As the name implies, the *equal sign* refers to things that are the same. In what sense some things are the same is a philosophical question and initially we are bound to this: What equality points to must be understood by the context in which the sign is used. With this understanding of = we can study some basic properties of our numbers and then later return to more precise meanings of the sign.

Common ways of expressing = is

- “equals”
- “is the same as”

There are many ways a number can be defined, however, in this book we shall stick to two ways of interpreting a number; a number as an *amount* and a number as a *placement on a line*. All representations of numbers rely on the understanding of 0 and 1.

Talking about an amount, the number 0 is connected to “nothing”. A figure showing nothing will therefore equal 0: [ =0 ] 1 we’ll draw like a box:

In this way, other numbers are defined by how many one-boxes (ones/units) we have:

When placing numbers on a line, 0 is our starting point:

Other numbers are now defined by how many one-lengths (ones/units) we are away from 0:

Numbers which are a whole amount of ones are called *positive integers*. The positive integers are
[ 1, 2, 3, 4, 5 \text{ and so on.} ]
Positive integers are also called *natural numbers*.

Some authors also include 0 in the definition of natural numbers. This is in some cases beneficial, in others not.

Our numbers consist of the *digits* $ 0, 1, 2 , 3, 4, 5, 6, 7, 8 $ and $ 9 $ along with their *positions*. The digits and their positions define the *value* of numbers.

Let’s, as an example, write the number *fourteen* by our digits.

We can now make a group of 10 ones, then we also have 4 ones. By this, we write fourteen as [ \text{fourteen}=14 ]

Sometimes we don’t have a whole amount of ones, and this brings about the need to divide “ones” into smaller pieces. Let’s start off by drawing a one:

Now we divide our one into 10 smaller pieces:

Since we have divided 1 into 10 pieces, we name one such piece *a tenth*:

We indicate tenths by using the *decimal mark*: .

In a lot of countries, a comma is used in place of the period for the decimal mark.

- 3,5 (other)
- 3.5 (English)

The value of a number is given by the digits $ 0, 1, 2, 3, 4, 5, 6, 7, 8 $ and $ 9 $ and their position. In respect to the digit indicating ones,

- digits to the left indicate amounts of tens, hundreds, thousands, etc.
- digits to the left indicate amounts of tenths, hundredths, thousandths, etc.

Integers with 0, 2, 4, 6, or 8 on the ones place are called *even numbers*.
Integers with 1, 3, 5, 7, or 9 on the ones place are called *odd numbers*.

Two number lines can be put together to form a *coordinate system*. In that case, we place one number line *horizontally* and one *vertically*. A position in a coordinate system is called a *point*.

In fact, there are many types of coordinate systems, but we’ll use the *cartesian coordinate system*. It is named after the French mathematician and philosopher, RenĂ© Descartes.

A point is written as two numbers inside a bracket. We shall call these two numbers the *first coordinate* and the *second coordinate*.

- The first coordinate tells how many units to move along the horizontal axis.
- The second coordinate tells how many units to move along the vertical axis.

In the figure, the points $ (2, 3) $, $ (5, 1) $, and $ (0, 0) $ are shown. The point where the axes intersect, $ (0, 0) $, is called *origo*.