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Chapter 1: Introduction to Algebra

# The Basics

Algebra is the branch of mathematics that deals with numbers and their relations. Algebra is used throughout of people’s daily lives from buying groceries in the store to scientific researches. Algebra is so useful that NASA is using binary numbers to communicate to the possible extraterrestrial lives. So learning algebra is so important because that people’s lives are depended on algebra. Before you dive into the world of algebra, you need to know a few basics, and this chapter is dedicated to that purpose.

Algebra deals mainly with numbers. There are two types of numbers.

The first type of number is a real number. Real numbers are the numbers we see in everyday lives and most scientific researches.

Real numbers can also be divided into two kinds of numbers. The first kind is called rational numbers. An integer is a whole number (not a fractional number) that can be positive, negative, or zero. A natural number is a number that occurs commonly and obviously in nature, a whole, non-negative number. Rational numbers are numbers determined by the ratio of an integer to a nonzero natural number. Examples of these numbers include 5, 1/5 and 1/3. The decimal expansion of a rational number is either finite or eventually periodic (such as .333333333etc.)

The other kind of real number is called an irrational number. An irrational number is a real number that cannot be reduced to any ratio between an integer and a natural number. Examples of these numbers are the square root of 2, the cube root of 3, the circular ratio pi, and the natural logarithm base e. The square root of 2 and the cube root of 3 are examples of algebraic numbers. Pi and e are examples of special irrationals known as transcendental numbers. The decimal expansion of an irrational number is always nonterminating (it never ends) and nonrepeating (the digits display no repetitive pattern).

The second type of number is an imaginary number. Imaginary numbers do exist; they were named before they were fully understood. They are part of a complex number system. An imaginary number is a number whose square is negative. Every imaginary number can be written as ix, where x is a real number and i is the positive square root of -1.

** To correct a couple of mistakes, the above paragraphs have been edited by the ThinkQuest editor.

After you get to know the numbers, you need to know the ways to express the numbers or algebraic expressions. The algebraic expressions are made up of four different kinds of symbols: Variables, Numbers, Grouping Symbols, and Operation Signs. The variables are the unknown values that need to be found out through numeral steps of calculations. The numbers are the known values that are used to find out the variables. The grouping symbols are the signs that put a group of numbers or variables together so that they can be calculated first. The operation signs are the signs that actually do the calculations. There are several basic properties of algebraic expressions. By knowing them, you can do many algebra problems quickly and accurately. These properties are listed below:

Trichotomy Properties: For all real numbers a and b, one and only one of these statements is true-a< b, a= b, a> b.

Transitive Property of Order: For all real numbers a, b, and c, if a< b and b< c, then a< c.

Commutative Property of Addition: For all real numbers a and b, a+ b=b+a.

Commutative Property of Multiplication: For all real numbers a and b, ab=ba.

Associative Property of Addition: For all real numbers a, b, and c, (a+b)+c=a+(b+c).

Associative Property of Multiplication: For all real numbers a, b, and c, (ab)c=a(bc).

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