What is a number?

Question: What are the different types of numbers?

Our goal today is to answer the question “What is a number?” More specifically, “number” in this case refers to “real number”, since this includes pretty much all the numbers that appear in secondary school mathematics — natural numbers, integers, fractions, and irrational numbers.

How do mathematicians define real numbers?

Mathematicians define real numbers by outlining exactly the properties that the real numbers must satisfy, in the form of axioms. Here is an excerpt from a text by Dana Ernst at Northern Arizona University about the foundations of mathematics:

[The] axioms for the real numbers fall into three categories:

1. Field Axioms: These axioms provide the essential properties of arithmetic involving addition and subtraction.
2. Order Axioms: These axioms provide the necessary properties of inequalities.
3. Completeness Axiom: This axiom ensures that the familiar number line that we use to model the real numbers does not have any holes in it.

Dana Ernst, An Introduction to Proof Via Inquiry-Based Learning

We are definitely not going to introduce numbers to our students in this way! However, the axioms are just a way to make precise some intuitions that we and our students have developed from an early age. We will base our definition of numbers on this “ingrained knowledge” instead.

What are we starting from?

We start with the following three building blocks:

1. Counting 1, 2, 3, 4, … This is one of the very first mathematical activities we learn – we learn it at an early age, and it sticks.
2. Basic ideas about space. We live in physical world, and our intuitions about physical object and how they behave are hardwired into our brains. For example:
   • Two objects can’t occupy the same place at the same time.
   • If I pick an object up and move it, the size doesn’t change.
3. Points and Lines. Even though we can’t find actual physical examples of these in the world (they come from the more abstract world of geometry), we can still use our basic ideas about space to reason about them.
   • If two points occupy the sample place, they are the same point.
   • I can make a line segment by identifying two points on a line.
   • I can slide a line segment along the line without changing its length.
   • I can divide a line segment up into equal pieces.
   • A line doesn’t have any holes in it.

From these beginnings, we are going to define the concept “number” (in particular, real number). Once we have done that, we’ll be able to talk about different types of numbers – natural numbers, integers, fractions, etc.

Before we get to numbers, we start with a simple, essential picture – a line, and two points. (Even though we haven’t defined “number” yet, we still use the word number in the name of this picture – it’s called a Number Line).

Since we usually choose number lines to be horizontal, we’ll freely talk about the left direction and right direction. The point 1 is to the right of the point 0.

NOTE: we have just defined explicitly what a number is, namely, a point on the number line. This may not seem like much until you recall that, in TSM, the word number is bandied about repeatedly and yet nobody can say precisely what a number is.

NOTE: If this is our definition of number — a number is a point on a line — then when we want to talk about doing various things to numbers, like adding, subtracting, multiplying, dividing them, we better give definitions of those concepts in terms of this number line idea. This may take some getting used to!

Let’s start by just identifying on the line some common, familiar numbers.

Question: How do we place the number 5 on the number line?

Question: How do we place the number -7 on the number line?

We can use the unit distance (that is, the length of the interval $[0,1]$) to build the entire sequence of whole numbers on the number line.

At its heart, this process relies on:

• sliding line segments (so that one endpoint of the new segment occupies the same point as the the other endpoint of the original segment)
• repeating this process, and counting how many times we repeat it.

If we repeat this process to the left as well, we get the integers.

We call the resulting collection of points an infinite sequence of equidistant points, which gives us the whole numbers (to the right of 0) and the integers (along the whole line).

If a number is a point on the line, how do we talk about size, or distance? These are properties that apply to intervals $[a,b]$

Definition. If $d$ is a number, then we say an interval $[a,b]$ has length $d$ if, when we slide the interval $[a,b]$ so that $a$ coincides with $0$, we then have $b$ coincides with $d$.

Question: If a number is a point on a line, how would we define the addition of two numbers? What’s a visual way to describe addition?

How do we place the fractions on the number line?

NOTE: We will assume that we can divide a given segment into any number of segments of equal length.

When we think of “the whole” as in “parts of the whole”, we always mean the length of the unit segment $[0,1]$. NOTE: The whole is not the unit segment [0,1], but the LENGTH of the unit segment [0,1].

How do we place the fractions with denominator 3? e.g. $\frac{2}{3}, \frac{5}{3}, \frac{6}{3}$ etc.

• Divide the unit segment $[0,1]$ into three equal segments.
• Divide also each of $[1,2]$, $[2,3]$, $[3,4] \cdots$ also into three equal segments.
• These division points, together with the whole number themselves, form an infinite sequence of equidistant points, called the sequence of thirds.
• We call the first of these short segments, the segment with left endpoint $0$, the standard representation of $\frac{1}{3}$.
• The number $\frac{1}{3}$ is what we call the right endpoint of this segment.
• What is the number $\frac{5}{3}$?
• In general, what is the fraction $\frac{m}{3}$, where $m$ is a whole number?

QUESTION: How do we locate the fraction $\frac{7}{4}$ on the number line?

What can we do with this definition?

Theorem: For any whole numbers $k,n$ with $n\neq 0$, $\frac{kn}{k}=n$.

Challenge: Explain why this theorem works, in terms of the definition of fractions.

NOTE: The need for precision about what the unit is cannot be overstated! (it’s impossible to say which point is what fraction until the we fix 0 and 1).

Some different concepts of a fraction $\frac{m}{n}$

• Multiplication: $\frac{m}{n}$ means $m \times \frac{1}{n}$.
  • Take one (single unit), divide it into $n$-many equal pieces. 
  • Then count $m$ of these pieces.
  • This results in the number $\frac{m}{n}$
    • Divide 2 bananas among 3 people
    • Could do this with candy bars?
• Division.  $\frac{m}{n}$ means $m \div n$.
  • Take the line segment $[0,m]$.
  • Chop it up into $n$ many equal pieces.
  • The fraction $\frac{m}{n}$ is the size of one piece (you can find it on the number line by finding it the right endpoint of the first piece)
    • Divide 3 lbs sugar among 5 people
• Ratio. Top number forms a certain portion, or percentage, of bottom number.  When we take “fraction *of* number”, we are asking for the same portion, or percentage, of the number.
  • A man’s son is 2/5 of the age of the man.  If the man is 35, how old is the son?
  • A man’s son is 2/5 of the age of the man. If the son is 18, how old is the father.

Question: Which of these concepts most closely resembles our definition of fractions?