something like that. I mean, aren't we all noobs reading this?

Coloring
I colored the rest of this little guy using the Multiple Mesh technique. Here's what he looks like: http://img324.imageshack.us/img324/291/penguin1wn.png I'll probably add a section on how to do so, if nobody minds.
 * Sorry for making all of you guys wait, other things have come up. I've got to register with Wikipedia to upload the pics, which will slow down any work I'm going to do. If I can't do it for some reason I'll let you guys know.

-- Hi, My Name is Craig, i havnt got a clue how to colour my penguin, i have made him to how the tut says to, but im a bit puzzeled about how to colour him, could somone please add something onto this tut about colouring.

Thanks

'''EDIT:

i have now coloured him and i dont need any help but it still may be a good idea to post how to colour it for other people.

i used the same technique from the dice tut.'''

Fixed the Limit selection to visible picture
It was pointing to the wrong button before, so I replaced it with a picture of my own. -Cleobod

Step 1
to put cursor in middle from the box it says, selection-cursor, but it is cursor-selection instead

Step 3 – the head
Shouldn't limit selection to visable be turned on here?

Step 4 - The Wings
At the end of this step, you have to select the two backmost vertices, but I couldn't find the ones so it is the same as on the picture

Step 5
Says: Now select the bottom most vertices in the same fashion, and press SHIFT + FKEY to Fill the hollow bottom of the poor bottomless penguin.

Shouldnt it just be FKEY for new face?

 * Nope, it would work if there were only 3 or 4 vertices, but FKEY for new face can't cope with so many at once. It makes at most a single new face, and blender's faces cannot have more than 4 vertices. Shift-FKEY tries to work out a way to connect all the vertices using as many faces as it needs. -- Darloth is terrible at explanations, but hopefully that helped.

Making the Neck
Could someone explain this in a little more detail I'm having trouble forming the neck?

Me too. I get up to the part that says " repeat (selection, etc.)" but what part do i repeat do I unselect and do the same thing over or continue on with the top 3 already selected. Instead of repeating I ended up extruding the middle then rotating it a little:

I did this by:

From the picture directly right of Rotating the neck:

1. Extrude the top 3 rings up (just less than double the size) 2. Rotate them 45 Counter-Clockwise 3. Extrude the top 2 rings (double their size) 4. Rotate the extruded part 50 Counter-CLockwise

I ended up with this: http://img103.imageshack.us/img103/8686/penguinfv0.jpg (sorry can't upload images to wiki for 4 days)

Start:  http://img232.imageshack.us/img232/7466/penguin1kp7.jpg Stage 1: http://img232.imageshack.us/img232/1289/penguin2dh6.jpg Stage 2: http://img403.imageshack.us/img403/3517/penguin3zf2.jpg Stage 3: http://img103.imageshack.us/img103/8977/penguin4vy2.jpg

In the final step where you move neck to the left, if you follow the Noob Note, there is a keypress missing and the neck skews along the X-axis. You need to press G, Y, -0.4. The "Y" constrains the movement along the Y-axis and you end up with what is in the picture. --Blonewolfs (discuss • contribs) 15:36, 20 April 2013 (UTC)

neck pointing in oppisite direction to the photos
when I made the neck from the side view in 2.49b up .3 on the Z hittig the blue arrow worked fine, but rotating by typing 30, made my neck point in the oppisite direction to the pictures.. lol Pearts (talk) 22:06, 9 July 2010 (UTC)

Instead of typing "30", you need to type "-30" to rotate counter-clockwise. --Blonewolfs (discuss • contribs) 15:23, 20 April 2013 (UTC)

evidently the even though I made it in side view, i had to AKEY twice, then NUM7, then Space -> Transform -> Rotate -180 to get mine to look like the photos Pearts (talk) 22:17, 9 July 2010 (UTC)

but now I've learned I could have pressed ctrl + num 3 and saw it from the other side! Pearts (talk) 15:23, 12 July 2010 (UTC)

Extruding the wings
User comment: I really need to do this by hand? Is there no way to tell Blender to do the same on the other side?

reply:You can select CTRL+SPACE->scale and while in front view, use the red square. Just make sure the cursor is between the wings. After you select the faces to extrude, just SHIFT+SKEY->Cursor->Selection.

User reply: Modifiers-> Add Modifier -> Mirror. Use the X-axis. Click "apply" when done with the symetrical stuff.

User response: adding a mirror modifier resulted in the surface of the penguin looking corrupt (response to self - because the penguin body was being mirrored on top of itself, but even when half of the penguin vertices are removed an unsightly seem is left).

Final user response: You can fix the 'corruption' by pressing Ctrl+W and choosing "Remove Doubles" after mirroring.

Appendix to Final user response: "Remove Doubles" will only get rid of the distortion if you select half of the faces in Front view (NUM1), hit (XKEY) and select "Faces" in the delete menu on the side to be mirrored, then the mirror modifier won't leave a seam, or distortion due to overlap.

Another User reply: Or you can select the bottom faces on both sides, and only then extrude.

Last User comment: Switch to Edit Mode. Select all penguin vertex by pressing AKEY twice. In "Mesh tools" tab click in "Rem Doubl" button.

Another Last User Comment: If you select both faces and hit EKEY>Individual Faces and then use "1.4" it works just fine.

Creating the beak
I prefer to set the MMB to pan the view instead of rotate it, but until just recently I didn't know how to center the camera again, and I'm posting this to help other Blender newbies who prefer this. I found that if you press "C" you can easily move the view to the center of the scene.

[For those who use versions earlier than 2.49b {the version I use as well as the only one I've ever used}, I don't know if this will work; still, try it and see if it works]. This made things immensely easier for me, since I now have better control over the camera. [In case this wasn't covered earlier, you don't need the MMB to rotate. With NUM lock on, NUM pad 4 rotates {the scene, not the view} left, NUM pad 2 rotates down, NUM pad 6 rotates right, and NUM pad 8 rotates up. These keys rotate the view right, up, left, and down, respectively.]

Smoothing the wings
(User comment on grid snapping: If you hold CTRL+SHIFT while doing a manipulation, it allows you to use smaller fractions as grid snap than with CTRL. With translation this is 1 for just CTRL, and 0.1 for CTRL+SHIFT)

(User comment on using the arrows: I mentioned earlier that the grid snap seemed inconsistent. It now seems that when I’ve got the 3d manipulators enabled (the arrows), the grid size is 1.0 units. When I have them turned off, the grid size is 0.1 unit. I don’t know why, and I haven’t found where to change this. The previous picture shows where to turn the 3D manipulators on/off, and which arrow to drag (using LMB) to achieve Z-axis movement.)

(User comment: if you spin the view it becomes 1.0 snapping, go back by pressing NUM1 and you should get 0.1 if not make sure View → Orthographic is checked and not Perspective this can also be done with NUM5)

(User comment: I've had trouble with the three axis arrows. When I click on the 'hand' icon, the three lines come up but with little blocks instead of arrows at the ends, which I believe means that they will scale instead of move. So, I have had to use GKEY for 'grab' and then ZKEY or YKEY or whatever to move along an axis.)

(Note from another user: Next to the hand click the triangle to go to Moving Mode).

(Reply to previous comment: You can change the size of the 3d manipulator arrows in the preferences area so that they are easier to click.)

Culling the underside
Note 1: Instead of using SHIFT+FKEY to make a zigzag surface, you could extrude the circle of edges (EKEY), constrain the extrusion to the Z-axis (ZKEY) and enter 0 (0KEY + ENTER) as the distance. This effectively duplicates the circle. Then do a merge (ALT+MKEY → At Center) to create a nice surface.

Note 2: Press CONTROL+NUM7 to view the penguin from the bottom. Now, without deleting anything, select four vertices, top, bottom, left, and right in the second ring out from the center vertex. Then snap your cursor to the selection. SHIFT+SKEY → Cursor -> Selection This places the cursor at the center of those vertices. Finally, select the center vertex, expand the selection to include the first ring (CONTROL+NUM+) then merge as above (ALT+MKEY) but at cursor rather than at center. This achieves the same result but without having to delete or extrude anything.

Alternative Method - Extrude & Scale:'' From a side view. Box select (BKEY) the bottom vertex & the 2 bottom most rings, then delete (DELETEKEY or XKEY → Vertices). Now box select (BKEY) the remaining bottom most ring. Initiate an Extrude (EKEY → Region) command, and escape terminate (ESCAPEKEY) said command. It will appear as if nothing has happened, but you have created a new (overlapping) bottom ring and is currently selected. Change the view to Orbit Down (NUM2KEY). Now Scale (SKEY) the selection, press the number 0 to close, then press the Enter key to commit. The bottom should have closed with a number of triangles, all pointing to a common center vertex. As a good practice, you should now remove doubles' (WKEY → Remove Doubles).

Extruding a tail

 * Note 1: I found it easier to select the center most, lowest vertex, and move it along the Y axis only. This creates a crude but very effective tail (that doesn't look all that bad when subsurfed).
 * Note 2: to do this correctly (if I've properly understood the previous user) you can also select the two faces in the bottom row (CTRL+TAB→Faces), which are connected to the three middle vertices and their projection on the bottom line, then split them into triangles (SPACE→Edit→Faces→Convert To Triangles or using CTRL+TKEY), then select the right two of them (it was the right quad face before) and flip the triangles' orientation (SPACE→Edit→Faces→Flip Triangles Edges or using CTRL+FKEY). So, there will be the five edges connected to the middle point in the second row from the bottom and only three to the vertex, mentioned by previous user. Then just move this lowest, center most, vertex by Y on 0.4
 * ''Note 3: don't do that extruding of the tail with only the edges or you will get an so called 'non-manifold mesh', cause you will have an edge with outgoing three faces, so blender can't recalculate the surface and you can't use the decimate modifier. this can be very nasty if you're modeling more complex things later. better select the whole two faces at the back end and extrude that and then shrink it along the z-axis.

