Dimensions are the way we see, measure and experience our world, by using up and down, right to left, back to front, hot and cold, how heavy and how long, as well as more advanced concepts from mathematics and physics. One way to define a dimension is to look at the degrees of freedom, or the way an object can move in a specific space. There are different concepts or ways where the term dimension is used, and there are also different definitions. There is no definition that can satisfy all concepts.
In a vector space formula_1 (with vectors being "arrows" with directions), the dimension of formula_1, also written as formula_3, is equal to the cardinality (or number of vectors) of a basis of formula_1 (a set which indicates how many unique directions formula_1 actually has). It is also equal to the number of the largest group of straight line directions of that space. "Normal" objects in everyday life are specified by three dimensions, which are usually called length, width and depth. Mathematicians call this concept Euclidean space.
Dimensions can be used to measure position too. The distance to a position from a starting place can be measured in the length, width and height directions. These distances are a measure of the position.
In some occasions, a fourth (4D) dimension, time, is used to show the position of an event in time and space.
In modern science, people use other dimensions. Dimensions like temperature and weight can be used to show the position of something in less simple spaces. Scientist study those dimension with dimensional analysis.
Mathematicians also use dimensions. In mathematics, dimensions are more general. Dimensions in mathematics might not measure things in the world. The rules for doing arithmetic with dimensions in mathematics might be different than usual arithmetic rules.
Vectors are used to show distances and directions. Vectors are often used in engineering and science, and sometimes in mathematics.
A vector is a list of numbers. There is one number for each dimension. There are arithmetic rules for vectors.
For example, if Jane wants to know the position of Sally, Sally can give Jane a vector to show the position. If Jane and Sally are in the world, there are three dimensions. Therefore, Sally gives Jane a list of three numbers to show her position. The three numbers in the vector Sally gives Jane might be:
Distance
Distance is how far one thing is from another thing. It is also a measure of the space between two things. It can be measured along any path. Thus, someone who goes around in a circle has traveled a distance, even though his position has not changed.
In geometry, the distance between two points "A" and "B" is sometimes written as formula_1. Pythagorean theorem is often used in the calculation of distance. Distance is a scalar, and thus is different from displacement. Displacement is a vector that measures distance with a straight line (and in only one path). Displacement is the shortest way to travel the distance.
Depth
In math, the distance between the nearest end and farthest end of an object is its depth.
For example, Jane measures a box. When she measures the distance between the end of the box closest to her and the end of the box farthest away, Jane measures the box's depth.
"See"
For liquids, the distance between the top or surface of the liquid and the bottom of the liquid is the liquid's depth.
For example, water is a liquid. If you fill a container with water, the distance between the top of the water and the bottom of the container is the water's depth. If the depth is big we say the water is deep.
Dictionary
A dictionary is a type of book which explains the meanings of words or, more precisely, lexemes. The words are arranged in alphabetical order so that they can be found quickly. The word "dictionary" comes from the Latin "dictio" ("saying").
There are several types of dictionaries: dictionaries which explain words and how they are used, dictionaries which translate words from one language to another, dictionaries of biography which tell about famous people, technical dictionaries which explain the meanings of technical words or words connected to a particular subject (sometimes called a thesaurus). Some of these come close to being an encyclopedia, but an encyclopedia gives a lot of extra information about things (knowledge) and does not explain the use of the language. An encyclopedic dictionary gives less information about the topic than a real encyclopedia does, but more than a simple dictionary.
Dictionaries which explain what words mean will give a clear "definition" of the word (e.g. hippopotamus : a hoofed mammal with thick skin, large mouth and short legs that lives in rivers and lakes of Africa.)
A big dictionary will also give more information about the word. It will explain how it is pronounced. Usually the International Phonetic Alphabet is used for this. It will explain how the word is used. This is not a problem for a word like "hippopotamus", but a word like "put" has so many different meanings that a large dictionary may have a whole page or more to explain how it can be used. It will also explain the origin of the word (e.g. Greek "hippos" horse and "potamus" river).
A dictionary may also give the form of the word in different tenses, plural form etc.
There are also dictionaries which translate words into foreign languages. Often one volume (one book) will translate both ways; for example, half the book might be translating from English to Dutch and the other half from Dutch to English.
When using a dictionary to find out how to say something in another language one has to be careful to choose the right word. A word like "right" has two basic meanings in English: 1) "correct", and 2) the opposite of "left". Other languages have different words for these different meanings, but they have homonyms of their own. A word like "put" has many meanings. A good dictionary will have a large list of these meanings to help people find the word they want. In many languages, for example, the word “put” will be different according to whether something is being put onto something (e.g. a table) or into something (e.g. a cupboard).
Dictionaries need to be updated frequently because of the way language changes. New words are often brought into a language (e.g. lots of computer terms) or words change their meanings (e.g. "gay" or "cool"). In this sense, the most famous English Dictionary is the Oxford English Dictionary (or OED). Words are always being added to the OED. They are never taken out even if they are obsolete (not used any more). The OED can be accessed online (with a subscription).
Definition
A definition is an exact word or phrase of the meaning, nature, or limits of something. A definition usually answers the question "what". "Defining" means giving a definition. Other words with this meaning are "description" and "explanation". It describes what a word means and explains to the person when and where it can be used.
In mathematics, a definition is an exact way of saying what something is. It might not be the easiest way to say what it is, but it is used because it is exact. It can be used in a mathematical proof about the thing.
Denmark
Denmark (), officially named the Kingdom of Denmark, is a Nordic country in Northern Europe. It is the furthest south of the Scandinavian countries, to the south of Norway and south-west of Sweden (which it is connected to by a bridge). It has a south border with Germany. It borders both the North Sea to the west and the Baltic Sea to the east. Denmark is a developed country with a large welfare state; In 2006 and 2007, surveys ranked Denmark as "the happiest place in the world," based on standards of health, welfare, and education.