# Notation

### Dot dot dot \ldots

When you see \dots it means “continue the same pattern”. For instance 1,2,3 \ldots, 10 means 1,2,3,4,5,6,7,8,9,10.

### Braces { }

A pair of curly brackets/braces { } is used to denote sets. For example, {1,2,3} is the set containing the numbers 1, 2 and 3.

### Approximately Equal \approx

The symbol \approx means approximately equal. \sqrt(2) \approx 1.41 says "the square root of 2 is approximately equal to one point four one."

### Not Equal to \ne

The symbol \ne means not equal to so p\ne2 means "p is not equal to 2".

### Line over part of a decimal e.g. 0.1\overline{6}

An overbar over part of a decimal means the digits repeat forever. For example \frac{1}{6} = 0.1\overline{6} means that just the 6 is repeated giving 0.16666666666\ldots, but \frac{2}{7} = 0.\overline{285714} means that the whole sequence 285714 is repeated: 0.285714285714285714\dots.

### Absolute Value |a|

The absolute value |a| is the magnitude of a, that is, the number with any negative sign removed. For example, |3|=3 and |-3|=3. Note that any calculation within the absolute value is done first and then the sign ignored eg |5-8| = |-3| = 3.

# Types of Number

### Natural Numbers

The natural numbers are the numbers {1, 2, 3, 4, 5, ...}.  This is another name for the positive integers. (Although in this course we will define the smallest natural number to be 1, please note that the natural numbers are sometimes considered to include 0.)

### Whole Numbers

The whole numbers are the numbers {0, 1, 2, 3, 4, ...}.  This is another name for the non-negative integers. (Although in this course we will define the smallest whole number to be 0, please note that the whole numbers are sometimes considered to begin at 1.)

### Integers

The integers consist of the positive integers, negative integers {-1,-2,-3,-4,...} and 0. That is, the integers is the set {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}.

### Negative Integers

The negative integers is the set {-1, -2, -3, -4, \ldots}.

### Non-negative Integers

The non-negative integers is the set of positive integers and zero, also known in this course as the whole numbers.

### Terminating Decimals

Decimal numbers that do not go on forever are called terminating decimals. For example 0.3 is a terminating decimal but \frac{1}{3} = 0.\overline{3} is not as the 3 is repeated forever. The decimal representation of \sqrt(2) is not terminating either. We might approximate it to a terminating decimal, but however many decimal places we use, it will never be exactly \sqrt(2).

### Repeating Decimals

Decimals which do not terminate but repeat the same digit or sequence of digits over and over again are called repeating decimals. An overbar over the repeating digit or sequence of digits is used, for example, \frac{1}{6} = 0.1\overline{6} and \frac{2}{7} = 0.\overline{285714}.

### Rounding

When we want to approximately express a decimal which does not terminate as a terminating decimal, we round to a certain number of decimal places. For example \frac{1}{3} \approx 0.333, \frac{1}{6} \approx 0.167, and \sqrt(2) \approx 1.412 all rounded to 3 decimal places.

### Rational Numbers

Rational numbers are numbers which can be written as a fraction where the numerator (top) and denominator (bottom) are both integers, and the denominator (bottom) is not 0. For example, 2/3 is a rational number since 2 and 3 are both integers, and 3\ne0. Note that the integers are rational numbers since they can be written as a fraction with denominator (bottom) 1. In decimal form they are represented by either terminating or repeating decimals. For example, \frac{3}{10} = 0.3, \frac{1}{3} = 0.\overline{3} and \frac{2}{7} = 0.\overline{285714}.

### Irrational Numbers

Irrational numbers are real numbers which are not rational! They can not be written as a fraction with integer numerator (top) and denominator (bottom), and denominator (bottom) which is not zero. They can not be written as terminating decimals. They can not be written as repeating decimals. Examples of irrational numbers are \pi, \sqrt(2). Note that if you multiply or divide irrational numbers by rational numbers, you get an irrational number. If you add or subtract a rational and irrational number, you get an irrational number. However, if you multiply irrational numbers together you may get a rational number eg \sqrt(2)\cdot\sqrt(2) = 2.

### Real Numbers

The real numbers consist of all rational and irrational numbers. The numbers which are not real are called imaginary numbers. (To get the imaginary numbers we have to define the square root of -1 which we call i. Weird, huh? - This is introduced later in the course.)

### Set

A set is a collection of objects, for example, {1,2,3} is the set of numbers 1, 2 and 3.

### Subset

If all the elements of one set are contained in another set, then the first set is a subset of the second set. (Formally, A is a subset of B is every element in A is contained in B.) For example,
{1,2} is a subset of {1,2,3}. The empty set is a subset of all other sets as it has no elements.

# Algebra Terminology

### Variables

A variable is a symbol, generally a letter, that serves as a placeholder for a number.  The numeric value of a variable may change, depending on the context in which it is used.  For example, in the expression 2x+3, x can take on any value.  If we decide to let x equal 4, for example, then we can get a value for the expression: 2(4)+3=11.
In an equation like 2x+3=7, however, x represents a certain number whose value we just don't know yet.  If we modify the equation to get x by itself on one side, then we can find out that value: x=2.

### Constants

A constant is a number whose value is fixed.  Examples of constants include 3, -108, \sqrt 2, and -\frac {13}{2}, but sometimes constants are represented by letters or other symbols.  For instance, \pi and e are both constants.

### Algebraic Expression

An algebraic expression combines constants and variables using addition, subtraction, multiplication, and division.

### Variable Terms

A variable term is a term whose value is not fixed, due to the presence of a variable in it.  In other words, a variable term will have one or more variable factors.  In the expression -5xy^2+1000, -5xy^2 is a variable term.

### Constant Term

A constant term is a term whose value is fixed, since it does not have any variables in it (i.e. it does not have any variable factors).  In the expression -5xy^2+1000, 1000 is a constant term.

### Coefficient

A coefficient is the numerical factor of a variable term.  For instance, for the term -5xy^2, the coefficient is -5, and for the term x, the coefficient is 1.

### Evaluate

Evaluating an algebraic expression means substituting numeric values for each of the variables in the expression.  If we have the expression 3x-2, and we let x=5, then the value of the expression will be 3(5)-2, or 13.

### Numerator

In a fraction, something written in the form \frac {a}{b}, the number or expression that makes up the top of the fraction, a,is called the numerator.

### Denominator

In a fraction, something written in the form \frac {a}{b}, the number or expression that makes up the bottom of the fraction, b, is called the denominator.

### Factors

The factors of a term or expression are numbers, variables, or combinations of numbers and variables that, when multiplied together, produce that term or expression.  For example, 2 and 3 are factors of 6, and -1, 5, x, and y are factors of -5xy

### Prime Number

A prime number is a positive integer greater than 1 that only has two factors, which are itself and 1.  Examples of prime numbers include 2, 3, 5, 7, 11, and 17. Note that 1 is not a prime number.

### Composite Number

A composite number can be written as the product of two or more prime numbers.  Examples of composite numbers include 4, 6, 8, 9, 10, and 12.

### Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 is a prime number or can be written as the product of prime numbers in precisely one way (the prime factorization), as long as we don't take the order of the factors into account.  For example, the prime factorization of 30 is 2\cdot3\cdot5.

### Equation

An equation is a mathematical statement of the form A=B, meaning that the two expressions on either side of the equals sign, A and B in this case, have the same numeric value.  For instance, the equation 2x-3=5 means that the expression 2x-3 is equal to 5