What is a number?
When I considered what people generally want in calculating, I found that it always is a number. Mohammed ben Musa al-Khowarizmi. |
The individual is what he is and has the significance that he has not so much in virtue of his individuality, but rather as a member of a great human community .... Albert Einstein. |
To paraphrase Albert Einstein, a number in and by itself has no significance and only deserves the designation of number by virtue of its being a member of a group of objects with some shared characteristics. The most common characteristic of numbers is that they can be added and multiplied to produce other numbers in their group. However, not all objects that can be added or multiplied are designated as numbers.
As a matter of fact, there are many different kinds of numbers.
- rational and irrational
- real and complex
- imaginary
- algebraic and transcendental
- perfect
- surreal
- hyperreal numbers
- square and triangular numbers
Let's talk a little about each of these in turn.
Rational and Irrational numbers
A number r is rational if it can be written as a fraction
((√5 + 1)/2)^{2} + ((√5 - 1)/2)^{2}
may not at first look rational but it simplifies to 3 which is
Using only arithmetic methods it's easy to prove that the number √5 is not rational. Just to remind, √5 stands for the number whose square equals 5. Thus Arithmetic can show that, when squared, no rational number gives 5. It's impossible to derive in Arithmetic that such a number actually exists.
Proof that for no rational number r = p/q, (p/q)^{2} = 5
Assume that a rational number r exists such that
Thus 5q^{2} = p^{2} so that p^{2} is divisible by 5. 5 is a prime number (it has no other divisors but itself and 1). Therefore I can use a theorem by Euclid (Elements, VII.30) (see also Ref [2]) to claim that since 5 is a factor of p^{2} it also divides p, i.e.
(In fact for any integer n, which is not a square of another integer, √n is irrational. The above proof is commonly used to prove the irrationality of √2. This is because √2, being the length of the diagonal in a unit square, was the first number proved not to be rational. There is in fact great many ways to establish this result with various degrees of intuitive appeal.)
Intuitively the number √5 should exist. Indeed, apply the Pythagorean Theorem to the right-angled triangle with sides 1 and 2. Then (hypotenuse)
As often the case in Mathematics, there are many ways to define irrational numbers. We'll use the one suggested by Georg Cantor (1845-1918), the father of modern Set Theory. Incidentally, Cantor's doctoral thesis was titled "In mathematics the art of asking questions is more valuable than solving problems." (see Ref [4])
Without giving a rigorous definition let me say that Cantor defined irrational numbers as limits
of convergent sequences of rational numbers (see Ref [1]). The idea is quite intuitive. For simplicity, I'll handle only numbers
between 0 and 1. Start with writing rational
numbers as decimal fractions
a_{1} = 0.1, a_{2} = 0.14, a_{3} = 0.142, ...
The sequence converges because
Lemma 1
- Between any two different rational numbers a and b there is at least one other rational number.
- Between any two different irrational numbers a and b there is at least one other rational number.
- Between any two different rational numbers a and b there is at least one other irrational number.
- Between any two different irrational numbers a and b there is at least one other irrational number.
Corollary
In all four statements assertion of existence of a single number can be replaced with the assertion of existence of infinitely many numbers.
Proof of Lemma 1
a < (a + b)/2 < b and (a + b)/2 is rational.
Let decimal expansions of two irrational numbers a and b
(a < b) first differ in the n^{th} digit. Consider the number b_{n} obtained from b by cutting all the digits after the n^{th}. Thena < b_{n} < b. Of course, being a finite decimal expansion, b_{n} is a rational number.Let the decimal expansions of two rational numbers a and b
(a < b) first differ in the n^{th} digit. Then, as before, consider b_{n}:a < b_{n} < b. Assume b has non-zero digits after the n^{th}. Think of how to amend the proof if it does not. Let b_{m} be the first non-zero digit of b after b_{n}. Append to b_{n}m - n zeros and then any random sequence of digits. Every time you append a digit you obtain a new rational number between a and b. The sequence is convergent to a limit between a and b. If you make an effort to avoid periodic sequences the limit will be irrational.Will Rose suggested a different way of growing b_{n} into an irrational number. Pick any number r known to be irrational. It could be the square root of 2, e, π or any other irrational number. Since the product of two rational numbers is rational, r·10^{M} is bound to be irrational for any integer M. Take N so large as to make r·10^{-N} smaller than b_{m}10^{-m}. Then use the digits of this number in lieu of the random selections as above.
