beero1000
Veteran Member
Here is my overview of the standard number systems of mathematics. For those interested, you should be able to find the details in any book on undergraduate analysis. The underlying theme in the development of number systems is 'completion' - the notion that we are missing numbers to describe certain situations and want to find a number system that includes them.
We start with the integers Z = {...,-2,-1,0,1,2,...} (Z for 'Zahlen', which is German for 'numbers'). I'm assuming you all already know the arithmetic properties of the integers - specifically, addition, subtraction, and multiplication - as of yet, division is not well defined on integers because the result might not be an integer (a number system with these operations is called a 'ring'). All other arithmetic properties of the rationals, reals, and complex numbers will ultimately derive from the analogous properties in the integers. Like Kronecker said "God made the integers; all else is the work of man". The positive integers are sometimes called the 'natural numbers' and are denoted by N = {1,2, ...}.
Since division is not well-defined in the integers, we want to define a bigger number system that lets us perform all four operations (except dividing by zero) while remaining in the number system (this is called a 'field'). The smallest field that includes the integers is the rational numbers. One way to define rational numbers Q (Q for quotient) is as an ordered pair (a,b) where a is an integer and b is a natural number (specifically, b is not zero). We have addition (a,b) + (c,d) = (ad + bc,bd) and multiplication (a,b)*(c,d) = (ac,bd), and the standard representation of (a,b) is a/b. Addition gives rise to well-defined subtraction, and multiplication gives rise to well-defined division (by non-zero rationals, where (a,b) is zero if a = 0). Note that each of the four arithmetic operations in the rationals is entirely defined by operations for integers.
Suspiciously, the rationals don't contain the integers explicitly. Remember that by our definition an integer is a single number, while the rationals are pairs of numbers (a,b), and they aren't the same. However, the rationals contain numbers that act very much like the integers. If we look at (a,1), then (a,1) + (b,1) = (a+b,1) and (a,1)*(b,1) = (ab,1). We say that the integers are isomorphic to a sub-ring of the rationals, and they are essentially integers in all but name, which is why treating 3/1 as 3 is not a problem even though we are actually switching between number systems.
Even more suspiciously, we note that there are multiple ways of writing the same rational. In particular (a,b) = (c,d) if and only if (a,b) - (c,d) = 0 (where 0 is the analog of the zero integer in the rationals, i.e. (0,b) for some b in N). If you work out the subtraction you get that (a,b) = (c,d) if and only if ad - bc = 0 (this operation now lives entirely in the integers). That means that there many equivalent ways of writing the same rational number, so 6/2, 3/1, and 45/15 all represent the same rational number. If we look at all rationals equal to (a,b) we get what is called the 'equivalence class of (a,b)' in the rationals. Usually, we don't care which representative of the class is used (in arithmetic 6/2 and 45/15 are really the same for all intents and purposes), but if we want to pick the 'nicest' version, we'll pick the rational (c,d) with smallest d. This always exists, and is unique, and is (a,b) in 'simplest form'. Keep in mind this fact that people are perfectly OK with these numbers having multiple representations when we get to the 0.999... = 1 fiasco.
So we have the rationals, but the rationals are still missing numbers that we'd like to talk about. In particular, Greek mathematicians noted that the diagonal of a square with side length 1 cannot be a rational number. Since rational numbers contain the integers, it can't be an integer either, so we don't have a description of the number that is the length of that line segment. So we want to 'complete' the rationals in this new sense. Keep in mind that we can't just stick in the square root of 2 into our number system, because we would mess up the arithmetic operations (i.e. if you include \(\sqrt{2}\) you need to include \(1 + \sqrt{2}\) and \(\frac{3}{2}\sqrt{2}\), etc in order to make sure that we remain in our new field.
So, just like before, define a new kind of number. We'll say that this new kind of number is an ordered pair (a,b) of rational numbers, and give it the arithmetic representation \(a + b\sqrt{2}\). This is a new field of numbers that now includes the number we were missing (it is usually called Q(\(\sqrt{2}\)) and is known as a 'field extension'). But for us, it still isn't good enough, as it is still missing \(\sqrt{3},\sqrt{5},\sqrt{\sqrt{2}}\), etc, etc. If we continue to add all those numbers, we get what is called 'the algebraic closure of Q' and is also known as the set of algebraic numbers (i.e. numbers that are solutions to polynomial equations with integer coefficients).
