Chapters

## Real Number Definition

If you’re interested in operations involving real numbers, it is important to understand what a real number is. Real numbers are **defined** as all numbers that are:

**Whole**numbers and**Integers****Rational**numbers**Irrational**numbers

Take a look at some of the **definitions** below.

Whole Number | Integer | |

Definition | Positive number that doesn’t have a decimal or fraction | Positive or negative whole numbers |

Example | 0, 2, 30, 160 | -4, -90, 0, 3, 88 |

Now that you know the **difference** between whole numbers and integers, let’s compare rational and irrational numbers.

Rational | Irrational | |

Definition | Positive or negative number that has a decimal or fraction | Positive or negative numbers that can’t be written as a decimal or fraction |

Example | 0.55, -0.88, | , , |

## Basic Maths Operations

Basic math **operations** involve the following:

- Addition
- Subtraction
- Multiplication
- Division

Addition is when two numbers are combined as a **new sum**.

**away**from another.

**times**are specified.

**divided**into groups of another.

## Operations with Positive and Negative Numbers

There are a couple of **rules** you should remember with respect to signs. Signs tell us whether a number is positive (+) or negative (-).

When you perform addition or subtraction, signs can change depending on the **absolute value** of the numbers. Take for example two scenarios:

Rule | Example | Absolute Value |

A | 10 + (-5) = 5 | 10 > |-5| -> 10 > 5 |

B | 10 + (-11) = -1 | 10 < |-11| -> 10 < 11 |

Multiplication and division **rules** are much easier.

Rule | Example |

A | 4 x 4 = 16 |

B | 3 x -2 = -6 |

C | 4 2 = 2 |

D | 15 -3 = -5 |

## Order of Operations

When we have a combination of operations in a single expression, it can be confusing to understand what operation we should do first. This is where the acronym **PEMDAS** comes in handy.

Description | Symbol | |

P | Parenthesis | () |

E | Exponent | |

M | Multiplication | x |

D | Division | |

A | Addition | + |

S | Subtraction | - |

In order to solve this, let’s take a look at an example:

The steps are below.

Step | Operation | Why | Remaining |

1 | (5 + 3) = 8 | P | |

2 | 2 * 2 = 4 | M | |

3 | 8 4 = 2 | D | |

4 | 3 + 1 = 4 | A | |

5 | 2 - 4 = -2 | S |

## Commutative Law

The commutative law states that order **does not matter** in an operation. Note that this only applies to addition and multiplication.

### Addition

In addition, the order in which you **add** each number does not matter. This can be written as the following:

### Multiplication

In multiplication, the order in which you **multiply** each number does not matter. This can be written as the following:

## Associative Law

The associative law states that how you group numbers that have the **same operation** does not matter. Like the commutative law, this does not apply to subtraction and division. The associative law only applies when we have the same operation between multiple numbers.

### Addition

In addition, the order in which you **add** each number does not matter no matter how you **group** them. Grouping numbers usually comes in the form of putting parentheses around them. This can be written as the following:

### Multiplication

In multiplication, the order in which you **multiply** each number does not matter no matter how you **group** them. This can be written as the following:

## Distributive Law

The distributive law can be thought of as a **combination** of the other laws. It can be applied to:

- Multiplication
- Addition
- Subtraction

In order to understand the distributive law, we can look at the following image:

**summarizes**the steps:

Step 1 | You start with an operation between some numbers, here its addition | a + b |

Step 2 | You multiply this operation with a number | (a+b) x c |

Step 3 | You ‘distribute’ the c to every number inside the parenthesis | Distribute c to a and b |

Step 4 | Add the result | ac + bc |

As an **example,** take a look at the image below.

## Multiplicative Identity

A multiplicative identity sounds complicated, but is really just a number that, when multiplied, leaves the **original number** as it is. For rational numbers, the multiplicative identity is 1.

As you can see, when you multiply any rational number by 1, it is simply **itself.** See an example below.

## Multiplicative Inverse

A multiplicative inverse is the reciprocal of a number. To understand what a **reciprocal** is, take the image below as an example.

As you can see, a reciprocal can be thought of the inverse of a number. Since 3 can be written as , it’s inverse is simply . Notice that when a number and its inverse are multiplied together, they equal to **one.** Take a look at some examples below.

Number | Reciprocal | Result |

5 | 5 * = 1 | |

* = 1 | ||

4 | 4 * = 1 |

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