## Chapter 2 : Algebra

### By Lund University

Algebra is one of the main and most important part of mathematics. Chapter 2 will cover the algebra needed for this course, such as rules of algebra, fractions, powers, inequalities and logarithms. Most of what you will see in this chapter will be known to you. However, unless you had studied mathematics at the University level, it might be a good idea to go through this chapter. First of all, it will give you a chance to refresh and review material that you have done before. Second, the structure of the material in this chapter is more formalized with a clear separation of definitions and results.

## Number systems

This section introduces different types of numbers: natural numbers, integers, rational numbers and real numbers with a focus on the real numbers. We look at addition and multiplication as binary operations on the real numbers and ≤ as binary relation on the real numbers.

## Rules of algebra

This section contains the basic rules of algebra. We will cover the commutative and the associative laws of addition and multiplication. We will also look at the four types of inequalities. Next, we have all the sign rules, manipulating addition and multiplication of positive and negative numbers. We cover a bunch rules related to the numbers zero and one (for example, multiplying any number by zero will take it to zero). The distributive law is one of the most important rules in algebra and we will cover this law as well as extensions of the distributive law dealing with quadratic identities. We can often use quadratic identities “in reverse” to factorize and algebraic expression, that is, decomposing the expression into a product of expressions.

## Rules of algebra: problems

Algebra exercises

## Fractions and powers with integer exponents

We have two separate topics in this section: fractions and powers. In this section, we only consider powers where the exponent is an integer. This course assumes that you are familiar with fractions and that you know how to add, multiply and divide fractions. The focus here is instead to provide you with a complete list all the rules related to fractions and a bunch of problems allowing you to refresh your skills. The same is true for powers. It is assumed that you know the basic power rules such as multiplying two powers with the same base.

## Fractions and integer powers: Problems

Exercises on fractions and powers

## Powers with rational exponents

This section focuses on powers when the exponents is not necessarily an integer. The main focus is on the case when the exponent is one half which is the square root. We will also look at cube roots and extensions to an arbitrary nth root. From this, we can define a power with an arbitrary rational exponent. It is also important to know which combinations of a base and an exponent is allowed (for example, you cannot take the square root of a negative number if you are working with the real numbers).

## General powers: Problems

Exercises on square roots and powers

## Inequalities and sign diagrams

We begin this section by looking at some terminology related to inequalities, namely reflexivity, anti-symmetry, transitivity and totalness. We then look at sign diagrams, a helpful method for figuring out the sign of an expression containing a variable. The next topic is double inequalities which are useful for specifying a range of real numbers. We also cover the absolute value of a number in this section, mostly because the definition of an absolute value uses inequalities. Finally, we look at intervals, and concepts related to intervals, such as open and closed intervals, unbounded intervals, boundary points and the interior and the closure of an interval.

## Inequalities: Problems

Exercises on inequalities and sign diagrams

## Logarithms

We begin the section on logarithms by looking at the common logarithm. Next, we look at the logarithmic identities and the logarithm of a product, a ratio and a power. We move on to the natural logarithm and the rules governing the natural logarithm and we conclude section with logarithms with an arbitrary base.