Alas, even the algebraic numbers have gaps. For example the number e is not algebraic, and neither is \(\pi\). So we seem to be stuck again. We can continue extending our numbers, but there's no reason to believe that this procedure will actually complete our field. So we need a different approach. The idea is to look at limits.
We can define a sequence of numbers as a function from the natural numbers. In other words, the 'list' of numbers a1, a2, a3, ... is just shorthand for a function f that takes entries in N and returns those numbers, specifically f(1) = a1, f(2) = a2, ..., f
= an, ..., etc. As a shorthand for the shorthand, we write {a}n to stand for the sequence a1, a2, a3, ...
We define a 'metric' (distance function) by taking the distance between numbers x and y to be the absolute value of their difference, so d(x,y) = |x - y|. A set with a well-defined metric is called a metric space, and one particular property of a metric is that the distance between two points is 0 if and only they are the same point. Once we have a metric space, we can start talking about convergence.
We say a sequence {a}n converges to a value A if all ai with big enough index are arbitrarily close to A. The formal statement is that for any positive epsilon \(\epsilon > 0\), there exists an N such that \(|a_i-A| < \epsilon\) for every i > N. We say that the limit of the sequence is A, or \(\lim_{n\to\infty} a_i = A\). We say that a sequence is Cauchy if all entries with large enough index are arbitrarily close to each other. The formal statement of that is that for any positive epsilon \(\epsilon > 0\), there exists an N such that \(|a_n-a_m| < \epsilon\) for every n,m > N. Keep in mind that even though the sequences have varying indices, they are fixed mathematical objects (each is just a function). The limits, even though they seem like varying objects are just numbers, again, fixed mathematical objects.
Now, the hope is that we can find a field such that every Cauchy sequence converges to a number in the field. The idea being that in Q, the sequence 1, 1.4, 1.41, 1.414, ... is Cauchy, but does not converge to a rational number. Similarly, in the algebraic numbers 3, 3.1, 3.14, 3.141,... is Cauchy, but does not converge to an algebraic number.
So here is the completion idea - take the collection of all Cauchy sequences of Q. Define those to be the real numbers. It's almost too naive an idea to work, but it does work. Now, the sequence 3, 3.1, 3.14, .... is defined to be the real number \(\pi\). We can correctly define addition and multiplication of sequences index by index, and therefore we can do subtraction and division (being careful about zeros) so what we get is a field. Now, we get the nice property that every Cauchy sequence of real numbers converges to a real number. Two Cauchy sequences are equal if their difference converges to 0. Every real number has at least one decimal expansion - just take your epsilons small enough to fix each digit, and every decimal expansion corresponds to a real number. But, we need to be careful. Just like the rationals, that means that multiple different Cauchy sequences converge to the same real number - in particular the sequence 0, 0.9, 0.99, ... and the sequence 1,1,1,1,... have differences 1, .1, .01, .001, ... which converges to 0. Therefore, the two Cauchy sequences refer to the same real number. Keep in mind that the decimal expansion .99999... is just a representation of the real number corresponding to that specific Cauchy sequence, just like 6/2 is a representation of the rational numbers.
One thing we can prove is that the real numbers R have no more 'gaps'. If we imagine the number line, the integers, rationals, and algebraic numbers all had missing points (this despite the fact that the rationals and algebraic numbers are 'dense' - meaning there are no non-zero gaps). The real numbers are 'completed' in the sense that any sequence of reals that converges converges to a real. The reals contain the rationals as a subfield (again, in the isomorphic sense) because if r is rational, the Cauchy sequence r,r,r,... converges to r. That is why it is OK to write 0.3333333... as 1/3 and vice versa. In the same way, the reals contain the integers.
The next step in completion is algebraic - we want to be able to take square roots of any number and get a number back. So we go back to field extensions and extend R by the only square root that's missing, which is \(\sqrt{-1}\). This extension of the reals gives the complex numbers, which are complete in the sense that every complex polynomial has a complex root.
The completeness of the reals is absolutely vital for all of the mathematics of continuity - like calculus. In particular, statements like the intermediate value theorem are not true on the rationals or the algebraic numbers, we really need a number line with no 'holes'. In particular, once we can talk about limits of converging sequences like numbers in the real field without worry, we can talk about continuity of functions. A function f is continuous at a point x if every sequence {x}n that converges to x has the sequence f({x}n) converging to f(x). The paradox in the other thread is, in essence, a discontinuity of the cardinality function. The problem is that the limit of the cardinalities of the sets is not equal to the cardinality of the limit of the sets because one does not converge (it 'goes to infinity') and the other can be any countable cardinal number.