Adding the feet
User note: I merged the three vertices at the bottom one,and at the folding point I merged the middle and top vertices at the middle one, and then I used the red arrow to translate the newly merged vertex about 0.04 BU's toward the inner part of the penguin, and the green arrow to translate it about 0.1 backwards.

Normals not pointing the right way
Even after recalculating them, a patch of mesh along all the back of the penguin gets the normals wrong, on the opposite side of the rest of the mesh.

I had to flip them manually so that all of them would point in the same direction, but if I recalculate them again they go the wrong way again. I'm not sure what am I doing wrong or why this is happening. http://i.imgur.com/kY8tUSs.png 186.125.95.40 (discuss) 06:28, 9 August 2013 (UTC)


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A-level English/Wise Children/Character Profiles

Melchior Hazard
Serious but charismatic, Melchior Hazard is the most eminent Shakespearean actor in Britain. Like his father, Melchior has a "gift of gravitas". Melchior personifies legitimate, Shakespearean theatre. He idolises Shakespeare and pays homage to him by traversing the globe, spreading the word of Shakespeare. he also idolises his father and this is another reason for his love of shakespeare.To him, Shakespeare is a God-like figure. Melchior is seduced into Hollywood, showing the merging of high and lowbrow culture. He is a loveless and ambitious character with a cold heart, preferring his paper crown to his family. Melchior was also one of the wise men in the Bible. Both men follow a star: In Bible it leads Melchior to baby jesus, in Wise Children it symbolises Melchior following in his father Ranulph's footsteps trying to reach the same 'stardom'.

Peregrine
A carnival character with bright red hair, Peregrine is full of laughter and life. A man with a low boredom threshold Peregrine does not have the patience to pursue an interest for long. He performs magic tricks, spends time in the secret service and collects butterflies in his later life. Peregrine means Roving or wandering and the name is derived from the word pilgrim. This is exactly what Peregrine does moving from interest to interest. Although Peregrine is not Dora and Nora's father he passes for their father as he acknowledges the girls when Melchior repudiates them. He represents the carnivalesque through his frequent disappearances, his magic, his different incarnations in various roles, his cuckolding of his brother. He is exuberant and generous, never seems to age, but to grow. His lechery and slightly menacing aspect associate him with the big bad wolf of fairytale.

Dora and Nora
Dora and Nora are described as being "like as two peas", the only way in which the girls could be distinguished from each other was the perfumes they used. Unlike their father and his brother Dora and Nora are similar. As Dora describes: "identical we may be, but symmetrical -- never. For the body itself isn't symmetrical. One of your feet is bound to be bigger than the other, one ear will leak more wax. Nora is fluxy; me, constipated"

However there are some differences between the girls. While Nora is impulsive and emotive Dora is cautious, thoughtful and reserved: "She said: 'Yes!' to life and I said, 'Maybe...' ". Nora is described as throwing her heart away like a used bus ticket" whereas Dora states that "The more I saw of love the less I liked the look of it"

Saskia and Imogen
“Being unique amongst mammals, a cold blooded cow” “So rich, so connected, so legitimate”

These twins could be seen to be the opposite of the lucky Chances, if Dora and Nora grew up on the illegitimate wrong side of the tracks then Saskia and Imogen are the legitimate "darling buds of may". Their mother Lady Atalanta Hazard is a lady herself.

Throughout the novel Dora shows contempt for Saskia. She says that Saskia made “a career out of piggery” when commenting her cookery programme. Their dislike of Saskia and Imogen is also evident when they are taken to a show in which Dora and Nora star. Nora is unwilling to walk about, when she does Saskia “howls like a banshee” clearly enjoying mocking Dora and Nora’s performance.

Saskia and Imogen storm out of their twenty-first birthday party after receiving gifts they dislike from Peregrine and hearing that Melchior is marrying ‘Lady Margarine’. Saskia’s best friend. It is suggested that Saskia makes Lady A sign her house away and that she is responsible for Lady A’s fall down the stairs.

Although Peregrine is their father Saskia especially adopts many of the characteristics of Melchior. It is evident that Dora is jealous if the legitimacy that Saskia enjoys - “So rich, so connected, so legitimate”. Even the red hair that Saskia and Tristram have only when the legitimate side of the family.

Tristram and Gareth Hazard
"Both, in their different ways, carrying on the great tradition of the Hazard family -- the willing suspension of disbelief. Both of them promise you a free gift if you play the game"

Tristram and Gareth are children from Melchior's third marriage to 'My Lady Margarine'. These brothers appear complete opposites. Tristram is the host of game show called 'Lashings of Lolly' whereas Gareth becomes a missionary. Tristram is expelled from Beadles for smoking pot. He is a much more radical character than Gareth. Dora says that he discovered sex at the same time that Gareth discovered God. However, Tristram is not quite the man his father was; although he continues the Hazard tradition of celebrity on his game show he is a flashy character and the low-brow game show is in contrast to Melchior’s high-brow Shakespeare. However both Tristram and Melchior share their inability or unwillingness to father their own children. Tristram states, “I’m not ready to be a father”, a remark which could be described as typical for a Hazard male. Although Tristram shows concern for the well being of Tiffany after it is believed that she is dead. This is coupled with the news that she is pregnant because of Tristram and that he is having an incestuous affair with Saskia. In this part of the novel Dora and Nora’s contempt for Tristram matches that of Saskia.

Tristram and Gareth may seem opposites but as Dora reflects on them: "Both of them in show business”, "Both, in their different ways, carrying on the great tradition of the Hazard family -- the willing suspension of disbelief. Both of them promise you a free gift if you play the game". Tristram refers to Dora and Nora as his aunties although they are half-sisters. This is because Peregrine allowed everyone to believe that Dora and Nora were his illegitimate children rather than Melchiors.

Both Tristram and Gareth father illegitimate children. However the mother of Gareth’s child is never described as Dora refuses to give this information to the reader. As a Catholic missionary Gareth shouldn’t have been having sex let alone fathering an illegitimate child. Dora says “put it down to liberation theology”, she also admits that Gareth fathering a child sounds somewhat farfetched in the line “hard to swallow, huh?”

Grandma Chance
“The boarding house clung to respectability by the skin of its teeth…you could have said the same of her”

Grandma Chance is an eccentric character. Despite her eccentricities she gives Dora and Nora stability after the death of their mother Pretty Kitty. Grandma Chance’s love is the constant and everyday whereas Peregrine, although showing care for the girls financially, does have a tendency to come and go throughout Dora and Nora’s childhood. Grandma Chance has extreme views and she is extremely loving.

Grandma Chance’s views include:

 * A belief that war is caused by old men who want to rid the world of all the young men.
 * A belief that plants scream when they are cut
 * A strong belief in animal rights well before this was a mainstream movement
 * Nudism – she is responsible for Dora and Nora’s nudity as children

Grandma Chance raised Dora and Nora “out of pure love”, however her origins are a mystery, this is another one of her eccentricities, and she refuses to talk about her past to Dora and Nora. As Dora says: “we never found who put her up at Bard Road”.

Dora and Nora show their appreciation to Grandma in the line “we owe her everything”. To Dora and Nora she was their mother figure as well - “She was our air raid shelter; she was our entertainment; she was our breast”. Of course Grandma Chance was the name she used of respect of the dead Kitty. She was not biologically related to the girls. However Dora states “Grandma invented the family through sheer force of personality”. This is shown in the way that she takes in ‘Our Cyn’ when she turns up at the house in Bard Road. This further shows her capacity to love.

It is Grandma Chance who is responsible in part for Dora and Nora’s love of the theatre. She decides to take them to a theatre show on their seventh birthday called ‘Lady Be Good’. This is where the theatre captivates Dora and Nora, it is also where Grandma Chance announces, “that man is you father”, she is responsible for the girls longing for their father Melchior Hazard.

Throughout Wise Children, Peregrine and Grandma Chance get on well together. Both are larger than life characters. Later in the novel there is tension between the girls and Grandma. Dora laughs at her nudity and Grandma disapproves of a fur coat Dora is given. However, she continues to show her capacity to love by keeping a scrapbook of the girl’s theatre performances right from their first performance in ‘Babes in the Wood’ to the day that she dies.

Grandma Chance comes to Hollywood when Dora and Nora are to marry. Her greeting “Who the fuck are you” is in stark contrast to the artifice of the Hollywood episode.

Grandma Chance died during World War II buying alcohol; she is defiant against Hitler interfering with her drinking habits in the same way that her beliefs are defiant against the rest of society. Later when her funeral is described in greater detail Dora admits that Grandma Chance’s death meant a link with the past was lost. Throughout chapter four of the novel the girls feel lonely without Grandma Chance living with them. However, her ghost returns to tell Dora and Nora that “memory lane is a dead end”. This section is one of the novel’s many manifestations of magical realism.

Lady Atalanta Hazard nee Lynde
'''“I live mainly in the past, these days. I find it’s better”'''

Lady Atalanta Hazard is the first Hazard wife of three. Even after the end of their relationship Lady still loves Melchior, this is evident as a large portrait of Melchior hangs in her Sussex home. Even in old age Lady A or Wheelchair is obsessed with Melchior. This is shown in the way that she takes an interest in Melchior’s appearance on the game show ‘Lashings of Lolly’- “She perked right up”.

