February 25, 2021

Chapters

## Real number

While it may seem like we only have and use one type of number in our daily lives, there are actually many **different types** of numbers. Take a look at some of the numbers in the image below.

**real numbers**. Real numbers are any of the following:

- Whole numbers
- Integers
- Rational numbers
- Irrational numbers

To understand what these different **types** of numbers are, take a look at the table below.

Examples | Definition | |

Whole | 0, 5, 65, 100, 788 | Positive numbers that are not fractions or decimals |

Integer | -5, -75, -340, -800, 55, 90 | Positive and negative numbers that aren’t fractions or decimals |

Rational | , 0.67 | A number that is made from the division of two integers |

Irrational | , | A number that can’t be written as a fraction or decimal |

## Number Line

In order to better understand the differences between these different types of numbers, it can be helpful to use a **number line**. A number line can help you identify what types of numbers you’re dealing with. Take the following set of numbers as an example, which give 5 numbers. Let’s see if we can figure out what **kind** of number they are.

Number | Type | |

1 | 3.3 | ? |

2 | 5 | ? |

3 | -6 | ? |

4 | -5.3 | ? |

5 | 1.4142135623….. | ? |

A number line is a **plot** of numbers on a straight line. We can use a number line to help us decide what kind of numbers we have. Take the following as an **example.**

**positive**and

**negative**numbers, rational and irrational numbers and more. Let’s plot the numbers given in the table above.

**type**of number each of the 5 numbers are.

Number | Type | Why | |

1 | 3.3 | Rational | A number that is made by dividing 33 by 10 |

2 | 5 | Whole | A positive number that’s not a decimal |

3 | -6 | Integer | A negative number that’s not a decimal |

4 | -5.3 | Rational | A number made by dividing -53 by 10 |

5 | 1.4142135623….. | Irrational | A number that can’t be written as a fraction |

## Irrational Number

As stated in the first section, an irrational number is different from a rational number. While a rational number is any number that can be written as a decimal or a fraction, an irrational number **can’t be expressed** as a ratio of two integers. Find some examples in the table below.

In Decimal or Fraction Form | Description | |

5 10 | A ratio of 5 to 10 | |

.0066 | A ratio of 66 to 10 000 | |

1.7320508075688772... | Cannot be expressed as a ratio |

Generally, there are many **roots** that are irrational numbers. Recall that roots can be squares, cube and more. However, keep in mind that **not all** roots are irrational numbers. Take the following examples:

Result | Type | |

2 | Rational | |

3 | Rational | |

4.3267487109... | Irrational |

## Examples

In this section, we’ll go over some examples of irrational numbers. These irrational numbers are the ones you are likely to encounter in other classes, as they are very **famous.**

### Pi

Pi is the most common example of an irrational number. It is represented by the Greek letter **pi:**

**cannot**be written as a decimal or a fraction. In fact, some people have tried to calculate pi to

**hundreds**of decimal places without finding any sort of pattern. Take a look at pi below, which is cut off at the 15th number.

**close**to pie using some fractions. Take a look at the table below, which gives some examples.

Fraction | Decimal |

22/5 | 4.4 |

22/6 | 3.666667 |

22/7 | 3.142857 |

22/8 | 2.75 |

### Euler’s Number

Euler’s number is the second most **common** irrational number that you’ll encounter in math. Euler’s number is similar to pie in that it can be calculated to many decimal places but still **does not** show any pattern, or any sign of stopping.

Euler’s number is **written** as an , which can be seen below.

**natural base.**Take a look at the table below, which describes the difference between when you should use the two terms.

Description | Example | |

Euler’s | Using only | 5 * , |

Natural Base | Using it as a base for a logarithm |

When we use Euler’s number as the natural base, we can call the logarithm a **natural logarithm,** which is written as .

**same**as having the natural logarithm of x. This means that the natural base is the inverse of the natural logarithm. Take a look at the table below, which has some of the rules for natural logarithms.

Example | |

### Golden Ratio

The golden ratio is represented by the Greek letter and can be found in many different aspects of life. It can often be found in **art** and **architecture.** To find the golden ratio, let’s start with a rectangle.

**golden ratio**is present when the long part of some line, a, divided by the smaller part, b, has the same result as adding the small and long part, a + b, and dividing it by the larger part, a.

This sounds a bit complicated, but we can try with an **example.**

Result | |

1.6180341 | |

1.6180341 |

As you can see, this **approximates** the golden ratio.