Algebra is the branch of mathematics that deals with numbers and their relations, properties of operations and the structures these operations are defined on. Elementary Algebra that follows the study of arithmetic is mostly occupied with operations on sets of whole and rational numbers and solving first and second order equations.
Based on Quantum mechanics researchers made a break through by inventing quantum computers which are far better and faster than the classical computers. The work by researchers have proved that the matrix like simulation of reality would consume less memory on quantum computers than on classical computers.
A combination lock is a type of padlock in which a sequence (permutation) of numbers or symbols is used to open the lock. The sequence may be entered using a single rotating dial which interacts with several discs or cams, by using a set of several rotating discs with inscribed numerals which directly interact with the locking mechanism, or through an electronic or mechanical keypad.
The word algebra is a Latin variant of the Arabic word al-jabr. This came from the title of a book, Hidab al-jabr wal-muqubala, written in Baghdad about 825 A.D. by the Arab mathematician Mohammed ibn-Musa al-Khowarizmi. The al–jabr part of this title means "reunion of broken parts", the wal-muqubala part of this title translates as "to place in front of, to balance, to oppose, to set equal". Together they describe symbol manipulations common in algebra: combining like terms, moving a term to the other side of an equation, etc. Robert Recorde (ca. 1512–1558), the inventor of the symbol "=" of equality, was the first to use the term algebra in its mathematical meaning.
An important development in algebra that took place in the 16th century was the introduction of symbols for the unknown and for algebraic powers. The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2. The Hindus recognized that quadratic equations have two roots and included negative as well as irrational roots. Brahmagupta was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable. Bhaskara recognized that a positive number has two square roots.
The term "matrix" was first introduced by J.J. Sylvester in 1848. It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations. The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. The text "Nine Chapters on the Mathematical Art" gives the first known example of matrix methods to solve simultaneous equations. Arthur Cayley used the matrices to write simultaneous equations. Two great mathematicians Jacobi and Cauchy contributed to the development of 'determinants' in matrices. Almost sixty years after the invention of matrices, Heisenberg a famous physicist, used the matrices for his work on "Quantum Mechanics". Today, matrix theory is used in the study of physical sciences, engineering, statistics, economics, sociology, designing computer games and graphics.
A Set theory is the branch of mathematics that studies sets, which are essentially a collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. Although any type of objects can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Because of its generality, set theory forms the foundation of every other part of mathematics. Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source – "The Theory of Sets". Set theory was introduced by George Cantor in 18th century, later it was developed by John Venn and De Morgan. Cantor has given the basic idea about set theory, Venn has given the pictorial form of sets and De Morgan proposed laws on the sets.
An arrangement is called a Permutation. It is the rearrangement of objects or symbols into distinguishable sequences. When we set things in order, we say we have made an arrangement. When we change the order, we say we have changed the arrangement. So each of the arrangement that can be made by taking some or all of a number of things is known as Permutation. The word arrangement is used to emphasize that the order of the things is important. There are many applications of permutations in everyday life. One is a license plate. License plates are a permutation because the order matters. Another application of permutations is telephone numbers. For example: if we dialled 123-4567 we'd get a different person than when we dialled 765-4321.
A combination is a way to arrange items or numbers when order does not matter. Now we only have to figure out if the order does not matter. If we were to grab a yellow marble and a red marble from a bag of red and yellow marbles, order might not matter. As long as we draw two marbles, we don't need to pick the yellow first and the red second. Either way, we have both marbles. When order does not matter, it is a combination problem.