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
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
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.
The value of a number is given by the digits
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, René Descartes.
A point is written as two numbers inside a bracket. We shall call these two numbers the first coordinate and the second coordinate.
In the figure, the points