Although she has long ceased to have the wealth of aristocracy she still retains the manner of a lady. This can be seen in the way that she informs Brenda of Tiffany’s ‘death’.

Lady A is most unlike the two daughters Saskia and Imogen. When she took her girls to see a theatre show which Dora and Nora happened to be appearing in, Lady A sends the girls forget-me-nots whereas Saskia howls like a banshee and Imogen falls asleep. Lady A is also mortified by Saskia’s sarcasm in quoting the line from ‘A Winter’s Tale’ “Which some call nature’s bastards”. Lady A even manages to maintain calm after the news that Daisy Duck, Melchior’s second wife is pregnant, although this news later turns out to be false.

Lady A descends from great wealth to poverty. She at one time lives like aristocracy but after being abandoned she ends up living in the basement of the house in Bard Road. Lady A refuses to talk about the accident that left her in a wheelchair. “Did she fall or was she pushed?” is the question that Dora asks herself. However it is implied that the accident was the fault of Saskia and Imogen who also make Lady A sign her house away leaving her poor in Brixton.

At the end of the novel Lady A gives a dramatic speech in which she accuses Melchior of leaving her “womb empty” because he was too busy perusing his acting career to concentrate on family life. As Dora puts it “she got it all of her chest in one go”. However she still manages not to lose self-control during this outburst.

Delia Delaney nee Daisy Duck
“The classic thirties blonde”

Daisy is unlike Lady A in that she gains wealth by marrying Melchior. Although she has a somewhat extreme character, typical of Hollywood she still retains some moral standards. She unwilling to tell Genghis Khan that the baby she is carrying is his as she knows that it is Melchiors. However it is later revealed that she is not pregnant.

She is unsympathetic to the unnamed first wife of Genghis Khan telling her to “drop dead" when she calls. Peregrine is much more sympathetic uttering “poor cow” when he picks up her call. However she is a somewhat brash character.

In fact the role of ‘The Dream’ is that it is a vehicle to showcase Daisy’s talents. Despite playing Titania it is evident she is not appropriate for playing Shakespeare. She lacks class. The first pages of her descriptions informs the reader than she has heart-shaped pubic hair. Her first line in the novel is “When does the orgy begin?”

In the Dream episode full of artifice, money is the only reality. Daisy asks Genghis Khan for $1,000,000 dollars. Genghis gives this money but takes it back the next day. This cruel trick is what causes Daisy to gain revenge in a relationship with Melchior.

Daisy falls in love with Melchior. They are described as together as skin wrapped around sausage – they are that close. However the marriage is short lived and they divorce after only a few days of marriage.

Daisy later appears at Melchior’s 100th birthday party. She is described as looking like “$1,000,000, dollars. Even if in well used notes”. This is the second time this figure of money and Daisy have been mentioned together.

Her entrance to Melchior’s party in which guests stand on chairs shows that she still has something of the Hollywood glamour in her, even in old age. At the party she manages to reconcile with Melchior who she had last seen at the filming of ‘The Dream’.

My Lady Margarine
“I’m going to marry my Cordelia”

Comparatively little is said about ‘My Lady Margarine’ in comparison with the other Hazard wives. She takes her name from advertisements that she appears in advertising margarine. Dora mocks this in the line “To butter or not to butter”.

She is first mentioned when Melchior announces that he is to marry “my Cordelia”, as Lady Margarine played Cordelia to his Lear. This parallels Ranulph and Estella hazard earlier in the novel. Saskia and Imogen react badly to this news as it effectively disinherits them from their father’s wealth. Saskia is also angry, as ‘Lady Margarine’ was her best friend before this news.

Her only other mentions are at Melchior’s 100th birthday party where she is described as wearing “a smile so fixed”. She isn’t happy throughout this chapter as she is looking for her son Tristram – the party occurs on the day the Tiffany is reported dead but Tristram still attends the party.

At the end of the novel Lady Margarine donates a pram to Dora and Nora so they can look after Gareth’s babies.

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


text:
User:Schreiber555

 * German-English Dictionary

title: User:Schreiber555


text:
Talk:Old English/Adjectives

Untitled
I was under the impression that the Instrumental case was only found in the Neuter and Masculine gay.

Strong declension - German Ein
It might be worth pointing out the parallels between the strong declension of adjectives and the declension of ein in German. They are very close. Wodorabe (discuss • contribs)

title: Talk:Old English/Adjectives


text:
User:Htchien

w:User:Htchien

title: User:Htchien


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Wikijunior:Kings and Queens of England/The House of Windsor

George V (1910–1936)
The first monarch of the House of Windsor was George V. However, he started out as being of the House of Saxe-Coburg-Gotha, and so he can be read about in the previous chapter.

Edward VIII (1936)
Edward VIII was born at White Lodge, Richmond upon Thames on 23 June 1894. Edward was the eldest son of George V and Queen Mary. He was King of the United Kingdom of Great Britain and Northern Ireland and the British Dominions beyond the Seas, and Emperor of India from 20 January 1936 until his abdication on 11 December 1936, after which he was Prince Edward and then Duke of Windsor until his death on 28 May 1972. During World War II he was the Governor and Commander-in-Chief of Bahamas after spending a great deal of time in Bermuda. He was known to his family and close friends, by his last Christian name, David.

Prince of Wales
In 1910, George V made Edward Prince of Wales. When the First World War broke out in 1914 Edward joined the army, serving with the Grenadier Guards. Although Edward was willing to serve on the front lines, the British government refused to allow it, because of the harm that the capture of the heir to the throne would cause. However, Edward did witness at first hand the horror of trench warfare, and visited the front line as often as he could. His role in the war, although limited, led to his great popularity among veterans of the conflict.

Throughout the 1920s the Prince of Wales represented his father, King George V, at home and abroad on many occasions. He took a special interest in visiting the poorest areas of the country. Abroad the Prince of Wales toured the British Empire, undertaking 13 tours between 1919 and 1935.

In 1928, King George V gave Edward a home, Fort Belvedere, near Sunningdale in Berkshire. There Edward had relationships with a series of married women, including the American Wallis Simpson. Simpson had divorced her first husband in 1927 and later married Ernest Simpson. Edward's relationship with Wallis Simpson further weakened his poor relationship with his father, King George V. The King and Queen refused to receive Mrs Simpson at court, and his brother, Prince Albert, urged Edward to seek a more suitable wife. Edward, however, had now fallen in love with Wallis and the couple grew ever closer.

Reign
King George V died on 20 January 1936, and Edward became King Edward VIII. The next day he broke royal protocol by watching the proclamation of his own accession to the throne from a window of St James's Palace in the company of the still-married Mrs Simpson. It was also at this time that Edward VIII became the first British monarch to fly in an aeroplane, when he flew from Sandringham to London for his Accession Council.

It was now becoming clear that the new King wished to marry Mrs Simpson, especially when divorce proceedings between Mr and Mrs Simpson were brought at Ipswich Crown Court. Powerful figures in the British government deemed marriage to Mrs Simpson impossible for Edward, even if Wallis obtained her second divorce, because he had become the Supreme Governor of the Church of England which prohibited remarriage after divorce.

On 16 July 1936, an attempt was made on the King's life. Jerome Bannigan produced a loaded revolver as the King rode on horseback at Constitution Hill, near Buckingham Palace. Police spotted the gun and pounced on him, and he was quickly arrested. At Bannigan's trial, he alleged that "a foreign power" had paid him £150 to kill Edward, a claim the court rejected.

On 16 November 1936, Edward met with Prime Minister Stanley Baldwin at Fort Belvedere and expressed his desire to marry Wallis Simpson when she became free to do so. The Prime Minister responded by presenting the King with three choices: he could give up the idea of marriage; marry Wallis against his ministers' wishes; or abdicate. It was clear that Edward was not prepared to give up Wallis. By marrying against the advice of his ministers, it was likely that he would cause the government to resign, prompting a constitutional crisis. The Prime Ministers of the British dominions had also made clear their opposition to the King marrying a divorcée; only the Irish Free State was not opposed to the idea of marriage. Faced with this opposition, Edward chose to abdicate. Edward duly signed an instrument of abdication at Fort Belvedere on 10 December 1936. The next day, he performed his last act as King when he gave Royal Assent to His Majesty's Declaration of Abdication Act 1936 which applied to the United Kingdom and all the dominions except the Irish Free State. The Free State passed the equivalent External Relations Act, which included the abdication in its schedule, the next day.

On the night of 11 December 1936, Edward, now reverted to the title of The Prince Edward, made a broadcast to the nation and the Empire, explaining his decision to abdicate. He famously said, "I have found it impossible to carry the heavy burden of responsibility and to discharge my duties as king as I would wish to do without the help and support of the woman I love."

After the broadcast, Edward left the United Kingdom for France, though he was unable to join Wallis until her divorce became absolute, several months later. His brother, Prince Albert, Duke of York succeeded to the throne as King George VI. George VI made Edward the Duke of Windsor. The Duke of Windsor married Mrs Simpson, who had changed her name by deed poll to Wallis Warfield, in a private ceremony on 3 June 1937 at Chateau de Candé, Monts, France. None of the British royal family attended.

World War II
In 1937, the Duke and Duchess visited Germany as personal guests of the Nazi leader Adolf Hitler, a visit much publicised by the German media. The couple then settled in France. When the Germans invaded the north of France in May 1940, the Windsors fled south, first to Biarritz, then in June to Spain. In July the pair moved to Lisbon. In August a British warship took them to the Bahamas, where the Duke of Windsor was made Governor. He held the post until the end of World War II in 1945.

