Important sets of numbers

Here we will review important sets of numbers: natural numbers, integers, rational numbers, real numbers, and at the end we briefly look at complex numbers. We will list some properties of these sets, in more detail than needed for the purposes of Math Tutor, but to a patient reader it might be an interesting glance at the development of the idea of number sets and some abstract reasoning behind things we use every day.

Natural numbers are 1, 2, 3, 4, 5, and so on. The set of all natural numbers is denoted by ℕ. The first "leg" of the letter is doubled (or otherwise emphasized), which is a mathematical way of suggesting that it is not just a set, but a more complicated structure, namely that we also have some operations acting on this set and it can be ordered.

Addition.
As we all know, we can add natural numbers. This operation satisfies some useful properties. The first thing one should check when introducing an operation to a set is whether the set is closed under this operation, which here means that if we take two natural numbers and add them, we again get a natural number (we cannot get "outside" the set by using the operation). Of course, this is true here. We also have these properties:

• commutative law, that is, x + y = y + x for any two natural numbers;
• associative law, that is, (x + y) + z = x + ( y + z) for any three natural numbers;
• cancellation law, that is, if we have x + y = x + z, then necessarily y = z.

In algebra we would say that the structure (ℕ,+) forms a "commutative semigroup", but we won't worry about it here.

Multiplication.
Again, we know that the set of natural numbers is closed under multiplication. It has one more property than addition.

• It is commutative.
• It is associative.
• It has identity, which is an element e such that x ⋅ e = e ⋅ x = x for all x. Of course, this identity element is the number 1.
• It satisfies the cancellation law.

In algebra we would say that the structure (ℕ,⋅) forms a "commutative monoid", but again we won't worry about it here. Note that addition on natural numbers does not have an identity. We will return to this problem below.

These two operations also cooperate nicely, namely they satisfy the

• distributive law, that is, x ⋅ (y + z) = x ⋅ y + x ⋅ z and ( y + z) ⋅ x = y ⋅ x + z ⋅ x for any three numbers.

Ordering.
The set of natural numbers can be naturally ordered by two binary relations. The relation "<" satisfies these properties:

• transitive law, that is, if x < y and y < z, then necessarily x < z;
• trichotomy law, that is, for any two elements x and y, exactly one of the following three is true: x < y, y < x, x = y.

This relation works well with addition and multiplication:

• If x < y, then x + z < y + z and x ⋅ z < y ⋅ z for any z.

The second relation is the relation "≤". It satisfies the following properties:

• reflexivity, that is, x ≤ x for all x;
• antisymmetry, that is, if x ≤ y and y ≤ x, then x = y;
• transitivity;
• comparability, that is, for any two elements x and y, exactly one of the following two is true: x ≤ y or y ≤ x.

The first three properties mean that this relation is a partial ordering, when we add the fourth, we get that (ℕ,≤) is a linearly ordered set. Note that the last property is not as obvious as it seems, for instance the relation of being a subset (A ⊆ B) is a partial ordering (between sets), but it does not satisfy the last property.

This ordering also works well with addition and multiplication.

Problems:
Although natural numbers form a wonderful structure, there are some things missing, in trying to rectify them we will end up with other sets of numbers. What are the problems?

1. No identity element for addition.
This is solved easily, we consider the set {0,1,2,3,4,...}, which here we will denote ℕ₀. Now also the addition has an identity element, so from an algebraic point of view this set is "better". Indeed, some authors take this set as natural numbers. Why didn't we do it here? There are two reasons.

It is more convenient this way. There are situations where you want the set {1,2,3,...} and at other times you need the set {0,1,2,3,...}. It would be handy to have a simple notation for both. The way we introduced natural numbers here we do have a simple notation also for the second set, as we just saw. Indeed, the little symbol X₀ indicates in general that we added "0" to the set X. If we also considered 0 to be a natural number, then for the set {1,2,3,...} we would have to write something unnatural or more complicated like ℕ − {0}.

The second reason is questionable but I find it even more compelling than the first. Numbers 1,2,3,... are indeed natural, once you start thinking about the world around you, sooner or later you come up with them. On the other hand, 0 is a much more advanced concept and the mankind "discovered" it comparatively late, it started to be used around 850 AD in India and as late as in the 1600's it was encountering resistance in Europe.

2. Impossibility to solve equations.
Given two natural numbers a and b and an equation a + x = b, can we always find a natural number x such that the equality becomes true? The answer is simple: Only if we are lucky. We can solve 13 + x = 15, but not 13 + x = 5.

