# The concept of number

The natural numbers, or positive integers, are the numbers used for counting. They are the **positive whole numbers**.

$$1$$, $$2$$, $$3$$, $$\dots$$ are positive integers. $$3.2$$, $$1/4$$ and $$-1$$ are not integers.

$$\Tred{0}$$ is not a positive integer.

The integers are zero and the positive and negative whole numbers.

$$\dots$$, $$-2$$, $$-1$$, $$\Tred{0}$$, $$1$$, $$2$$, $$\dots$$ are integers. $$-1/3$$ and $$2.4$$ are not.

The symbol for the **set of the integers** is $$\mathbb{Z}$$.

The symbol for the **set of the positive integers** is $$\mathbb{Z}^+$$ or $$\mathbb{N}$$.

There are **infinitely many** positive integers and infinitely many integers.

A rational number, or fraction, is any number that can be written as the **ratio** of two integers.

$$2$$, $$-2/5$$ and $$3.1$$ are rational numbers. $$3.1$$ is a rational number because it can be written as the ratio $$31/10$$.

All integers are rational numbers. For example, $$5$$ can be written as the ratio $$5/1$$.

Examples of numbers that are not rational are $$\sqrt{2}$$, $$\pi$$, or the cube root of $$2$$.

The symbol for the **set of the rational numbers** is $$\mathbb{Q}$$.

Basic operations on rational numbers (addition, subtraction, multiplication and division) always give a rational number.

There are numbers that are **not rational**. These are called irrational numbers.

$$\sqrt{2}$$, the number whose square is $$2$$, is an irrational number.

Another irrational number is $$\pi$$, which is the area of a circle with radius $$1$$.

The collection of numbers that combines all the rational and irrational numbers is called the real numbers. The set of the real numbers is written as $$\mathbb{R}$$.

The numbers have a natural **ordering**. We can always say that a number is **greater than**, **smaller than**, or **equal to** another one.

You can use a temperature scale to say that an object is hotter or colder than another, using the same ordering as the real numbers.

You can check the ordering of two numbers $$x$$ and $$y$$ using **subtraction**.

$$x$$ is greater than $$y$$ if $$x-y$$ is **positive**.

$$x$$ is less than $$y$$ if $$x-y$$ is ** negative**.

$$10$$ is greater than both $$0$$ and $$3.5$$ but less than $$101$$.

$$-20$$ is less than $$-2$$, and $$-1$$ is less than $$5$$.

Symbol | Meaning | Example |
---|---|---|

$$$\lt$$$ | Less than | $$$-5 \, \Tred{\lt} \, 8$$$ |

$$$\le$$$ | Less than or equal to | $$$-2 \, \Tred{\le} \, -2$$$ |

$$$\gt$$$ | Greater than | $$$1.2 \, \Torange{\gt} \, 1.1$$$ |

$$$\ge$$$ | Greater than or equal to | $$$13 \, \Torange{\ge} \, -8$$$ |

Even though a negative number like $$-500$$ looks bigger than a number like $$-2$$ or $$15$$, it is always considered to be less than these numbers, mathematically.

We write $$-500 \lt -2 \lt 15$$.

All numbers can be ordered. This is represented by the number line, or real line. **Smaller numbers are on the left**. Larger numbers are on the right.

This is a method to allow you to put a number in exactly the right place on a number line.

- Write the
**integers first**, equally spaced. Use a ruler to find the distance between $$0$$ and $$1$$, for instance $$\Tblue{2}\ucm$$. This distance is called the scale. - Compute numbers in
**decimal form**, e.g. with a calculator. $$$\frac{1}{2} = \Tred{0.5},\quad \frac{37}{14} \simeq \Tred{2.64},\quad 2\sqrt{2} \simeq \Tred{2.83}\quad \pi \simeq \Tred{3.1416}.$$$ - Use a ruler to put the number at a distance from $$0$$ that is the number times the scale. So, $$\Tred{0.45}$$ will be $$\Tblue{2}\times \Tred{0.45}\ucm = 0.90 \ucm$$ away from $$0$$.