Death
After the war, the couple returned once again to France in Neuilly near Paris, where they spent much of the remainder of their lives in retirement. The Royal Family never accepted the Duchess and would not receive her formally, although the former King sometimes met his mother and a brother after his abdication. The Duke died of throat cancer in 1972 in Paris, and his body was returned to Britain for burial at Frogmore near Windsor Castle. The Duchess travelled to England to attend his funeral, staying at Buckingham Palace during her visit. The Duchess, on her death a decade and a half later, was buried alongside her husband in Frogmore simply as "Wallis, his wife". The Duke and Duchess had no children.

George VI (1936–1952)
George VI was born at York Cottage on the Sandringham Estate in Norfolk on 14 December 1895. He was the second son of George V and Queen Mary. He was King of the United Kingdom of Great Britain and Northern Ireland and the British dominions beyond the seas from 11 December 1936 until his death on 6 February 1952. George VI was also the Emperor of India (until 1947) and King of Ireland (until 1949). George VI came the throne after the abdication of his brother, Edward VIII. He was king during the Second World War. He was known as Bertie, after his first name, Albert, to his family.

Early life
As a child, George often suffered from ill health and he was described as "easily frightened and somewhat prone to tears". Albert developed a severe stammer that lasted for many years as well as chronic stomach problems. He also suffered from knock knees, and to correct this he had to wear splints, which were extremely painful. He was also forced to write with his right hand although he was a natural left-hander.

In 1909, Bertie joined the Royal Navy and served as a naval cadet. Despite coming in at the bottom of the class, Albert moved to Dartmouth and served as a midshipman. He was still in the Navy when Edward VII died on 6 May 1910. Albert was now second in line for the throne. Albert served during World War I. He saw action aboard HMS Collingwood in the Battle of Jutland in 1916. In 1917 Albert joined the Royal Air Force but did not see any further action in the war. After the war, Albert studied history, economics and civics for a year at Trinity College, Cambridge, from October 1919. In 1920, he began to take on royal duties, representing his father, King George V.

Marriage and children
Albert had a great deal of freedom in choosing his wife. In 1920 he met Lady Elizabeth Bowes-Lyon, the youngest daughter of Claude Bowes-Lyon, 14th Earl of Strathmore and Kinghorne and set his sights on marrying her. She rejected his proposal twice before she said yes. They were married on 26 April 1923. They had two children: Elizabeth, born in 1926, and Margaret, born in 1930.

Reign
The Duke and Duchess lived a relatively sheltered life at their London residence, 145 Piccadilly. When Edward VIII he abdicated on 11 December 1936, however, Albert became King George VI.

The growing likelihood of war in Europe would dominate the reign of King George VI. Initially the King and Queen took an appeasement stance against the German dictator Adolf Hitler, and supported the policy of Neville Chamberlain. In 1939, the King and Queen visited Canada from where they made a shorter visit to the United States of America. George was the first reigning British monarch to visit either of these countries. The aim of the tour was mainly political, to shore up Atlantic support for Britain in any upcoming war. The King and Queen were extremely enthusiastically received by the Canadian public. They were also warmly received by the American people, visiting the 1939 New York World's Fair and staying at the White House with President Roosevelt. When war broke out in 1939, George VI with his wife chose to stay in London and not flee to Canada, as had been suggested. The King and Queen officially stayed in Buckingham Palace throughout the war, although they often escaped to Windsor Castle to avoid bombing raids. George VI and Queen Elizabeth narrowly avoided death when a lone German bomber despatched to bomb Buckingham Palace attacked. The bomb exploded in the courtyard, shattering windows in the palace.

Throughout the war, the King and Queen provided morale-boosting visits throughout the UK, visiting bomb sites and munition factories. On Victory in Europe (VE) Day, the Royal Family appeared on the balcony of Buckingham Palace to celebrate the end of the war in Europe. After the war, George VI's reign saw the start of the break up of the British Empire, in particular when India became an independent dominion, with George VI giving up the title of Emperor of India.

Death
The war had taken its toll on the King's health. This was made worse by his heavy smoking and he got lung cancer. Increasingly his daughter Princess Elizabeth, the heir to the throne, would take on more of the royal duties as her father's health got worse. On 6 February 1952, George VI died aged 56 in his sleep at Sandringham House in Norfolk. He was buried in St George's Chapel in Windsor Castle. In 2002, the body of his wife Elizabeth and the ashes of his daughter Princess Margaret were interred in a tomb alongside him.

Elizabeth II (1952–2022)

Elizabeth II was born at 17 Bruton Street in Mayfair, London on 21 April 1926. She was the eldest daughter of George VI and Queen Elizabeth. Elizabeth II was Queen of the United Kingdom of Great Britain and Northern Ireland and of many other independent nations. She became Queen on 6 February 1952 at age 25, and ruled until her death on 8 September 2022. Her 70-year reign was the second-longest after King Louis XIV of France (72 years), and the longest of any British monarch ever.

Early life
Elizabeth was thirteen years old when World War II broke out. She and her younger sister, Princess Margaret, were evacuated to Windsor Castle, Berkshire. There was some suggestion that the princesses be sent to Canada, but their mother refused to consider this, famously saying, "The children could not possibly go without me, I wouldn't leave without the King, and the King will never leave". In 1940 Princess Elizabeth made her first radio broadcast, addressing other children who had been evacuated.

In 1945 Princess Elizabeth joined the Women's Auxiliary Territorial Service. She was the first and so far only female member of the royal family actually to serve in the military. Elizabeth made her first official visit overseas in 1947, when she went with her parents to South Africa. On her 21st birthday she made a broadcast to the British Commonwealth and Empire pledging to devote her life to the service of the people of the Commonwealth and Empire.

Marriage and children
Elizabeth married Prince Philip of Greece and Denmark on 20 November 1947, who was created the Duke of Edinburgh on the same date. The Duke was Queen Elizabeth's second cousin once removed, as they both descended from Christian IX of Denmark. The couple were also third cousins. After their wedding Philip and Elizabeth took up residence at Clarence House, London. But at various times between 1946 and 1953, the Duke of Edinburgh was stationed in Malta as a serving Royal Navy officer. Philip and Elizabeth lived in Malta for a period between 1949 and 1951. On 14 November 1948 Elizabeth gave birth to her first child, Charles. They later had three other children: Anne, Andrew and Edward.

Reign
King George's health declined during 1951 and Elizabeth frequently stood in for him at public events. She visited Greece, Italy and Malta (where Philip was then stationed) during the year. In October she toured Canada and visited President Truman in Washington, DC. In January 1952 Elizabeth and Philip set out for a tour of Australia and New Zealand. They had reached Kenya when word arrived of the death of her father on 6 February 1952. They were staying at the Treetops Hotel, where Prince Philip was the one who informed Elizabeth that she was now queen.

After the coronation, Elizabeth and Philip moved to Buckingham Palace in central London. It is believed, however, that like many of her predecessors she dislikes the Palace as a residence and considers Windsor Castle, west of London, to be her home. She also spends time at Balmoral Castle in Scotland and at Sandringham House in Norfolk.

At the time of Elizabeth's accession there was much talk of a "new Elizabethan age". As nations have developed economically and in literacy, Queen Elizabeth has witnessed over the past 50 years a gradual transformation of the British Empire into its modern successor, the Commonwealth. She has worked hard to maintain links with former British possessions, and in some cases, such as South Africa after the end of Apartheid, she has played an important role in retaining or restoring good relations. Queen Elizabeth has become the most widely travelled head of state in history and has visited many Commonwealth and non-Commonwealth countries. Her son Charles, the Prince of Wales, welcomed his first child Prince William in 1982.

In 2002, Queen Elizabeth celebrated her Golden Jubilee, marking the 50th anniversary of her accession to the Throne. The year saw a large tour of the Commonwealth Realms, including numerous parades and official concerts. Then in 2012, the Queen celebrated her Diamond Jubilee, marking the 60th anniversary of her accession to the Throne. A year later, her grandson William had his first son Prince George. Queen Elizabeth surpassed her great-great-grandmother Queen Victoria to become the longest-reigning British monarch in 2015, and she later became the first British monarch to celebrate her Sapphire Jubilee (65 years) and Platinum Jubilee (70 years) on the throne.

In 2021, the Queen's husband Prince Philip, the Duke of Edinburgh, died age 99, after 73 years of marriage. Her health started to decline around the same time, and it became more common for the Queen to cancel royal engagements. At the State Opening of Parliament in 2021, she could not attend, and Prince Charles had to give the speech from the throne in her place. Queen Elizabeth was also absent from the celebrations of her Platinum Jubilee during the summer of 2022. Seven months later, on 8 September 2022, she died at Balmoral age 96, bringing the second Elizabethan era to an end.

Charles III (2022 onwards)

Charles III was born at Buckingham Palace in London on 14 November 1948. He is the eldest son of Elizabeth II and Prince Philip, the Duke of Edinburgh. Charles III is King of the United Kingdom of Great Britain and Northern Ireland and 14 other independent nations, as well as Head of the Commonwealth. He became King on 8 September 2022 after the death of his mother. He was only 3 years old when his mother became Queen, but 73 when she died. This makes Charles both the longest-serving heir to the British throne, and the oldest new king in British history.

Before his accession, Charles (at the time known as the Prince of Wales) worked extensively on charity, particularly for the Prince's Trust. His personal life drew a large amount of public attention, particularly his marriages to the late Lady Diana Spencer and later to Camilla, the Duchess of Cornwall. In the final years of his mother's life, Charles began taking over some of her royal duties.