We have the same problem with multiplication, for instance we can solve 3 ⋅ x = 15, but not 3 ⋅ x = 5. We will try to fix these problems by introducing richer sets of numbers.

We arrive at the notion of integers by trying to fix the equation solving problem for addition. In fact, to be able to solve all of them it is enough to be able to solve a + x = 0. In order to be able to solve such an equation we need negative numbers, which in the algebraic context means inverses with respect to addition.

We therefore try to add those needed elements to natural numbers (and 0), we define the integers as {0,1,−1,2,−2,3,−3,...} and denote them as ℤ. Again, the bold stroke suggests that integers is not just a set, but we also have a structure (operations, ordering). By the way, this adding of elements is not as simple as it seems, which will be manifested quite clearly when we get to last parts of this section. If you are curious where the catch is, take a look here.

The set of integers is closed under addition and addition now satisfies these properties:

• It is commutative.
• It is associative.
• It has identity, here e = 0.
• It has inverse or reciprocal for each element, that is, for every x there is an element y such that x + y = y + x = 0. This inverse element is denoted (−x).

In algebra we would say that (ℤ,+) is a commutative group, which is pretty much the best one can say about an operation. Once we have inverse elements, we can solve equations a + x = b for any a and b, we also have the cancellation law.

In practical use we talk of an operation called "subtraction", but from an algebraical point of view there is no such thing, when we write x − y, we in fact mean x + (−y).

Multiplication still has the same properties as before with one exception, the cancellation law now does not work. We can cancel, but only if x is not zero. However, this property is not all that important and we did not lose it entirely, so it is a small price to pay. Addition and multiplication are still tied by the distributive law. Considering all the properties, in algebra we would say that (ℤ,+,⋅) is a commutative ring with multiplicative identity, or that it is an integral domain.

Addition of the extra elements did not spoil the main properties of the two orderings above, "<" still satisfies the laws of transitivity and trichotomy, "≤" is again a linear ordering. However, now we have a little problem with the relations and operations. They still work well with addition, but with multiplication we have to be more careful.

Let x be an integer. We say that it is positive if 0 < x. We say that it is negative if x < 0. The elements of integers thus split into three distinct groups: {0}, positive integers (which is in fact the set of natural numbers) and negative integers. Elements of these groups now behave very differently with respect to multiplication.

• If x < y, then x ⋅ z < y ⋅ z for z positive and y ⋅ z < x ⋅ z for z negative. Similarly for "≤".
• x ⋅ y is positive if x and y are positive or if x and y are negative.
• x ⋅ y is negative if x and y are one positive and one negative.

We now fixed the problem of solving equations with addition, but not with multiplication. That's the next step. Before we move on, we introduce another general notation that is sometimes used. If X is a set of numbers, then by X ⁺ we denote the set of elements from X that are positive and by X ^- we denote the set of elements from X that are negative. For instance, ℤ⁺ = ℕ.

We arrive to rational numbers by trying to solve the equation a ⋅ x = 1. Obviously, if a = 0 then there is no solution possible. Thus we cannot expect to get something as nice as we had before for addition, but we will do our best. If a is not zero, then we have a chance to solve such an equation, it would yield an inverse element to a with respect to multiplication. We do not have them in the set of integers, but it is possible to add all needed inverses into this set and define operations so nicely that we do not get any contradiction or worsen the nice properties we had before.

Thus we define the set of rational numbers by starting with integers, then we add abstract elements denoted by 1/a for every non-zero a and to make the set closed under multiplication, we also add all possible products of the form b ⋅ (1/a), we denote them b/a for short and call them fractions. Then we have to define how addition and multiplication act on these new elements, see this note on extending to integers. Thus we get the set of rational numbers ℚ. By the way, one has to somehow deal with the fact that one element can be obtained in several ways ("cancelling" in fractions), but this can be done as well. Since everything is done in a natural way, we did not spoil properties that we had before and even have some extra ones.

The set of rational numbers is closed under addition, and addition is still commutative, associative, has an identity and inverse elements.

The set of rational numbers is closed under multiplication, and multiplication satisfies these properties.

• It is commutative.
• It is associative.
• It has identity, here e = 1.
• It has inverse or reciprocal for every non-zero element; that is, for every x not equal to zero there is an element y such that x ⋅ y = y ⋅ x = 1. This inverse element is denoted x⁻¹ or 1/x.

