My background on this topic is that I’ve taught intermediate algebra for over 20 years.
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
We assume the empty set exists and call this 0. We define 1 to be the set containing 0. So 1 = {0}. We define 2 to be the set containing 0 and 1. So 2 = {0, 1}.
Let’s look at this set: {a, b}. I know this set has size 2 and not 1 because I can map {a, b} to {0, 1} in a one-to-fashion. I can’t map {a, b} to {0} in a one-to-one fashion. We say any set has size 2 if it can be mapped to {0, 1} in a one-to-one fashion.
First define natural numbers as sysops did (or you can use peanos axioms).
Then add the negative numbers (I actuallly don't remember how this is done, ig it's usually hand waved as trivial). The negative and natural numbers together make up the integers.
The rationals are introduced as a pair of numbers (a, b) where a is an integer and b is a positive integer. (a, b) is considered the same rational as (c, d) if a * d = b * c.
The reals are finally introduced as sets of rationals with a certain property, namely that if p is in S, then all smaller rationals must also be in S. Edit: there are a few more properties, see https://en.wikipedia.org/wiki/Dedekind_cut
(Real numbers are my favorite construction because I was enamored with Cantor's diagonal argument when I first learned it. It's quite clever and hints at the magic mathematicians are capable of. Although most mathematicians (algebra peeps) seem to like the classic "root 2 is irrational" proof more.)
This vsauce video is actually a really accurate (and entertaining!) introduction to set theory: https://youtu.be/s86-Z-CbaHA
So it's like learning data structures by coding in assembly, which is what Donald Knuth thinks is the right thing to do anyway, but some other teachers would disagree. But if you want to see some high level construction, you could look to eg. Tarski's synthetic construction of reals
https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_t...
Which doesn't build reals using other theories as building blocks; real numbers are real numbers.
Anyway, from the analytic constructions of reals, I'm most partial to
https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...
Which uses integers rather than sets as the building block, and is simpler than many constructions. And, of course integers themselves can be constructed out of sets, but they can be constructed out of lambda calculus terms as well https://en.wikipedia.org/wiki/Lambda_calculus#Encoding_datat... among many other constructions - but when we finally define integers, we can abstract away the implementation details (and that's really the crux of the question!)
Anyway there's a discussion of analytic vs synthetic mathematics in this post, https://golem.ph.utexas.edu/category/2015/02/introduction_to... (it seems that part 2 wasn't written unfortunately)
We use the label 'number' to refer to a broad swathe of mathematical objects, objects that are different but also so similar they often appear interchangeable (for example counting numbers and fractions).
In the formal mathematical sense, a specific type of number is a group of objects which have been defined to have specific properties. I'm going to leave object and property as defined in the usual sense, I think most people have a good idea about what those are and not sure I can add anything to them.
There are no rules as to what properties you are allowed to give to objects, nor what group of objects you want to include, but generally if you are learning about some specific thing it's because people find them useful or interesting; the definitions we have for different types of numbers are the ones we have found useful or interesting.
Remember that the different types of number appear very similar. It is common to build a 'hierarchy' of differnt types of numbers, where we start with a simple type of number and then add new properties and objects when we find limitations we don't want.
The first type of number in this hierarchy are the natural numbers (sometimes called counting numbers). The most common properties defined for these today are called the Peano axioms [0]. There are quite a few of them, and the history of how we came to the formalisation is very interesting (a lot of it is about avoiding inconsistencies/contradictions) but the key ideas are:
- there is a natural number called 0
- every natural number has a successor, which is also a natural number - we can write S(n) is the successor of n
Most of the other axioms define what it means for two natural numbers to be equal (=).
Just having these objects isn't particularly useful, we typically want to do things like add, multiply, and compare numbers. To do that we include some operations: addition (+), multiplication (*), and total ordering(<=).
These are defined as, taking a, b, c as natural numbers:
- a + 0 = a
- a + S(b) = S(a + b) (this is recursive, so if we define 1 as 1:=S(0) then we have 1+1 = 1+S(0) = S(1+0) = S(1))
- a * 0 = 0
- a * S(b) = a + (a * b)
- a <= b if (and only if) there exists some c such that a + c = b
Importantly, using these definitions, we can say that the natural numbers are closed under addition and multiplication; whenever you add or multiply two natural numbers together you get another natural number.
To continue building the hierarchy we notice that there are operations we would like to do but are not possible for every natural number (please note I am skipping over the formalisations from hereon and talking about the motivation for different types of numbers).
We notice that if we can add two numbers together we should be able to subtract them again. If a + b = c, then c - b = a. However (for example) 0 - 1 is not a natural number. So we extend the natural numbers to the integers, such that the integers are closed under subtraction.
If we can multiply it makes sense to try and divide, but 2 / 3 is not an integer so we extend integers to the rationals (ratios of integers) which are closed under division. We add an object called 2/3 so that now when we can say 2 / 3 = 2/3.
