Introduction to ALGEBRA

Algebra, branch of mathematics in which symbols represent relationships. Classical algebra grew out of methods of solving equations; it represents numbers with symbols that combine according to the basic arithmetical operations of addition, subtraction, multiplication, division, and the extraction of roots. However, arithmetic cannot generalize mathematical relations such as Pythagoras' theorem, which states that in any right-angled triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides. Arithmetic can produce only specific instances of these relations (for example, 3, 4, and 5, where 3² + 4² = 5²). Algebra, by contrast, can make a purely general statement that fulfils the conditions of the theorem: a² + b² = c². Any number multiplied by itself is termed squared, and is indicated by a superscript number 2. For example, 3 × 3 is notated 3²; similarly, a × a is equivalent to a².

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. As such, in its most general form, algebra may fairly be described as the language of mathematics.