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Algebra: Introduction

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Algebra can be described as the branch of mathematics that is concerned with the relations between quantities, which are indicated by letters and other characters.

Mathematical operations are essential in these relations. Examples are addition, subtraction, multiplication, division, and exponentiation.

The most common mathematical operations

In the addition #234+567#, the numbers #234# and #567# are called terms. The terms are also called summands.
In the substraction #234-567#, the numbers #234# and #567# are called the terms. Because the substraction can be regared as an addition #234+(-567)#, #234# and #-567# can be called the summands of the expression as well.
In the multiplication #234\times567#, which can be written as #234\cdot567#, the numbers #234# and #567# are called the factors. Notice the difference between multiplication and the decimal point as in #234.567#.
In the division#\frac{234}{567}#, which can be written as #234/567#, the number #234# is called the numerator and the number #567# is called the denominator.
In the exponentiation #234^{567}#, which sometimes can be written as #{234}\widehat{\phantom{i}}{567}#, the number #234# is called the base and #567# the exponential.

Often we will represent numbers by variables, so the numbers in the expression are replaced by letters.

First we will discuss the basic rules: order of operation, calculating with exponentials, eliminating brackets and bringing factors outside the brackets.

Next we will consider special products and rational expressions.

Finally, we will discuss the binomial theorem and the inherent binomial coefficients. The binomial theorem tells us how powers like#(a+b)^3#, #(a+b)^4#, and #(a+b)^5# can be written as sums of products of #a# and #b#. The binomial coefficients appear as the coefficients of these products. They can be described in terms of factorials, that is, numbers of the form #1\cdot 2\cdots (n-1)\cdot n#, where #n# is a natural number. Factorials and binomial coefficients are often seen in combinatorics, probability theory, and statistics. Combinatorics can be considered as the part of mathematics that is concerned with the systematic solving of counting problems. We will provide a few examples.