We start with the integers Z = {...,-2,-1,0,1,2,...} (Z for 'Zahlen', which is German for 'numbers'). I'm assuming you all already know the arithmetic properties of the integers - specifically, addition, subtraction, and multiplication - as of yet, division is not well defined on integers because the result might not be an integer (a number system with these operations is called a 'ring'). All other arithmetic properties of the rationals, reals, and complex numbers will ultimately derive from the analogous properties in the integers. Like Kronecker said "God made the integers; all else is the work of man". The positive integers are sometimes called the 'natural numbers' and are denoted by N = {1,2, ...}.
Since division is not well-defined in the integers, we want to define a bigger number system that lets us perform all four operations (except dividing by zero) while remaining in the number system (this is called a 'field'). The smallest field that includes the integers is the rational numbers. One way to define rational numbers Q (Q for quotient) is as an ordered pair (a,b) where a is an integer and b is a natural number (specifically, b is not zero). We have addition (a,b) + (c,d) = (ad + bc,bd) and multiplication (a,b)*(c,d) = (ac,bd), and the standard representation of (a,b) is a/b. Addition gives rise to well-defined subtraction, and multiplication gives rise to well-defined division (by non-zero rationals, where (a,b) is zero if a = 0). Note that each of the four arithmetic operations in the rationals is entirely defined by operations for integers.
Suspiciously, the rationals don't contain the integers explicitly. Remember that by our definition an integer is a single number, while the rationals are pairs of numbers (a,b), and they aren't the same. However, the rationals contain numbers that act very much like the integers. If we look at (a,1), then (a,1) + (b,1) = (a+b,1) and (a,1)*(b,1) = (ab,1). We say that the integers are isomorphic to a sub-ring of the rationals, and they are essentially integers in all but name, which is why treating 3/1 as 3 is not a problem even though we are actually switching between number systems.
Even more suspiciously, we note that there are multiple ways of writing the same rational. In particular (a,b) = (c,d) if and only if (a,b) - (c,d) = 0 (where 0 is the analog of the zero integer in the rationals, i.e. (0,b) for some b in N). If you work out the subtraction you get that (a,b) = (c,d) if and only if ad - bc = 0 (this operation now lives entirely in the integers). That means that there many equivalent ways of writing the same rational number, so 6/2, 3/1, and 45/15 all represent the same rational number. If we look at all rationals equal to (a,b) we get what is called the 'equivalence class of (a,b)' in the rationals. Usually, we don't care which representative of the class is used (in arithmetic 6/2 and 45/15 are really the same for all intents and purposes), but if we want to pick the 'nicest' version, we'll pick the rational (c,d) with smallest d. This always exists, and is unique, and is (a,b) in 'simplest form'. Keep in mind this fact that people are perfectly OK with these numbers having multiple representations when we get to the 0.999... = 1 fiasco.
So we have the rationals, but the rationals are still missing numbers that we'd like to talk about. In particular, Greek mathematicians noted that the diagonal of a square with side length 1 cannot be a rational number. Since rational numbers contain the integers, it can't be an integer either, so we don't have a description of the number that is the length of that line segment. So we want to 'complete' the rationals in this new sense. Keep in mind that we can't just stick in the square root of 2 into our number system, because we would mess up the arithmetic operations (i.e. if you include \(\sqrt{2}\) you need to include \(1 + \sqrt{2}\) and \(\frac{3}{2}\sqrt{2}\), etc in order to make sure that we remain in our new field.
So, just like before, define a new kind of number. We'll say that this new kind of number is an ordered pair (a,b) of rational numbers, and give it the arithmetic representation \(a + b\sqrt{2}\). This is a new field of numbers that now includes the number we were missing (it is usually called Q(\(\sqrt{2}\)) and is known as a 'field extension'). But for us, it still isn't good enough, as it is still missing \(\sqrt{3},\sqrt{5},\sqrt{\sqrt{2}}\), etc, etc. If we continue to add all those numbers, we get what is called 'the algebraic closure of Q' and is also known as the set of algebraic numbers (i.e. numbers that are solutions to polynomial equations with integer coefficients).
Alas, even the algebraic numbers have gaps. For example the number e is not algebraic, and neither is \(\pi\). So we seem to be stuck again. We can continue extending our numbers, but there's no reason to believe that this procedure will actually complete our field. So we need a different approach. The idea is to look at limits.