The next step in the hierarchy is a bit more complex. We notice that we can define a subset of rational numbers that all meet a certain criteria, for example all rational numbers that are less than 2. We call 2 an upper bound of that subset. Notice that 2 is a rational number, and that there are no rational numbers smaller than 2 that are also an upper bound of our subset - 2 is the least upper bound. Define a new subset, where we say a rational number x is in the subset if x * x < 2. We can easily see that 2 is an upper bound for this set, but so is the rational number 1.5, and 1.42, and 1.415. In fact, there is no least upper bound for this set that is a rational number. We extend the rational numbers to include a least upper bound for every subset of rationals, and we call this the real numbers. The real numbers have a lot of nice properties, most notably they are complete under the normal ordering, which essentially means that there are no gaps.
The reals don't have everything though! We notice that we can create polynomial equations, like x * x - 1 = 0 and that sometimes these can be solved (in this case x = 1 or x = -1 solves the equation) and in other cases they can't. For example, there are no real numbers that are the solution to the equation x * x + 1 = 0. We can extend the real numbers to the complex numbers by adding a new object called i, which has the property i * i = -1. A complex number has the form a + b * i, where a and b are real numbers. The complex numbers are called algebraically closed, and there is a really nice result that shows that all polynomials have solutions in the complex numbers.
The hierarchy actually keeps going, but hopefully you can see that numbers are just objects with properties that behave in useful and interesting ways under different operations. The formal definitions we have today have been refined over a long period of time to avoid contradictions and other issues, but there is nothing stopping you from making up your own numbers with their own properties. If they are useful or interesting other people will probably use them too!
If you don't know what a rational number is, it's the equivalence class of every pair of integers that can be simplified to the same fraction. For example, (4,6) and (2,3) are both rational numbers, and in fact are the same rational number: two thirds.
If you don't know what an integer is, it's the natural numbers, but with negative numbers.
If you don't know what a natural number is, it's either zero, or a number that follows a natural number. For example one is the number that follows the natural number zero, and two is the number that follows the natural number that is the natural number that follows zero.
The integers are the equivalence classes of differences of natural numbers, while the natural numbers are the equivalence classes of finite sets having the same number of elements (i.e. which may have a bijection between themselves), including the empty set.
How do you define the _number_ of elements of a finite set without defining natural numbers first?
You just need to be able to show an one-to-one correspondence between the elements of the two sets. If an one-to-one correspondence cannot exist, then the sets have different numbers of elements.
This relationship divides then the sets in equivalence classes. If you choose a representative of each equivalence class that you use to compare to other sets to see if they have the same number of elements and you give a name to each of those representatives, you have defined the so-called natural numbers.
This is actually how the numbers originated, for humans and for many other animals.
Nobody conceived a system of axioms and then thought about what could satisfy them. That came much later and is useful only for establishing which are the essential properties of some mathematical objects. Most of the definitions of various mathematical objects as equivalence classes correspond to their real historical origin, because recognizing that some things are equivalent according to some criterion is how abstract concepts are created based on concrete things.
When you see a red apple and a red rose, you understand that they have a common property, being red, and then you name this property "red" and you can recognize the same property in other objects.
When you see 5 sheep and 5 crows, you understand that these groups have a common property, having 5 members, and the same property characterizes the set of fingers of your hand. You name this property "five" and when you see another group of things you can compare it with the set of fingers of your hand to see if it also has 5 members.
[1] https://www.amazon.com/Numbers-Graduate-Mathematics-Heinz-Di...
However, I absolutely detest it when teachers just "sweep it under the rug", when they pretend that they just provided a definition when they evidently did not.
Like the commenter you replied to, this sort of stuff genuinely threw me off in high school and made me feel like I didn't understand mathematics.
I'm fine with having an intuition for sets, but I think reals really should be defined properly. At least, R should not be confused with the algebraic closure of Q.
But I must admit I haven’t read the whole post.
If the students haven't yet encountered complex numbers, infinitesimals, infinities etc. then it's perfectly reasonable to say that all numbers are assumed to be real (as follows strictly from the definition in the book).
But by definition there are no real numbers that are neither rational or irrational.
The intention of my post was to point out the complexity of not brain washing students at a low level. Your comments have enhanced my point by bring up considerations I didn’t want to get into!
K-12 was, of course, invented by the Germans in order to create good little factory workers that would get up early and work all day and not complain too much. The fact we still use the word Kindergarten is a nod to this origin story.
They weren't at all interested in the students gaining any "understanding" and most certainly not in them "winning" in any sense of the word.
I suppose you prefer things randomly defined by Euclid? Just kidding... kinda. Seriously though, randomly defining things and then working through the consequences of that definition is a totally valid way to do math. Those random definitions are called postulates.
But still most teachers give us the same standard set of axions. Why? What would happen if they dropped some of them or replaced them with others?
However almost any construction of the real numbers is challenging to give a simple explanation for. Even leading 19th century mathematicians didn’t truly understand the real numbers until Cantor.