(An Aside: Is number 1234567891011121314151617181920212223... periodic?).
The proof is a variation on the above.
Collectively, rational and irrational numbers are called real. Thus a real number is either rational or irrational. (In passing, Ian Stewart questions the wisdom of calling irrational numbers real: How can things be real if you can't even write them down fully?) From Lemma 1 it may appear that rational and irrational numbers play symmetric roles in the set of real numbers. One of the many Cantor's contributions was in showing that there are various kinds of infinities. While it is true that there are infinitely many rational numbers and infinitely many irrational numbers, in a well defined sense, the are more irrational numbers than rational.
Cantor began with the notion that counting or enumeration of objects brings finite sets into a 1-1 correspondence with finite segments of the set of integers. When you say you have two hands you mean that it's possible to assign the number 1 to one hand and the number 2 to another and no hands will be left uncounted. Right? Is it what you mean? Cantor applied the notion of the 1-1 correspondence to infinite sets such as the sets of integers, rational, irrational or real numbers.
With the notion of the 1-1 correspondence came quite a few surprises. (Actually the fact that handling infinities is wrought with surprises was already known to medieval thinkers, see Ref [3]. Check also Hilbert's paradox of the Grand Hotel.) For example, the sets of odd numbers, even numbers, complete squares or cubes, the set of integers greater than 1996 all can be brought into a 1-1 correspondence with the set of all integers (of which they are subsets.) Indeed, the following table summarizes how it's done.
Integers | Odd numbers | Even numbers | Squares | Cubes | Greater than 1996 |
---|---|---|---|---|---|
n | 2n-1 | 2n | n^{2} | n^{3} | n+1996 |
1 | 1 | 2 | 1 | 1 | 1997 |
2 | 3 | 4 | 4 | 8 | 1998 |
3 | 5 | 6 | 9 | 27 | 1999 |
4 | 7 | 8 | 16 | 64 | 2000 |
... | ... | ... | ... | ... | ... |
Let N, E, and O denote the sets of all counting numbers, all even and all odd numbers, respectively.
Then, on the one hand, N~E and N~O (where tilde signifies existence of a 1-1 correspondence). On
the other hand,
I'll call two sets A and B
A set A is called countable (or enumerable, or denumerable, and, sometimes, even numerable) if A~N, i.e. if it's equivalent to the set of all integers. Our second discovery can be formulated as
Lemma 2
Union of two countable sets is countable.
Proof of Lemma 2
Let us denote the two sets A and B. We are given that
It follows from Lemma 2 but also can be proven directly that the union of a finite number of countable sets is countable. A more exciting fact is given by
Lemma 3
Union of a countable number of countable sets is countable.
Proof of Lemma 3
For a proof please refer to a separate page.
Corollary
A Cartesian product of two countable sets is countable. (Cartesian product of two sets A and B consists of pairs (a, b) where
Cantor's Set Theory is a fascinating topic in its own right but what was the purpose of swaying away from the discussion of various sets of numbers? Besides being useful in the forthcoming discussion on algebraic and transcendental numbers, there was a more immediate goal.
Lemma 4
The set Q of all rational numbers is equivalent to the set N of all integers.
Proof of Lemma 4
It's a direct consequence of Lemma 3. To get an enumeration of Q, write in the diagram above 1/1 instead of 11, 1/2 instead of 12 and so
on. Put another way, the function
Another appealing proof can be found on a separate page. And a third one is based on enumeration of fractions on the Stern-Brocot tree.
