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

In algebra we would say that the structure

**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 for all*x*⋅*e*=*e*⋅*x*=*x**x*. Of course, this identity element is the number 1. - It satisfies the cancellation law.

In algebra we would say that the structure

These two operations also cooperate nicely, namely they satisfy the

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

**Ordering.**

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

**transitive law**, that is, if and*x*<*y* then necessarily*y*<*z*,*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
then*x*<*y*, and*x*+*z*<*y*+*z* for any*x*⋅*z*<*y*⋅*z**z*.

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

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

The first three properties mean that this relation is a partial ordering,
when we add the fourth, we get that
*A* ⊆ B)

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}.
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
*X*_{0} 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

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*,*x* such that the equality becomes true? The answer is
simple: Only if we are lucky. We can solve
*x* = 15,*x* = 5.

We have the same problem with multiplication, for instance we can solve
*x* = 15,*x* = 5.

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.

We therefore try to add those needed elements to natural numbers (and 0), we
define the integers as

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 This inverse element is denoted*x*+*y*=*y*+*x*= 0.(− *x*).

In algebra we would say that *a* + *x* = *b**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*,*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

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
*x*.**negative** if
*x* < 0.

- If
then*x*<*y*, for*x*⋅*z*<*y*⋅*z**z*positive and for*y*⋅*z*<*x*⋅*z**z*negative. Similarly for "≤". is positive if*x*⋅*y**x*and*y*are positive or if*x*and*y*are negative. is negative if*x*⋅*y**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,
^{+} = ℕ.

We arrive to rational numbers by trying to solve the equation
*a* ⋅ *x* = 1.*a* = 0*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 *a**a* and
to make the set closed under multiplication, we also add all possible
products of the form *b* ⋅ (1/*a*),*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 This inverse element is denoted*x*⋅*y*=*y*⋅*x*= 1.*x*^{−1}or1/ *x*.

Again, the distributive law holds.
In algebra we would say that *a* ⋅ *x* = *b**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**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

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

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**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**a*. However, there is a price to pay, complex numbers
cannot be reasonably ordered, so there is no inequality available here.