Arithmetic and Algebra
Arithmetic comes from the Greek word arithmetike, which combines the ideas of two words in Greek, arithmos, meaning "number," and techne, referring to an art or skill. It refers generally to the study of the nature and properties of numbers, measurement, and numerical computation (that is, the study of the algorithms of calculation with numbers, the fundamental operations of addition, subtraction, multiplication, and division, as well as raising to powers, and extraction of roots.)
The numbers used for counting are called the positive integers. They are generated by adding 1 to each number in an unending series, so that each number in the sequence is one more than its immediate predecessor. Different civilizations throughout history have developed different kinds of number systems. One of the most common is the one used in all modern cultures, in which objects are counted in groups of ten.
In the base 10 system, integers are represented by digits expressing various powers of 10. For example, take the number 2497. Every digit in this number has its own place value, and the place values increase by another power of 10 as they move to the left. The first place value is a unit value (here, 7); the second place value is 10 (here, 9 x 10, or 90); the third place value is 10 x 10, or 100 (here, 4 x 100, or 400); and the fourth place value is 10 x 10 x 10, or 1000 (here, 2 x 1000, or 1000).
Elementary arithmetic is concerned primarily with the effect of certain operations, such as addition or multiplication, on specified numbers; elementary algebra is concerned with properties of arbitrary numbers. For instance, the fact that 5 multiplied by 6 gives the same result as 6 multiplied by 5 is expressed in algebra by the formula a x b = b x a for all numbers a, b.
The word "Algebra" is derived from Arabic and refers to a branch of mathematics which may be defined as the generalization and extension of arithmetic. It deals with relations and properties of quantity by means of letters and other symbols. At its most elementary level, algebra involves the manipulation of real number constants and variables in simple equations whereas intermediate algebra extends the principles of elementary algebra to include complex numbers and abstract algebra involves the study of algebraic structures such as groups, rings and fields. The term algebra may also be used to refer to an algebraic structure or an algebra over a field.
Unlike arithmetic, which deals with specific numbers, algebra introduces 'variables' that greatly extend the generality and scope of arithmetic. A variable is a symbol that represents a number. Usually we use letters such as n, t, or x for variables. For example, we might say that s stands for the side-length of a square. We now treat s as if it were a number we could use. The perimeter of the square is given by 4 x s. The area of the square is given by s x s. When working with variables, it can be helpful to use a letter that will remind you of what the variable stands for: let d be the distance from my house to the park; let p be the number of people in a movie theater; let t be the time it takes to travel somewhere.
As in arithmetic, the basic operations of algebra are addition, subtraction, multiplication, division, and the extraction of roots. Arithmetic, however, cannot generalize mathematical relations such as the Pythagorean theorem: The square of the hypotenuse is equal to the sums of the squares on the other two sides. Arithmetic can only produce specific instances of these relations (for example, 3, 4, and 5, where 32 + 42 = 52 But algebra can make a purely general statement that fulfills the conditions of the theorem: a2 + b2 = c2 Any number multiplied by itself is termed squared (as in, the area of a square, which is the length of a side multiplied by itself) and is indicated by a superscript number 2. For example, 3 x 3 is notated 32; similarly, a x a is equivalent to a2.
What is the product of the following series? (x-a)(x-b)(x-c)...(x-z) Answer
Equations and inequalities involving variables are solved for the "unknowns." Systems of equations are used to solve practical problems. The solution of systems of linear equations leads to the study of linear algebra, in which the elements are matrices and vectors.
Classical algebra, which is concerned with solving equations, uses symbols instead of specific numbers and uses arithmetic operations to establish ways of handling symbols. Modern algebra has evolved from classical algebra by increasing its attention to the structures within mathematics. Mathematicians consider modern algebra to be a set of objects with rules for connecting or relating them.
Abstract algebra is a more general branch of mathematics that analyzes algebraic axioms and operations with arbitrary sets of symbols. Abstract algebra is the study of systems that satisfy certain sets of axioms. Some of these structures are fields, rings, groups, and domains. The elements used in abstract algebra may be numbers, vectors, or even geometric transformations.
Mathematical Number Types
A number system is any of various sets of symbols and the rules for using them to express quantities as the basis for counting, comparing amounts, performing calculations, determining order, making measurements, representing value, setting limits, abstracting quantities, coding information, and transmitting data. In this article, we'll briefly review the simple (basic) number concepts and then overview the other kinds of number.