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The concept of number

Natural numbers and integers

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.

This Babylonian clay tablet shows arithmetic using positive integers. Babylonians also used fractions and square roots, but no negative numbers or zero.

Rational numbers

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.

All integers are rational numbers.

Real numbers

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 set of the real numbers contains the rational numbers, the integers and the positive integers.

Ordering numbers

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

The number line

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.

A section of the number line. The number line extends forever in both directions.

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 $0$.
Putting $-1.32$, $0.45$ and $37/14\simeq 2.64$ on the number line