Early life
Charles was born when his grandfather George VI was on the throne, and his mother Elizabeth was first in line. When George VI died in 1952, his mother became Queen as Elizabeth II. Charles became first in line, and also gained the titles Duke of Cornwall and Duke of Rothesay (in Scotland only), which are titles that both automatically go to the heir to the throne.

He went to Hill House School in West London, and later the Cheam preparatory school in Berkshire, which the Duke of Edinburgh had also gone to. The Prince finished his education at Gordonstoun, a private boarding school in the north east of Scotland. His father, the Duke of Edinburgh, had previously attended Gordonstoun, becoming head boy. It is often reported that the Prince hated his time at the school, where he was a frequent target for bullies. In 1966 Charles spent two terms at Geelong Grammar School in Victoria, Australia during which time he also visited Papua New Guinea. On his return to Gordonstoun he followed in his father's footsteps by becoming head boy. In 1967 he left Gordonstoun with two A levels, in history and French.

Traditionally, the heir to the throne would go straight into the military after finishing school. However, Charles went to university at Trinity College, Cambridge where he studied Anthropology and Archaeology, and later History, earning a 2:2 (a lower second class degree). Charles was the first member of the British royal family to be awarded a degree. He also went to the University College of Wales, Aberystwyth to learn the Welsh language, making him the first English-born Prince of Wales ever to make a serious attempt to do so.

He was created the Prince of Wales and Earl of Chester in 1958, though his actual investiture did not take place until 1 July 1969. This was a major ceremony, held at Caernarfon Castle in north Wales, a place traditionally associated with the creation of the title in the 13th century. The Welsh borough of Swansea was granted city status to mark the occasion. He served on HMS Norfolk as Sub Luietenant Prince of Wales in the early seventies.

Marriages and children
In 1980, the Prince of Wales began dating Lady Diana Spencer (born 1961), a woman from the Spencer family of nobility. They announced their engagement on 24 February 1981, and their wedding took place at St Paul's Cathedral on 29 July 1981, before 3,500 invited guests. An estimated 750 million people watched it on television around the world, making it one of the most viewed events in television history. Almost immediately, Princess Diana became a star attraction, chased by the press, her every move (including every change in hairstyle) closely followed by millions. However, the marriage soon became troubled. Within five years of the wedding the marriage was already on the brink of collapse, and though they remained publicly a couple, they had effectively separated by the mid-1990s, with him living in Highgrove and her in Kensington Palace. They divorced on 28 August 1996. They had had two sons, Prince William born in 1982 and Prince Henry (known by the name "Harry") born two years later. Princess Diana died in a car accident in Paris, France in 1997.

During Charles and Diana's marriage, and even before it, rumors were circulating that Charles was having an affair with Camilla Parker Bowles (herself married to British Army officer Andrew Parker Bowles). Prince Charles confirmed this after his and Diana's divorce, explaining that this was one reason for why their marriage fell apart. As Camilla and her husband had also divorced, Charles and Camilla's relationship became publicly known. Over time, opinion shifted to a point where a civil marriage would be acceptable. They married on 9 April 2005 at Windsor Guildhall in Berkshire.

Out of respect for Diana, who in the eyes of the public had become closely associated with the title Princess of Wales, it was announced that Camilla would not use this title (even though she as the wife of the Prince of Wales was formally entitled to do so). Camilla was instead Her Royal Highness the Duchess of Cornwall. It was also decided at the time that when Prince Charles one day became king, Camilla would be known as princess consort rather than queen. At the Platinum Jubilee of Elizabeth II in 2022, the Queen announced that she wanted Camilla to be known as queen when Charles became king, which happened when Elizabeth II died later that year.

Personal interests
Charles is a keen horseman and huntsman. He served in the Royal Navy from 1971 to 1976, commanding HMS Bronington, a minehunter, from February 1976 until December 1976. He is also a watercolour artist and a published writer. He has exhibited and sold a number of paintings. The Prince's Trust, which he founded, is a charity that works mainly with young people, offering loans to groups, businesses and people (often in deprived areas) who had difficulty receiving outside support. Fundraising concerts are regularly held for the Prince's Trust, with leading pop, rock and classical musicians taking part

To put his ideas on architecture and town planning into practice, Charles is developing the village of Poundbury in Dorset. Prior to commencing work on Poundbury he had published a book and produced a documentary called A Vision for Britain. In 1992 he also established The Prince of Wales Institute of Architecture and began the publication of a magazine dealing with architecture, but the latter has since ceased independent operation after being merged with another charity to create The Prince's Foundation for the Built Environment in 2001. He is also keen on growing and promoting organic food, although he drew some ridicule when he joked about sometimes talking to his houseplants. Before becoming king, Charles was president of 16 charities, and raised over £100 million for charity in 2004.

Reign
Charles became king immediately when his mother died on 8 September 2022. The following day, he named his eldest son William as the new Prince of Wales. Charles III and his wife Camilla were coronated on 6 May 2023 in a ceremony at Westminster Abbey.


text:
Wikibooks:WikiProject Ada

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Visit and Contact  Dom th  e  dude  0  0  1  on Wikipedia! I have an account on Wikipedia, which I go to much more often than Wikibooks. Please contact me on Wikipedia instead of Wikibooks. However, I will post information about the books I contribute to on Wikibooks.

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Talk:Number Theory

I will soon have a large amount of free time and would like to expand the book on number theory, as it is my passion, and I wish there to be an accesible way for people to learn it.

I would like to follow a plan along the lines of:

Divisibility, Unique Factorization, Multiplicative Functions, Congruences, Congruence Equations, Reduced Residue Systems, Totient Function, Polynomial Congruences, Diophantine Equations, Rational Irrational and Transcendental numbers, Continued Fractions, Quadratic Fields, and a Collection of Theorems with proof sketches

Each area would be divided into several sub chapters and each subchapter would have examples. I would hopefully highlight many theorems and their proofs, at the same time covering many basic methods of proof, with questions at the end of each section.

The Collection of Theorems at the end would contain many theorems with proof sketches, and it can be left as an excercise to the student to fill them in.

After this is complete I would be interested in expanding it to include more advanced topics, however I believe this would be a good start.

I would like to assume minimal knowledge of Calculus in the first part. Robert Carr 00:17, 8 May 2006 (UTC)

Hi, I'm not sure if anybody has been planning anything for this book, but I would like to help any construction going on. I think I will add some sections about Gaussian Integers to the template, and maybe start to fill in some of the unwritten and partially written chapters during my freetime.

Hmm.. I think that some discussion of axioms would be a good idea.

Hopefully we'll see more people tack onto this book soon, because Number Theory is truly a marvelous subject. mac (talk) 19:31, 28 September 2008 (UTC)

title: Talk:Number Theory


text:
Number Theory

This book covers an elementary introduction to Number Theory, with an emphasis on presenting and proving a large number of theorems. No attempts will be made to derive number theory from set theory and no knowledge of Calculus will be assumed. It is important to convince yourself of the truth of each proof as you work through the book, and make sure you have a complete understanding. For those who wish to use this as a reference book, an index of theorems will be given.

Chapters

 * /Notation and Introduction to Proof/
 * /Axioms/
 * /Elementary Divisibility/
 * /Pythagorean Triples/
 * Unique Factorization
 * /Congruences/
 * /Diophantine Equations/
 * /Irrational Rational and Transcendental Numbers/
 * /Rings and Ideals/
 * /Gaussian Integers/
 * /Continued Fractions/
 * /Quadratic Fields/
 * Index of Theorems with Proofs
 * Additional Theorems with proof sketches
 * Exercises in Proof

 * 32-Bit Linear Congruential Generator

 * /Bibliography/

Power tables

 * /Powers modulo 11
 * /Powers modulo 19

title: Number Theory


text:
Number Theory/Notation and Introduction to proof

=Notation=

$$ \N \,$$ is the set of positive integers, also called the natural numbers.

$$ \Z \,$$ is the set of all integers.

$$ \mathbb{Q} \,$$ is the set of all rational numbers.

$$ \mathbb{A} \,$$ is the set of algebraic numbers.

$$ \R \,$$ is the set of real numbers.

$$ \Complex \,$$ is the set of complex numbers.

=Introduction to Mathematical Proof=


text:
Number Theory/Elementary Divisibility

=Elementary Properties of Divisibility=

Divisibility is a key concept in number theory. We say that an integer $$a$$ is divisible by a nonzero integer $$b$$ if there exists an integer $$c$$ such that $$a=bc$$.

For example, the integer 123456 is divisible by 643 since there exists a nonzero integer, namely 192, such that $$123456=643\cdot192\,$$.

We denote divisibility using a vertical bar: $$a|b$$ means "a divides b". For example, we can write $$643|123456 \,$$.

If $$b$$ is not divisible by $$a$$, we write $$a \nmid b$$. For example, $$5 \nmid 12$$.

The following theorems illustrate a number of important properties of divisibility.

Theorem 1
Suppose $$ a,b,d,r,\, $$ and $$s\,$$ are integers and $$d|a$$ and $$d|b \,$$. Then $$d|(ra+sb) \,$$.

Proof:

Let $$ a,b,d,r,\, $$ and $$s\,$$ be integers and suppose $$d|a$$ and $$d|b \,$$. Then there exist integers $$e \,$$ and $$f \,$$ such that $$a=de \,$$ and $$ b=df \,$$. Thus

$$ra+sb=rde+sdf =d (re+sf) .\,$$

We know that $$re + sf \,$$ is also an integer, hence $$d|(ra+sb) \,$$.

Corollary
Suppose $$a,b,d\,$$ and $$r \,$$ are integers and $$d|a$$ and $$d|b\,$$. Then $$d|(a+b), d|(a-b), \,$$ and $$d|ra \,$$.

