Some games are played with two or more dice and some use only one. Some games that use dice are Monopoly, Yahtzee, Risk, and many more. Dice with different numbers of sides, like those used in Dungeons & Dragons, are called polyhedral dice.
Weights can be put inside a die to make it only land on a certain number once it is rolled.
Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangle of numbers, arranged in rows and columns. The rows are each left-to-right (horizontal) lines, and the columns go top-to-bottom (vertical). The top-left cell is at row 1, column 1 "(see diagram at right)".
Matrices are often represented by capital roman letters such as formula_1, formula_2 and formula_3, and there are rules for adding, subtracting and "multiplying" matrices together, but the rules are different than for numbers. As an example, the product formula_4 does not always give the same result as formula_5, which is the case for the multiplication of ordinary numbers. A matrix can have more than 2 dimensions, such as a 3D matrix. Also, a matrix can be one-dimensional, as a single row or a single column.
Many natural sciences use matrices quite a lot. In many universities, courses about matrices (usually called linear algebra) are taught very early, sometimes even in the first year of studies. Matrices are also very common in computer science, engineering, physics, economics, and statistics.
The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with "m" rows and "n" columns is called an "m"-by-"n" matrix (or "m"×"n" matrix) and "m" and "n" are called its dimensions.
The places in the matrix where the numbers are is called "entries". The entry of a matrix "A" that lies in the row number "i" and column number "j" is called the "i,j" entry of "A". This is written as "A"["i,j"] or "a".
We write formula_6 to define an "m" × "n" matrix "A", with each entry in the matrix called "a" for all 1 ≤ "i" ≤ "m" and 1 ≤ "j" ≤ "n".
The matrix
is a 4×3 matrix. This matrix has m=4 rows, and n=3 columns.
The element "A"[2,3] or "a" is 7.
The sum of two matrices is the matrix, which ("i","j")-th entry is equal to the sum of the ("i","j")-th entries of two matrices:
The two matrices have the same dimensions. Here, formula_9 is true (and is true in general for matrices of equal dimensions).
The multiplication of two matrices is a bit more complicated:
So with numbers:
There are some matrices that are special.
A square matrix has the same number of rows as columns, so m=n.
An example of a square matrix is
This matrix has 3 rows and 3 columns: m=n=3.
Every square dimension set of a matrix has a special counterpart called the "identity matrix", represented by the symbol formula_15. The identity matrix has nothing but zeroes except on the main diagonal, where there are all ones. For example:
is an identity matrix. There is exactly one identity matrix for each square dimension set. An identity matrix is special because when multiplying any matrix by the identity matrix, the result is always the original matrix with no change.
An inverse matrix is a matrix that, when multiplied by another matrix, equals the identity matrix. For example:
formula_18 is the inverse of
formula_19.
The formula for the inverse of a 2x2 matrix, formula_20 is:
formula_21
Where formula_22 is the determinant of the matrix. In a 2x2 matrix, the determinant is equal to:
formula_23
A matrix, that has many rows, but only one column, is called a column vector.
The determinant takes a square matrix and calculates a simple number, a scalar. To understand what this number means, take each column of the matrix and draw it as a vector. The parallelogram drawn by those vectors has an area, which is the determinant. For all 2x2 matrices, the formula is very simple:
formula_24
For 3x3 matrices the formula is more complicated:
formula_25
There are no simple formulas for the determinants of larger matrices, and many computer programmers study how to get computers to quickly find large determinants.
There are three rules that all determinants follow. These are:
Hawaiian
Hawaiian can be:
British Isles
The British Isles are an archipelago (a group of islands) in north-western Europe. They have a long geographical and geological history. The first people were in Britain by 800,000 years ago (Happisburgh footprints).
The biggest islands are called Great Britain and Ireland, followed by the Isle of Man. Great Britain, the biggest island, is home to three countries. These are England, Wales and Scotland. The island of Ireland is divided into Northern Ireland and the Republic of Ireland. Great Britain is the biggest island of Europe and Ireland is the third biggest.
England, Wales, Scotland and Northern Ireland join together to make the United Kingdom.
Some people include the Channel Islands of Jersey, Guernsey, Sark in the term "British Isles", but they are a lot closer to France than to Great Britain. In politics, they are British islands but in geology they are French islands.
The term tends not to be used in the Republic of Ireland because many people feel it implies that Ireland is politically British.
The writings of the Roman cartographer Ptolemy used the name "Great Britain" for the larger island in the 2nd century AD. He used the name "Megale Britannia" or "Great Britain" () to show the difference between this island and Ireland. For Ireland, Ptolemy used the name "Mikra Britannia" or "Little Britain" ().
Mandarin Chinese
Mandarin Chinese or simply Mandarin (; ) is the language of government and education of the Chinese mainland and Taiwan, with the notable exceptions of Hong Kong and Macau where a local dialect of Chinese called Cantonese is more often used.
Mandarin is one of five major regional languages of China. It spreads wider than any other regional variety, from the whole northern part of China to Yunnan Province in the southwest corner of China. In that big area there are many regional differences in vocabulary, so somebody who moves from Beijing to Yunnan could not understand people there who were speaking their own dialect, Yunnanhua. The problem is bigger than for a person in Great Britain or the United States to go to Australia. Therefore, starting in the 1920s, the Chinese government set up a national language based on the Beijing dialect and on the most widely understood words and pronunciations.
Mandarin is a standard language. It is nobody's native language, but a good average between various language forms and a common language everyone can understand and communicate with. Although it is based on the Beijing dialect, it is not the same as Beijing dialect.
Schools use a dialect called Standard Mandarin, "Putonghua" (普通话/普通話) meaning "common (spoken) language" or "Hanyu" (汉语/漢語) meaning "language of the Han". In places such as Malaysia, it is known as "Huayu" (华语/華語). In Taiwan, it is known as "Guoyu" (国语/國語) meaning "national language." There are some minor differences in these standards.
Mandarin is spoken by over 800 million people around the world, more than any other language. Most people emigrating from the Greater China region now speak Mandarin, while in past centuries most spoke Cantonese or Taishanese, another local Chinese dialect.
Standard Mandarin is one of the six official languages at the United Nations. The others are English, French, Spanish, Russian and Arabic.
Mandarin is written with Chinese characters called "Hànzì" (漢字 or 汉字) which literally means "Han characters". Each "Hànzì" has its own pronunciation and meaning. An ordinary dictionary will contain about 10,000 characters. Spoken Mandarin uses very many compound words, words that combine meanings the way English does in such terms as "machine gun," "fire truck," "playground," etc.