Lemma 5
The set Q of all rational numbers is equivalent to the set N of all integers. The set I of all irrational numbers is not countable.
Proof of Lemma 5
The proof is a variant of what is known as the diagonal argument. Again I shall only consider numbers between 0 and 1. Irrational numbers have been defined as decimal (non-periodic) fractions. Assume it is possible to enumerate all such decimals. Let choose an enumeration and list the decimals in the corresponding order:
a_{1} = 0.a_{11}a_{12}a_{13}a_{14}...
a_{2} = 0.a_{21}a_{22}a_{23}a_{24}...
a_{3} = 0.a_{31}a_{32}a_{33}a_{34}...
...
where a_{mn} stands for the n^{th} digit of the m^{th} decimal. The approach I employ is Cantor's famous diagonal process. To remind, we made an assumption that all the decimals between 0 and 1 have been listed in the sequence above. I'll get a contradiction by showing that at least one decimal is missing from the list. The decimal
(An absolutely delightful argument of a recent vintage the does not refer to the diagonal process is well worth looking into.)
From Lemma 1 it appeared that the distributions of rational and irrational numbers have much in common. Between any two numbers of one kind there always existed numbers of the same and of the other kind. However, rational numbers form a countable set whereas irrational form a set which is not countable.
To conclude the discussion of rational and irrational numbers:
- There are infinite sets for which no 1-1 correspondence exists. There are various kinds of infinite sets. This topic belongs to the realm of the Set Theory and, more particularly, to the Theory of Transfinite Numbers (Cardinals).
- The property expressed in Lemma 1 is known as density of both rational and irrational numbers on the straight line and, as such, belongs to the General Topology. Sets could be dense without being equivalent.
- Expectations based on experience with finite sets should not be extrapolated to the sets infinite.
Algebraic and Transcendental numbers
Now let us consider polynomial equations with integer coefficients:
P_{n}(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{1}x + a_{0} = 0
Real roots of such equations are said to be algebraic. In other words, a number a is called algebraic if
it satisfies an algebraic equation: P_{n}(a)=0 for some polynomial P_{n}(x) with
integer coefficients. Rational numbers and integers are all algebraic. Indeed, for a given rational
number
If a_{n} ≠ 0 then the polynomial is said to be of order n.
Polynomials of order 1 have two integer coefficients each, a_{1} and a_{0}. It follows from Lemma 3 that the set of all polynomials of order 1 is countable. Directly or by induction, one can prove a general assertion. For every n the set of polynomials of order n is countable. The Fundamental Theorem of Algebra claims that a polynomial of order n has exactly n (perhaps complex) roots. Thus the set of all numbers that satisfy some equation of order n is countable. The subset of this set that consists of real numbers is as well countable. Applying once more Lemma 3 we get
Lemma 6
The set of all algebraic numbers is countable.
Real numbers that are not algebraic are called transcendental. Thus the set R of all real numbers is the union of two sets - algebraic and transcendental. The set R is not countable while there are countably many algebraic numbers. Therefore the set of all transcendental numbers is not countable either. Given the difficulty of establishing whether a given number is algebraic or not, this was one of Cantor's early surprising results.
It's known that π and e are transcendental.
e was proven to be transcendental by Hermite in 1873, and π by
Lindemann in 1882. e^{π} is transcendental from the Gelfond's Theorem.
At least one of πe and
A note
Note that although most of the real numbers are transcendental (since there is only countably many algebraic numbers), transcendentality is proven a number at a time.
Real and Complex numbers
The union of rational and irrational numbers forms the set of real numbers. Complex numbers
are pairs
Complex numbers have unusual properties especially when it comes to differentiation. With real numbers differentiation is a starting point of Calculus and Real Analysis. With complex numbers it leads to the Analytic Function Theory.
Imaginary numbers
As was just defined, a complex number c is a pair
Complex number (x, y) is associated with a point
In the plane the distance between two points
(1) | (x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}. |
This is an equation of a real (i.e. one that could be drawn with a compass) circle in a real (although consisting of complex numbers) plane.