Proof: Letting $$r=1$$ and $$s=1$$ in Theorem 1 yields $$d | (a+b) \,$$. Similarly, letting $$r=1 $$ and $$s=-1 $$ yields $$d | (a-b) \,$$. Finally, setting s=0, yields $$d|ra \,$$.

Theorem 2
If $$a, b, c \,$$ are integers, and $$a|b \,$$ and $$b|c \,$$, then $$a|c \,$$.

Proof:

Let $$a,b,$$ and $$c$$ are integers, and suppose $$a|b \,$$ and $$b|c \,$$. Then $$b=ad \,$$ and $$c=be \,$$ for some integers $$d\,$$ and $$e\,$$.

Then $$c=ade=a\left(de\right) \, $$, and hence $$a|c \,$$.

Theorem 3
Let $$a,b,c \,$$ be integers. Then $$ a|b \, $$ if and only if $$ac|bc \,$$

Proof:

Let $$a,b,c \,$$ be integers. Suppose $$a|b\,$$.

Then there exists an integer $$d$$ such that

$$ b=ad$$.

So it follows that

$$ bc=\left(ac\right)d $$ and hence $$ ac|bc \,$$.

For the reverse direction, we suppose $$ ac|bc \,$$.

Then there exists an integer $$ d \,$$ such that

$$bc=\left(ac\right)d$$, so $$c(b-ad)=0.$$

We know that c is non-zero, hence

$$b-ad=0,\,$$ i.e., $$b=ad$$. Thus, $$a|b$$.

Prime and composite numbers
A natural number p greater than one is a prime number if it has exactly two distinct natural number divisors, itself and 1. The first eleven such numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. There are an infinite number of primes, however, as will be proven below.

A natural number greater than one that is not prime is composite. Thus, 4, 6, 8, 9, 10, 12, 14, 15 and 16 are the first few composite numbers.

Note that the number 1 is the only natural number that is neither prime nor composite.

Theorem 4
Fundamental Theorem of Arithmetic (FTA)

Every composite positive integer n is a product of prime numbers. Moreover, these products are unique up to the order of the factors.

Proof:

We prove this theorem by contradiction.

Suppose there exists a positive composite integer that is not a product of prime numbers; let N be the smallest such integer. Since N is composite, it can be written as N = a b for some integers a and b with a, b > 1.

Since N is not a product of primes, at least one of a and b is not a product of primes; without loss of generality, let's say a is not a product of primes.

Then, since a>1, a is composite, and hence a is a positive composite number that is not a product of prime numbers.

But, since a<N this contradicts our assumption that N is the smallest such number.

Hence, there does not exist a smallest such number, and therefore there do not exist composite positive integers which are not the product of prime numbers.

Proof of uniqueness is left to the reader.

Alternative Proof:

This is an inductive proof.

The statement "k is composite implies that k is a product of prime numbers" is true when $$k = 2$$.

Let $$N$$ be an integer $$\ge 2$$.

Suppose the statement is true for all $$k \le N $$.

$$N+1$$ is either composite or prime. If $$N+1$$ is prime, then the statement is true for $$k = N+1$$

If $$N+1$$ is composite, then $$N+1$$ is divisible by some prime $$p < N+1$$, so $$k = N+1$$ can be written k=pa where p is prime and a is an integer with $$1<a < N + 1$$. Then a is either prime, or composite; if it is composite, then it is a product of prime numbers by our induction hypothesis.

Hence $$N+1$$ can be written as a product of primes.

It follows that the statement is true for all $$k \le N+1$$ and hence by induction for all $$ \N $$.

Proof of uniqueness is left to the reader.

Theorem 5
There are infinitely many primes.

Proof:

Suppose that there are only $$k \, $$ primes.

Let these primes be: $$ p_1,p_2,...,p_k \,$$.

Let $$ n=p_1p_2...p_k+1. \, $$ Then either $$n$$ is prime, or it is a product of primes. If is is a product of primes, it must be divisible by a prime $$p_i$$ for some $$i$$. However, $$\frac{n}{p_i} = \frac{p_1p_2...p_k+1}{p_i} = p_1p_2...p_{i-1}p_{i+1}...p_k + \frac{1}{p_i}$$ which is clearly not an integer: $$n$$ is not divisible by $$p_i$$. Hence, $$n \,$$ is not a product of primes.

This is a contradiction, as by Theorem 4, all numbers can be expressed as a product of primes.

Therefore, either $$n \,$$ is prime or it is divisible by some prime greater than $$ p_k \, $$.

We conclude that the assumption that there are only $$k \,$$ primes is false.

Thus there are not a finite number of primes, i.e., there are infinitely many primes.

Theorem 6
The Division Algorithm (Division with smallest non-negative remainder)

Let a and b be integers where $$b>0$$. Then there exist uniquely determined integers q and r such that

$$a = bq + r$$

and $$0\leq r < b$$.

Proof:

We define the set

$$M = \{x\in \Z \mid x \leq \frac{a}{b} \}$$

which is nonempty and bounded from above. Hence it has a maximal element which we denote by q.

We set $$ r = a - bq$$. It is $$r = a - bq \geq a - b \frac{a}{b} = a-a = 0$$ and $$r<b$$, because otherwise

$$b \leq r = a-bq$$.

This implies

$$q+1 \leq \frac{a}{b}$$

which contradicts to the maximality of q in M.

We now prove the uniqueness of q and r:

Let $$q'$$ and $$r'$$ be two integers which satisfy $$a=bq'+r'$$ and $$0\leq r'<b$$. It is

$$ | b (q-q') | = | r' - r | < b $$

and thus $$|q-q'| < 1$$ which implies $$q=q'$$. This also shows $$r=r'$$ and we are done.

Index

title: Number Theory/Elementary Divisibility


text:
Number Theory/Unique Factorization and Multiplicative Functions

=The Euclidean Algorithm= Given two numbers not prime to one another, to find their greatest common measure.

The algorithm is based on the following two observations:

1. If b|a then gcd(a, b) = b.

This is indeed so because no number (b, in particular) may have a divisor greater than the number itself (I am talking here of non-negative integers.)

2. If a = bt + r, for integers t and r, then gcd(a, b) = gcd(b, r).

Indeed, every common divisor of a and b also divides r. Thus gcd(a, b) divides r. But, of course, gcd(a, b)|b. Therefore, gcd(a, b) is a common divisor of b and r and hence gcd(a, b)=gcd(b, r). The reverse is also true because every divisor of b and r also divides a.

Example Let a = 2322, b = 654.

2322 = 654*3 + 360             gcd(2322, 654) = gcd(654, 360)

654 = 360*1 + 294	       gcd(654, 360) = gcd(360, 294)

360 = 294*1 + 66	 	gcd(360, 294) = gcd(294, 66)

294 = 66*4 + 30	 	        gcd(294, 66) = gcd(66, 30)

66 = 30*2 + 6	 	        gcd(66, 30) = gcd(30, 6)

30 = 6*5	               gcd(30, 6) = 6

Therefore, gcd(2322,654) = 6.

For any pair a and b, the algorithm is bound to terminate since every new step generates a similar problem (that of finding gcd) for a pair of smaller integers. Let Eulen(a,b) denote the length of the Euclidean algorithm for a pair a,b. Eulen(2322, 654) = 6, Eulen(30, 6) = 1. I'll use this notation in the proof of the following very important consequence of the algorithm: Corollary For every pair of whole numbers a and b there exist two integers s and t such that as + bt = gcd(a, b). Example

2322×20 + 654×(-71) = 6. Proof

Let a > b. The proof is by induction on Eulen(a, b). If Eulen(a, b) = 1, i.e., if b|a, then a = bu for an integer u. Hence, a + (1 - u)b = b = gcd(a, b). We can take s = 1 and t = 1 - u.

Assume the Corollary has been established for all pairs of numbers for which Eulen is less than n. Let Eulen(a, b) = n. Apply one step of the algorithm: a = bu + r. Eulen(b, r) = n - 1. By the inductive assumption, there exist x and y such that bx + ry = gcd(b,r) = gcd(a,b). Express r as r = a - bu. Hence, ry = ay - buy; bx + (ay - buy) = gcd(a, b). Finally, b(x - uy) + ay = gcd(a, b) and we can take s = x - uy and t = y.

There is also a simple proof that employs the Pigeonhole Principle. Remark

Note that any linear combination as + bt is divisible by any common factor of a and b. In particular, any common factor of a and b also divides gcd(a, b). In a "reverse" application, any linear combination as + bt is divisible by gcd(a, b). From here it follows that gcd(a, b) is the least positive integer representable in the form as + bt. All the rest are multiples of gcd(a, b). The generalization of the Corollary to an arbitrary field is known as Bézout's identity or Bézout's Lemma after the French mathematician Éttiene Bézout (1730-1783), so it often happens that the result stated in the Corollary is also often referred to as Bézout's identity or Bézout's Lemma.

For coprime numbers we get existence of s and t such that as + bt = 1. This Corollary is a powerful tool. It appeared in the 3 Glass and Hour Glass problems. For example, let's prove the Euclid's Proposition VII.30 If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers.

Let a prime p divide the product ab. Assume pa. Then gcd(a, p) = 1. By Corollary, ax + py = 1 for some x and y. Multiply by b: abx + pby = b. Now, p|ab and p|pb. Hence, p|b.

Actually, this proves a generalization of the Proposition VII.30 I used several times on these pages: Let m|ab and gcd(a, m) = 1. Then m|b.

