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