Again, the distributive law holds. In algebra we would say that (ℚ,+,⋅) is a field. We can now also solve equations a ⋅ x = b for any non-zero a. And again, note that informally we use "division", but in algebra we do not recognize any such operation, all expressions of the form a/b are understood as a ⋅ b⁻¹.

Also the orderings we had above can be extended to compare fractions without losing any of their properties, including the division to positive and negative numbers. The notation stays the same, for instance ℚ⁺₀ is the set of all positive rational numbers and zero included.

Essentially we fixed all the problems listed above as far as possible. Does it mean that rational numbers are perfect? The answer is: Close, but no cigar. They are great and work well in most practical situations, but they have one problem. The equation x ⋅ x = 2 cannot be solved.

We concluded the previous part with a statement that is by no means obvious, but it is true. Ancient Greeks already proved that if we draw a right-angle triangle with adjacent sides equal to 1, then no rational number can express the length of the hypotenuse. This could be bad or not so bad, depending on from where you look at it. If you are a practical engineer, then you never work with precise measurements anyway, and once you work within a certain tolerance, you can always use rational numbers to express that hypotenuse length "almost exactly", so you may not worry. On the other hand, that problem shows that rational numbers have a serious flaw as a tool for precise theoretical description of nature, which is quite bad for math and physics among others.

This lack shows up in many ways. For instance, sequences that converge have a certain property called Cauchy property, it basically means that toward the "tail" of the sequence its elements do not change much. It would be nice if every sequence with this property converged (such a space is called complete), but in the world of rational numbers this is not true. For instance, if we approximate that hypotenuse using rational numbers with a better and better precision, we get an infinite sequence of rational numbers that gets closer and closer to a certain place, but that number (the precise length) is not available and thus this sequence does not converge. Thus the set of rational numbers is not complete.

It also spells trouble from the point of view of ordering. If we take a set that is bounded below, we would like to have an infimum, sort of like the least element of the set (see the section Functions - Theory - Topology of real numbers). However, consider the set M of all rational numbers larger than the length of that hypotenuse. The "smallest" number in this set should be that length, but in the world of rational numbers this length is not available. Big problem.

So from many points of view the set of rational numbers contains "holes". A natural idea would be to "fill them in" by adding (to the set of rational numbers) all the lengths that are missing. This can be done, such lengths are called "irrational numbers" and after carefully defining how operations and orderings work with such numbers we obtain the set of real numbers. Formally and precisely this is not all that easy, people mostly define real numbers as all potential limits of sequences formed by rational numbers that have the Cauchy property (that is, real numbers are all lengths that can be arbitrarily well approximated with rational numbers).

Anyway, now we have real numbers and their structure, we denote this set as ℝ. What can we say about it?

First, we did not spoil any of the nice properties we had for rational numbers. That is, the set of real numbers is closed under addition; on it the addition is commutative and associative, it has identity and inverses for all elements. Similarly, the set of real numbers is closed under multiplication; on it the multiplication is commutative and associative, it has identity and inverses for all non-zero elements. The distributive law still works, so (ℝ,+,⋅) is a field. Also ordering works as it did before. The difference is that the set of real numbers with the usual distance is a complete set, that is, all Cauchy sequences are convergent there. This in particular means that this set is rich enough to measure any length.

We will look at real (and also rational) numbers closer in the section Functions - Theory - Topology of real numbers. Here we conclude this part with a natural question: Are we happy now? Actually, not quite. Granted, we can solve equations involving addition and multiplication (with the exception of that zero case), we have completeness, but a mathematician is never happy and here we have a problem. One thing that we fixed by passing from rationals to reals is that now we can solve equations x ⋅ x = a for all a positive or zero. However, we cannot solve such an equation for negative a. This brings us to the last part.

We obtain complex numbers by adding to real numbers all square roots of negative numbers (they do not exist as real objects, but we may take them as some abstract things and give them names). Since we want a set closed under multiplication and addition, we have to first define how we add and multiply these imaginary objects and how they interact with ordinary real numbers, and then also include all possible things that we can get out of such operations. It turns out that it is enough to include all elements of the form "real number plus a square root of some negative number" and we already obtain a set closed under these operations. In fact, it even turns out that it is enough to add a hypothetical square root of −1 and all its linear combinations with real numbers and we get the same set.

This is called complex numbers and denoted ℂ and we will not look at them here, since Math Tutor is about real functions and related concepts. We just say that all algebraic properties we had for real numbers are still true for complex numbers (it is a field) and now we can also solve all equations of the form x ⋅ x = a for any complex number a. However, there is a price to pay, complex numbers cannot be reasonably ordered, so there is no inequality available here.

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