Proposition VII.30 immediately implies the Fundamental Theorem of Arithmetic although Euclid has never stated it explicitly. The first time it was formulated in 1801 by Gauss in his Disquisitiones arithmeticae. Fundamental Theorem of Arithmetic Any integer N can be represented as a product of primes. Such a representation is unique up to the order of prime factors.

Since, by definition, a number is composite if it has factors other than 1 and itself, and these factors are bound to be smaller than the number, we can keep extracting the factors until only prime factors remain. This shows existence of the representation: N = pqr..., where all p, q, r,... are prime. To prove uniqueness, assume there are two representations: N = pqr... = uvw... We see that p divides uvw... By Corollary, it divides one of the factors u,v,w,... Cancel them out. We can go on chipping away on the factors left and right until no factors remain.

Representation of a number as the product of primes is called prime number decomposition. The Fundamental Theorem of Arithmetic asserts that each integer has a unique prime number decomposition.

=The Fundamental Theorem of Arithmetic= =Multiplicative Functions=


text:
Number Theory/Congruences

= Notation and introduction =

We will call two integers a and b congruent modulo a positive integer m, if a and b have the same (smallest nonnegative) remainder when dividing by m. The formal definition is as follows.

Definition
Let a, b and m be integers where $$m>0$$. The numbers a and b are congruent modulo m, in symbols $$a \equiv b \pmod{m}$$, if m divides the difference $$a-b$$.

Lemma
We have $$a \equiv b \pmod{m}$$ if and only if a and b have the same smallest nonnegative remainder when dividing by m.

Proof:

Let $$a \equiv b \pmod{m}$$. Then there exists an integer c such that $$cm = a-b\,$$. Let now $$q,q',r,r'\,$$ be those integers with

$$a=qm+r,\quad  0\leq r<m$$

and

$$b=q'm+r',\quad 0\leq r'<m$$.

It follows that

$$cm = a-b = m(q-q') + (r-r')\,$$

which yields $$m|(r-r')\,$$ or $$m\ \mbox{divides} \ (r-r')\,$$ and hence $$r=r'\,$$.

Suppose now that $$r=r'\,$$. Then, $$a-b = m(q-q')\,$$, which shows that $$m|(a-b)\,$$.

= Fundamental Properties =

First, if $$a\equiv b \pmod{m}$$ and $$c\equiv d \pmod{m}$$, we get $$ac\equiv bd \pmod{m}$$, and $$a+c\equiv b+d \pmod{m}$$.

As a result, if $$a\equiv b \pmod{m}$$, then $$ a^p\equiv b^p \pmod{m}$$

= Congruence Equations =

= Residue Systems =

Chinese Remainder Theorem
= Polynomial Congruences =

title: Number Theory/Congruences


text:
Number Theory/Diophantine Equations

=Introduction=

The theory of Diophantine equations is an ancient subject that typically involves solving (a system of) polynomial equation in integers. Perhaps the most famous diophantine equation is Fermat's, who stated in the 17th century that the equation $$ x^n+y^n=z^n $$ has no solution in integers if $$ n $$ is greater than or equal to 3 (this is called Fermat's last theorem). Fermat never published his solution to this problem (that he actually had a valid solution is highly debated)and it took mathematicians the next 350 years to finally solve this problem.

=The Relation of Congruences to Diophantine Equations=

=Proofs involving Diophantine Equations=

Teoria de números/Equações diofantinas

title: Number Theory/Diophantine Equations


text:
Number Theory/Quadratic Fields

=Introduction= A quadratic field F is an extension of degree 2 of the rational number field Q. The solution of the quadratic equation shows that it is generated over Q by the square root of a rational number. Multiplying (or dividing) this rational number by the square of a suitable integer, we may assume that it is a “square free” integer m, that is, an integer m without square factors. Let F = Q(s) where s.s = m. Any element x of F is of the form x = a + bs with a and  b in Q. The mapping a + bs -->  a – bs is an automorphism of F over Q. Thus F is a normal extension of Q (with a cyclic group of order 2 as Galois group).

title: Number Theory/Quadratic Fields


text:
AvernumScript/Appendix/Basic Character Calls

void alert_char(short which_char_or_group)

 * Alerts the Creature/Group. This means that, if it is hostile, it will actively hunt the party down when they are nearby.

void award_char_xp(short which_char,short amount,short source_level)

 * Gives character which_char amount experience, adjusted as if it was gained by killing a creature of level source_level. You can’t give more than 1000 experience at any one time.

void award_party_xp(short amount,short source_level)

 * Gives each party member amount skill points, adjusted as if they were gained by killing a creature of level source_level. Source_level is relative level of party/monster giving experience. (1-10 for Beginner, 10-20 for Medium, 20-30 for High)

Note that a character can’t get more than 1000 experience in one lump.

short can_see_char(short which_char_or_group)

 * This can only be called in a creature or terrain script. Returns 1 if the creature/terrain spot whose script it is can see the given character or member of the given group, or 0 otherwise.

void change_char_xp(short which_char,short amt_to_change)

 * Changes experience character which_char by amount. Amt_to_change can be negative.

void change_pc_skill_pts(short which_char,short amt_to_change)

 * Changes the character which_char skill points by amt_to_change. Amt_to_change can be negative.

void change_spell_level(short which_char,short mage_or_priest,short which_spell,short amt_to_change)
Changes the skill character which_char knows spell which_spell by amt_to_change (which can be negative). If mage_or_priest is 0, changes mage spell. If 1, changes priest spell.

If the character is a party member, he or she can only learn the spell if his or her mage or priest spells skill is high enough. If the skill isn’t high enough, this call has no effect.

short char_attitude_to_char(short which_char,short char_to_evaluate)
Evaluates how character which_char feels about character char_to_evaluate. Returns 0 if the character likes the other character, 1 if the character is neutral about the other character, and 2 if the character hates the other character.

short char_has_trait(short which_char,short which_trait)
Returns 1 if character which_char has trait which_trait, 0 if it doesn’t. Traits are listed in the appendices. (in the original manual)

short char_ok(short which_char)
Returns 1 if this character is existing and alive, 0 otherwise.

short char_on_spot(short loc_x,short loc_y)
Checks if there is an NPC (a character not controlled by the player, numbers 6 or higher) on space {loc_x,loc_y}. Returns –1 if no character there. Otherwise, returns the number of the character there. If you also want to check for the presence of the player’s characters, use the call char_on_loc.

short char_status(short which_char)
Returns the current status of this character:

short creature_type(short which_char)
Returns the creature type of which_char.

void deduct_ap(short num_ap)
This can only be called in a creature script and only has effect in combat. Removes num_ap action points from the character whose script is running.

void end_combat_turn
This can only be called in a creature script and only has effect in combat. Removes all action points from the character. Note that this does NOT end the state. You need to call end right after this if you want to stop running script.

short enemies_nearby(short range)
This can only be called in a creature’s script. Returns the number of characters within range spaces that the character can see and which are hostile.

void erase_char(short which_char_or_group)
Makes the character/group stop existing. No treasure is dropped. The character’s killed flag is not set (so it can reappear when the party returns).

short friends_nearby(short range)
This can only be called in a creature’s script. Returns the number of characters within range spaces that the character can see and which are friendly.

short get_attitude(short which_char)
Returns the attitude of the character.

short get_char_skill_pts(short which_char)
Returns the character’s skill points.

short get_energy (short which_char)
Returns the character’s current spell energy. This call was added for Scenario Format Version 2, only use it with Mac version 1.1 or later or Windows version 1.0 or later.

short get_health(short which_char)
Returns the character’s current health.

short get_level(short which_char)
Returns the character’s level. Note: The ME paramater doesn't work with get_level. Use the my_number call instead.

short get_max_health(short which_char)
Returns the character’s maximum health.

short get_species(short which_char)
Returns the character’s species.

short get_spell_level(short which_char,short mage_or_priest,short which_spell)
Returns the skill level that character which_char knows spell which_spell. If mage_or_priest is 0, checks mage spell. If 1, checks priest spell.

short get_summon_level(short which_char)
Returns the summoning level of the current character (which can reveal whether it is summoned or not). The results are:

void give_recipe(short which_recipe)
Teaches the party recipe which_recipe. Recipes are listed in the appendices. (in original manual)

short has_recipe(short which_recipe)
Returns 1 if party knows alchemy recipe which_recipe, 0 if it doesn’t. Recipes are listed in the appendices.

void kill_char(short which_char_or_group, short death_type, short luck_helps)
Kills the character or every member of the group. Death_type determines the sort of death: If luck_helps is 1, the character’s luck can save it. If 0, there is no escape.

short my_ap
This can only be called in a creature script and only has effect in combat. Returns the character’s current number of action points.

short my_number
Can only be called in a creature script or terrain script. Returns this creature’s number (0 - 119) if in a creature script or the terrain script’s number (0 - 99) in a terrain script.

short party_can_see_loc(short loc_x,short loc_y)
This call only works correctly inside a town. Returns 1 if a party member can see location {loc_x,loc_y}, 0 otherwise.

short party_has_trait(short which_trait)
Returns 1 if some party member has trait which_trait, 0 if it doesn’t. Traits are listed in the appendices.

short party_size
Returns the number of living, present characters in the party.

void print_named_str(short which_char,char print_text)
Prints in the text area the name of character which_char, immediately followed by print_text.

