## Write basic expressions and equations

When it comes to solving problems, it is useful to be able to translate between written words and mathematical expressions and equations.

Consider the following problem: Bill is currently twelve years older than Janet, and in two years Bill will be twice the age of Janet. How old is Janet?

To solve such a problem we will need to come up with mathematical equations which will need to be solved. If we let \(\displaystyle{B}\) denote the current age of Bill and \(\displaystyle{J}\) denote the current age of Janet then the phrase "Bill is currently twelve years older than Janet" is mathematically expressed as

\(\displaystyle{B=J+12}\)

and the phrase " in two years Bill will be twice the age of Janet" is mathematically expressed as

\(\displaystyle{B+2=2 J}\).

We can now use the method of substitution to find Janet's age, but that will be discussed in a later section.

## Number Classification

### The real numbers system and its internal classification

#### Real number

A real number is any quantity, which can be either positive or negative, that can be used to represent the distance along a line. The set of all real numbers is denoted \(\displaystyle{\mathbb{R}}\).

#### Integer

An integer is a whole number which can be either positive, negative, or \(\displaystyle{0}\). The set of all integers is denoted \(\displaystyle{\mathbb{Z}}\).

#### Rational Number

A rational number is any real number which can be expressed as the ratio \(\displaystyle{\dfrac{a}{b}}\) of two integers \(\displaystyle{a,b}\) with \(\displaystyle{b\neq0}\). The set of all rational numbers is denoted \(\displaystyle{\mathbb{Q}}\).

Sometimes, the decimal expression of a real number involves a repeating decimal pattern.

### Introduction to the complex number system

## Basic operations with real numbers

### Compare and order real numbers

#### Compare real numbers

Comparing two real numbers means assessing which one is greater than the other. Given two real numbers *a* and *b* :

- If
*a*is greater than*b*, we write \(\displaystyle{a\gt b}\). - If
*b*is greater than*a*, we write \(\displaystyle{a \lt b}\).

Given two real numbers *a* and *b* :

- If
*a*is greater than*b*or equal to*b*, we write \(\displaystyle{a\geqslant b}\) - If
*b*is greater than*a*or equal to*a*, we write \(\displaystyle{a \leqslant b}\)

If \(\displaystyle{a\mbox{ and }b}\) real numbers, then \(\displaystyle{a\leq b}\) if and only if \(\displaystyle{a}\) is further to the left than \(\displaystyle{b}\) on the real number line.

### Addition, substraction, multiplication and division of real numbers

#### Addition

If \(\displaystyle{a\mbox{ and }b}\) are real numbers and \(\displaystyle{b\geq 0}\), then the number \(\displaystyle{a+b}\) is the number obtained by increasing the size of \(\displaystyle{a}\) by the size of \(\displaystyle{b}\).

#### Subtraction

If \(\displaystyle{a\mbox{ and }b}\) are real numbers with \(\displaystyle{b\geq 0}\) then the number \(\displaystyle{a-b}\) is the number obtained by decreasing the size of \(\displaystyle{a}\) by the size of \(\displaystyle{b}\).

If \(\displaystyle{a\mbox{ and }b}\) are any real numbers, then we can still interpret \(\displaystyle{a+b\mbox{ and }a-b}\). If \(\displaystyle{b}\) is negative then \(\displaystyle{-b}\) is positive, \(\displaystyle{a+b=a- \left(-b\right)}\), and \(\displaystyle{a-b=a+\left(-b\right)}\). Therefore we can interpret \(\displaystyle{a+b}\) as a subtraction and \(\displaystyle{a-b}\) as an addition.

#### Multiplication

If \(\displaystyle{a\mbox{ and }b}\) are real numbers with \(\displaystyle{b\geq 0}\) then the number \(\displaystyle{ab}\) is the number obtained by scaling the number \(\displaystyle{a}\) by the number \(\displaystyle{b}\).

#### Division

If *a* and *b* are real numbers (with \(\displaystyle{b\neq0}\) ) then \(\displaystyle{\dfrac{a}{b}}\) is the real number such that \(\displaystyle{b\cdot \dfrac{a}{b}=a}\).