Now, mathematicians did not stop at a simple equation
(2) | (x - x_{0})^{2} + (y - y_{0})^{2} = (ir)^{2}. |
which is really an imaginary circle with a real center c_{0} but an imaginary radius ir. Getting this far, it must be said that the polynomial equation (2) has no real solutions
(3) | x^{2} + y^{2} = (ir)^{2} |
then it'll become obvious that equations (2) and (3) do have solutions. For example, the pair
Perfect numbers
Every number n is divisible by at least 1 and n. The sum of all divisors of n is called
The numbers for which
Euclid in his Elements, IX.36 proved that if, for a prime p,
The first perfect number is
Surreal numbers
Surreal numbers have been invented by John Conway who, among many other things, created the famous Game of Life. Donald Knuth's brochure Surreal Numbers gives a very popular but rigorous introduction into the development of surreal numbers successively starting with literally nothing. This is the only aspect of the theory that I intend to mention here.
A surreal number (see a short introduction or in relations to games) is a pair of sets {X_{L}, X_{R}} where indices indicate the relative
position (left and right) of the sets in the pair. The first number to be constructed is
Martin Gardner (Ref [9]) describes application of surreal numbers to the theory of games also developed by John Conway.
Square and Triangular numbers
Every one knows how to compute a square of a given number especially if it's not too large.
A square with a side of length n can be visualized as comprising a grid of n×n smaller squares of size 1×1.
Triangular numbers owe their name to a similar construction, now of triangles. Actually
we already used triangular numbers in the proof of Lemma 3 above. Triangular numbers are in the
form 1, 3, 6, 10, 15, ... The general formula is that for every n the number
Of course it needs to be proven that the total number of crosses arranged as in the diagram is given by the formula
1 + 2 + ... + n,
where n is the number of crosses on the side of the triangle. Thus, along the way, we are going to prove that
1 + 2 + ... + n = n(n+1)/2.
Another diagram will make this statement obvious
Indeed, two triangles, each with n crosses on the side, together form an n×(n+1) rectangle. You may question whether pointing to the diagram proves anything. No, of course by itself the diagram does not prove anything. But it does help to visualize and perhaps discern the proof. Indeed, we seek a formula for the sum of integers from 1 through n. This sum represents the number of crosses in a triangle. Two triangles of which one is rotated 180° form a rectangle, i.e. a shape in which all rows have the same number of crosses. Rows in the first triangle count crosses 1,2,3... in the increasing order whereas in the second we have rows with n,
First take two sums (one in increasing, another in decreasing order)
Quite rigorously. Q.E.D.
It is obvious that all Euclid's perfect numbers are triangular.
References
- J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, NY, 1996.
- R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1966.
- H. Davenport, The Higher Arithmetic, Harper&Brothers, NY.
- W. Dunham, The Mathematical Universe, John Wiley & Sons, NY, 1994.
- Encyclopædia Britannica
- M. Gardner, Penrose Tiles To Trapdoor Ciphers, W.H.Freeman and Co., NY, 1989.
- D. Knuth, Surreal Numbers, Addison-Wesley Publishing Co., 1974.
- K. Kuratowski and A. Mostowski, Set Theory, North-Holand, Amsterdam, 1967.
- M. Gardner, Time Travel and Other Mathematical Bewilderments, W.H.Freeman and Co., NY, 1988.
- R. Rucker, Infinity and the Mind, Princeton University Press, Princeton, NJ, 1955
- H. Schwerdtfeger, Geometry of Complex Numbers, Dover Publications, NY, 1979.
- I.Stewart, Nature's Numbers, BasicBooks, 1995.
On the Web
- Abstract Algebra OnLine
- Ask Dr. Math
- Eric's Treasure Trove of Mathematics
- Triangular numbers (an online and iPod video by Julio de la Yncera)
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