Example: If character 17 is a Cute Puppy, the command print_named_str(17,” fires ray of doom.”);

puts the text “Cute Puppy fires ray of doom.” in the text area.

short random_party_member
Returns the number of a random character in the party (will be a number from 0 to 5).

void revive_party
Completely heals the party’s damage and restores all spell energy.

void set_attitude(short which_char_or_group,short which_attitude)
Sets the attitude of the character/group:

void set_char_alert(short which_char_or_group,short alert_level)
Sets the alert level of the Creature/Group which_char_or_group to value alert_level. If alert_level is 1, the characters are alerted and will, if hostile to the party, hunt the party down. If 0, they will no longer be alerted.

short set_char_script_mode(short which_char_or_group,short what_setting)
Adjusts when the scripts for the Creature/Group will be run by the game when the party is not nearby. (By default, the game does not call all the scripts all the time, to keep the game from slowing down.) Values for what_setting are: The default is always 0.

void set_char_trait(short which_char,short which_trait,short set_to)

 * Changes whether the character which_char, has the trait ''which_trait'.

Values for set_to are:

Values for which_trait can be found: here

Notes:
 * You should probably only use this ability to teach the character Ritual of Sanctification (trait 22). You really shouldn't mess with a character's other traits. This call only has a real effect on party members.

void set_creature_memory_cell(short which_char_or_group,short which_cell,short new_value)

 * Sets the memory cell which_cell of the creature or group which_char_or_group to the value new_value.

void set_level(short which_char_or_group,short what_level)

 * Changes the level of the creature or group which_char_or_group to level what_level. The character or character's health and statistics are Automatically adjusted to reflect their new level.

Notes:
 * You shouldn’t use this call on party members.
 * This call also completely heals and restores the character (or characters), so it should probably only be called when the party enters the area.(i.e. in the town script’s state INIT_STATE).

void set_mobility(short which_char_or_group,short mobile)

 * Sets whether the creature or group which_char_or_group can move or not.

Values for mobile are:

void set_name(short which_char_or_group,char new_name)

 * Sets the name of the creature or group which_char_or_group to name new_name.

Notes:
 * Maximum name length is 19 characters.

void set_special_ability(short which_char_or_group,short what_ability)

 * Changes the special ability of the creature of group which_char_or_group to special ability what_ability. The different special abilities are listed in the section on scripting creature types. If what_ability is 0, the creature gets no special ability.

void set_summon_level(short which_char_or_group,short new_level)

 * Sets the summoning level of creature or group which_char_or_group to new_level. Summoned creatures and illusions do not give experience or treasure when killed.

Values for new_level are:

short species_in_party(short which_species)

 * which_species is the number of a species.

Returns:

Values for which_species are:

Note:
 * This call does check any extra characters that are traveling with the party, so any species beyond those listed here will always return 0.


text:
Talk:Visual Basic/Coding Standards

24.47.80.195
Please log in so we can collaborate on changes. It's good to see someone taking an interest. --kwhitefoot 06:23, 8 May 2006 (UTC)

Procedure Calls
I was wondering why "This standard deprecates the use of the Call keyword."? What then should be the best approach for multiple arguments procedures? I think "Call MyFunction(arg1, arg2)" looks more like a function call than "MyFunction arg1, arg2", due to the parenthesizes.
 * Maybe because the call result must be stored in a variable to return a proper user output. JackPotte (discuss • contribs) 18:13, 22 October 2016 (UTC)

Line Numbers in VBA
Any thoughts about the conventional layout for line numbers. I am trying to add them to an indenting procedure but realize that I am not sure about the standard.

For example, should numbers fall below sub and function, or stand off to the left margin with sub and function in "next column", so to speak?

Thanks in advance for any input. Armchair (discuss • contribs) 10:48, 22 October 2016 (UTC)
 * As it's impossible to display the VBA line numbers in Excel, plenty of alternatives had been proposed. But I've never read any convention. JackPotte (discuss • contribs) 18:13, 22 October 2016 (UTC)

title: Talk:Visual Basic/Coding Standards


text:
Talk:False Friends of the Slavist/Map sutra

Untitled
Czech word is "zitra", slovak "zajtra", both means "tomorrow".


text:
Talk:False Friends of the Slavist/Map sbor

Untitled
Czech word is "sbor" - mmeaning "choir".

There is a Polish word "zbór" meaning "Protestant church"

here dont have bulgarian word "събор" I think this word like as "local holiday" and official meeting(like fair, convention, council, synod) --Николов (talk) 01:04, 9 May 2009 (UTC)


text:
Talk:False Friends of the Slavist/Map sklad

Untitled
Czech word is "sklad" - meaning "storehouse".

In Slovenian, sklad means "fund, fundation". I don't know how to add an eight meaning. If you can do it, please proceed. Viator slovenicus (talk) 01:18, 4 November 2010 (UTC)


text:
Talk:False Friends of the Slavist/Map sьrbъ

Untitled
Czech word is "Srb" - meaning "Sorbian" and "Serbian" too.


text:
Talk:False Friends of the Slavist/Map sьrbьskъjь

Untitled
Czech word is "srbsky" - meaning "sorbian" and "serbian" too.


text:
Talk:False Friends of the Slavist/Map utro

Untitled
Czech word is "jitro" - meaning "morning" and hist."acre".


text:
Visual Basic/Simple Graphics

This chapter is a simple introduction to drawing simple graphics in Visual Basic Classic. It is possible to do very sophisticated graphics programming in VB but here we will concentrate of the basics.

There are two ways of creating simple two dimensional graphics in VB:

 * draw lines, boxes, circles, and ellipses at run time by executing statements or

 * add shape objects to the form at design time.

If you know in advance exactly what to draw, then it might be simpler to use the second method, as you can then see exactly what you will get as you drag and drop shapes to the form. However if the picture is to be generated based on, say, some statistics, you might be better off using the drawing methods of a Form or PictureBox.

Shape and Line controls
There are two built-in controls in Visual Basic that can be used to create simple two-dimensional drawings:

 * Shape
 * Line

Here is a simple example showing some of the shapes that can be created simply by dropping Shape controls on a Form:

Note that you can modify the properties of Shape and Line controls at run time just as you can any other control. You can also create them at run time, either by using Control Arrays or by using the CreateObject function.

title: Visual Basic/Simple Graphics


text: File:VBClassicSimpleGraphicsExample.png

title: File:VBClassicSimpleGraphicsExample.png


text:
Umbraco/Samples and Articles/Templates

Templates

 * Validating User Input
 * Using Master Templates
 * Use different URL address
 * The ASPNET_FORM tag

title: Umbraco/Samples and Articles/Templates


text:
Umbraco/Samples and Articles/Templates/MasterTemplates

Introduction
A master template in Umbraco can be included by child templates in order to ensure that a set of pages have common elements (typically header and footer).

You will need to understand how to create Umbraco documents and template types before reading this article.

Example of a master template
In this very simple example we are going to create a master template that will create the opening HTML and HEAD tags, open the BODY tag, insert the child template and then close the HTML tag.

In Umbraco, create a template called 'Master' (can be called anything) with the following code.

<?UMBRACO_GETITEM field="pageName"/> <?UMBRACO_TEMPLATE_LOAD_CHILD/>

Important! a bug was reported in Umbraco 2.1 where whitespace in <?UMBRACO_TEMPLATE_LOAD_CHILD/> could cause the master/child template functionality to fail. Please copy and paste exactly as is.

Notes on the example master template
The above template should be fairly familiar HTML. There are two Umbraco tags that need your attention:

<?UMBRACO_GETITEM field="pageName"/>

The pageName attribute is a common element across all nodes in the Umbraco content XML. The code above simply inserts this name into the html title tag.

<?UMBRACO_TEMPLATE_LOAD_CHILD/>

The above code tells Umbraco that this is the place in the markup where a child template should be inserted. We will learn how to do this in the next step.

Creating the child template
In Umbraco, create a template called 'Child' or whatever you want.

In the left hand menu click on the template you just created.

The source of the template can be anything you want, but for this very simple example let's use the following:

I am output from the child template

Before you save the template, change the select box labelled 'Master Template' so the value is set to the master template you created in the first step of this tutorial.

Testing your work
Choose a document type from the settings menu, or create a new blank document type.

Add the child template you created in the second step of this tutorial to the Allowed templates for your document type.

Goto the content section of Umbraco and create a node of the document type.

Now preview the node with the Temlate type set to the child template you created in the second step of this tutorial.

View the source of the preview and the results should show your master template has been included around the child template as follows:

Node Title I am output from the child template

Further steps
You can now extend your master template to include a common stylesheet and other metadata in the HEAD element.


text:
Suomen kieli käyttöön/lukeminen

Introduction
The texts here will help you to understand Finnish. The texts are provided by teachers and are used to develop the learners' understanding of Finnish.

The material is sourced from an original Finnish-text article. Each Finnish-text article has a discussion page entirely in Finnish. These pages are subscribed to by people whose mother tongue is Finnish and can answer questions about the text. Please do not ask questions in English here!!

Each article can also have a number of parallel pages in the language of the learners. There pages are created and maintained entirely by the learners themselves. They record their understanding of the material and editors of these pages will often refer to the original Finnish discussion page to find answers to their questions. If they cannot find an answer to their question, they can try raising a question in the English translation article's discussion page in English, or can pose the question in Finnish on the Finnish article's Finnish discussion page.

Editing and correcting pages
Translation pages will, by their very nature, be incomplete until completed by the students! If you do some work on a translation article (for example by translating sections not yet translated) please publish your translation by updating the page of the book for the benefit of others!! DO NOT WORRY THAT YOUR UNDERSTANDING MAY BE WRONG!!! Others with a better understanding than you will soon discover your error and change it. If you find an error anywhere in any page, please change it immediately. If you are not certain about a change you would like to see, raise the matter in the article's discussion page.

title: Suomen kieli käyttöön/lukeminen