The equal sign, amounts, and number lines

The equal sign

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”

Amounts and number lines

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.

Numbers as amounts

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:

Numbers as placements on a line

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:

Positive integers

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.

What about 0?

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

Numbers, digits and value

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.

Integers larger than 10

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 ]

Decimal numbers

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

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

  • 3,5 (other)
  • 3.5 (English)

Base-10 positional notation

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.

Even and odd numbers

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.

Coordinate systems

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.