# Mathematics Glossary

Often with mathematics, common words have different meanings which can be confusing. On top of that there is all the specialised vocabulary. This page aims to cover, in everyday language as well as more formally, all the mathematical vocabulary used.

## 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.

### 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.

## Sets

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

### Empty Set

The set with no elements, { } is called the empty set, and has its own notation \emptyset.

### Element

The objects in a set are called elements. For example, 1 is an element in {1,2,3} but {1} is not.  However, { 1 } is an element of A={ { 1 }, 2, { 1,2,3 } } and 3 is not since A contains the elements { 1 }, 2 and { 1,2,3 }.

### 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.

## Number

### 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 express a decimal which does not terminate as 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.

### Positive Integers (Natural Numbers)

The Positive Integers, also called the Natural Numbers, are the numbers {1, 2, 3, 4, 5, ...}.
(Note that Natural Numbers sometimes include 0 so make sure you are always aware of which definition is being used in any books and webpages you're reading and in any courses you are taking.)

### 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.

### Integers

The Integers consists of the Positive Integers, Negative Integers {-1,-2,-3,-4,\ldots} and 0. That is, the Integers is the set {\ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots}.

### Real Numbers

The real numbers consist of all fractions, whole numbers and decimals. 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?)

### 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}0.

### 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 multiple 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.