We can define a sequence of numbers as a function from the natural numbers. In other words, the 'list' of numbers a1, a2, a3, ... is just shorthand for a function f that takes entries in N and returns those numbers, specifically f(1) = a1, f(2) = a2, ..., f
= an, ..., etc. As a shorthand for the shorthand, we write {a}n to stand for the sequence a1, a2, a3, ...We define a 'metric' (distance function) by taking the distance between numbers x and y to be the absolute value of their difference, so d(x,y) = |x - y|. A set with a well-defined metric is called a metric space, and one particular property of a metric is that the distance between two points is 0 if and only they are the same point. Once we have a metric space, we can start talking about convergence.
We say a sequence {a}n converges to a value A if all ai with big enough index are arbitrarily close to A. The formal statement is that for any positive epsilon \(\epsilon > 0\), there exists an N such that \(|a_i-A| < \epsilon\) for every i > N. We say that the limit of the sequence is A, or \(\lim_{n\to\infty} a_i = A\). We say that a sequence is Cauchy if all entries with large enough index are arbitrarily close to each other. The formal statement of that is that for any positive epsilon \(\epsilon > 0\), there exists an N such that \(|a_n-a_m| < \epsilon\) for every n,m > N. Keep in mind that even though the sequences have varying indices, they are fixed mathematical objects (each is just a function). The limits, even though they seem like varying objects are just numbers, again, fixed mathematical objects.
Now, the hope is that we can find a field such that every Cauchy sequence converges to a number in the field. The idea being that in Q, the sequence 1, 1.4, 1.41, 1.414, ... is Cauchy, but does not converge to a rational number. Similarly, in the algebraic numbers 3, 3.1, 3.14, 3.141,... is Cauchy, but does not converge to an algebraic number.
So here is the completion idea - take the collection of all Cauchy sequences of Q. Define those to be the real numbers. It's almost too naive an idea to work, but it does work. Now, the sequence 3, 3.1, 3.14, .... is defined to be the real number \(\pi\). We can correctly define addition and multiplication of sequences index by index, and therefore we can do subtraction and division (being careful about zeros) so what we get is a field. Now, we get the nice property that every Cauchy sequence of real numbers converges to a real number. Two Cauchy sequences are equal if their difference converges to 0. Every real number has at least one decimal expansion - just take your epsilons small enough to fix each digit, and every decimal expansion corresponds to a real number. But, we need to be careful. Just like the rationals, that means that multiple different Cauchy sequences converge to the same real number - in particular the sequence 0, 0.9, 0.99, ... and the sequence 1,1,1,1,... have differences 1, .1, .01, .001, ... which converges to 0. Therefore, the two Cauchy sequences refer to the same real number. Keep in mind that the decimal expansion .99999... is just a representation of the real number corresponding to that specific Cauchy sequence, just like 6/2 is a representation of the rational numbers.
One thing we can prove is that the real numbers R have no more 'gaps'. If we imagine the number line, the integers, rationals, and algebraic numbers all had missing points (this despite the fact that the rationals and algebraic numbers are 'dense' - meaning there are no non-zero gaps). The real numbers are 'completed' in the sense that any sequence of reals that converges converges to a real. The reals contain the rationals as a subfield (again, in the isomorphic sense) because if r is rational, the Cauchy sequence r,r,r,... converges to r. That is why it is OK to write 0.3333333... as 1/3 and vice versa. In the same way, the reals contain the integers.
The next step in completion is algebraic - we want to be able to take square roots of any number and get a number back. So we go back to field extensions and extend R by the only square root that's missing, which is \(\sqrt{-1}\). This extension of the reals gives the complex numbers, which are complete in the sense that every complex polynomial has a complex root.
The completeness of the reals is absolutely vital for all of the mathematics of continuity - like calculus. In particular, statements like the intermediate value theorem are not true on the rationals or the algebraic numbers, we really need a number line with no 'holes'. In particular, once we can talk about limits of converging sequences like numbers in the real field without worry, we can talk about continuity of functions. A function f is continuous at a point x if every sequence {x}n that converges to x has the sequence f({x}n) converging to f(x). The paradox in the other thread is, in essence, a discontinuity of the cardinality function. The problem is that the limit of the cardinalities of the sets is not equal to the cardinality of the limit of the sets because one does not converge (it 'goes to infinity') and the other can be any countable cardinal number.
