There is an excellent textbook on this subject: Grundlagen der Analysis by Edmund Landau. Pray, employ your favorite search-engine or on-line bookseller to locate a copy. However, it would be desirable to have the material on-line. Hence, I will summarize my recollection of the course that I took more than half of a century ago. If I commit any mistakes; please call them to my attention.
At least, at this time, I will omit most proofs; because they are so easy and I am lazy. However, upon request; I promise to work-out and post any specific proof. From time to time, I may elaborate. However, it is neither the purpose nor the intent to provide any computational algorithms here.
The natural numbers come from the Peano axioms.
We define addition recursively. 1 + 1 = s(1). For any a in N, 1 + a = s(a). For any a and b in N, s(b) + a = s(b + a).
Definition: We define two -- symbolized by the usual glyph -- as the successor of one; that is 2 = s(1). Please
observe that in the binary number-system, there is no such glyph.
Theorem: Addition is associative; that is (a + b) + c = a + (b + c).
Theorem: Addition is commutative; that is a + b = b + a.
Definition: We say that a is greater than b -- written as a > b -- iff (= if and only if) there exists a c such that a = b + c.
We also say that in such a case b is lesser than a -- written b < a. Thus, we have a trichotomy. For any a and b in N, exactly one of the three
conditions holds: a > b, a = b, xor a < b. Hence, the natural numbers are totally ordered.
Since we do not have an identity element, the natural numbers do not constitute a group under addition. We define multiplication as successive
additions.
Theorem: Mathematical Induction is equivalent to saying that any non-null subset of the natural numbers possesses a least
element.
Proof: Its contra positive is a subset of the natural numbers, which does not possess a least element, is null. Hence, it obviously is
equivalent to the Mathematical Induction axiom. QED.
Definition: A pair of natural numbers is said to be relatively prime; if their greatest common divisor
is one.
Definition: The Euler totient function phi(n) is defined as the cardinality
of the subset of natural numbers strictly less than n, each of which is relatively prime to n.
Lemma: The totient of a prime number p is phi(p) = p - 1.
Lemma: The totient of a power m of a prime number is phi(p^m) = p^(m - 1) (p - 1).
Lemma: The totient of a product of a pair p and q of distinct prime numbers is phi(p q) = phi(p) phi(q) = (p - 1) (q - 1).
Theorem: The totient of n is given by the formula phi(n) = n * product, over primes -- in the semi-closed interval (1, n] --
which divide n, of (1 - 1 / p).
Corollary: For each n in [3, oo), the totient function is even.
Consider ordered pairs of natural numbers (a, b). We say that two such pairs -- (a, b) and (c, d) -- are equivalent iff a + d = b + c. We call the equivalence classes of these ordered pairs the integers.
Theorem: The integers are totally ordered. We say that an integer (a, b) is strictly-positive iff a > b, strictly-negative iff a < b, xor zero iff a = b. This is a trichotomy.
We extend addition to the integers by the definition (a, b) + (c, d) = (a + c, b + d).
Theorem: The 0 = (1, 1) is the identity.
Theorem: For any (a, b), (b, a) is the additive inverse; that is, (a, b) + (b, a) = (1, 1).
Theorem: The integers are associative and commutative under addition.
Theorem: The integers constitute an Abelian (= commutative) group under addition.
We extend multiplication to the integers by the definition (a, b) * (c, d) = (a c + b d, a d + b c).
Theorem: The 1 = (s(1), 1) is the identity.
Theorem: The integers are associative and commutative under multiplication.
Theorem: The multiplication of integers is distributative over addition; that is (a, b) * ((c, d) + (e, f)) = (a, b) * (c, d) +
(a, b) * (e, f) ((c, d) + (e, f)) * (a, b) = (c, d) * (a, b) + (e, f) * (a, b) for left and right distributative, respectively.
Theorem: The integers constitute a commutative ring under multiplication.
Theorem: The product of two strictly-negative integers -- (a, b) and (c, d) -- is a strictly-positive integer.
Proof: Their product is (a c + b d, a d + b c). By hypothesis, we know that a < b and c < d. Hence there exist two natural numbered u and v
such that a + u = b and c + v = d. We need to show that 0 <? a c + b d - (a d + b c) = a c + (a + u) (c + v) - a (c + v) - (a + u) c = u v >
0. QED.
Subtraction is the inverse operation of addition. As such, it requires that all of the group axioms be satisfied. Thus, the natural numbers are not closed under subtraction. Hence, the first opportunity to define a useful concept of subtraction is for the integers.
Definition: The unary minus operator is defined as an indication of the additive-inverse of a given number (a, b); that
is -(a, b) = (b, a).
Theorem: The sum of a number (a, b) and its minus is the additive-identity; that is (a, b) + -(a, b) = -(a, b) + (a, b) = (a
+ b, a + b) = (1, 1) = 0.
Theorem: The solution of the equation (a, b) = (x, y) + (c, d) is (x, y) = (a, b) + -(c, d).
Corollary: The same (x, y) solves the equation (a, b) = (c, d) + (x, y).
Definition: Hence, we define subtraction as (a, b) - (c, d) = (x, y).
Consider ordered pairs of integers (a, b), usually written a / b and called fractions. The second element must not be zero.. We say that two such pairs -- a / b and c / d -- are equivalent iff a * d = b * c.
We call the equivalence classes of these ordered pairs the rational numbers. This is exactly the same as we did previously with the natural numbers to obtain the integers.
Theorem: The rational numbers are totally ordered.
We extend addition to the rational numbers by the definition a / b + c / d = (a * d + b * c) / (b * d).
Theorem: The 0 = 0 / 1 is the identity.
Theorem: For any a / b, -a / b is the additive inverse; that is, (a / b) + (-a / b) = 0 / 1.
Theorem: The rational numbers are associative and commutative under addition.
Theorem: The rational numbers constitute an Abelian group under addition.
We extend multiplication to the rational number by the definition (a / b) * (c / d) = (a * c) / (b * d).
Theorem: The 1 = 1 / 1 is the identity.
Theorem: For any a / b, b / a is the multiplicative inverse; that is, (a / b) * (b / a) = 1 / 1.
Theorem: The rational numbers are associative and commutative under multiplication.
Theorem: The rational numbers, less the zero element 0 = 0 / 1, constitute an Abelian group under multiplication.
Theorem: The multiplication of the rational numbers is distributative over addition.
Theorem: The rational numbers constitute a field under the two operations of addition and multiplication.
Division is the inverse operation of multiplication. As such, it requires that all of the group axioms be satisfied. Hence, since the fractions do not posses a multiplicative-inverse, they cannot be divided. For example, the nearest we can come to a multiplicative-inverse of the fraction 1/2 is 2/1. Their product is (1/2) * (2/1) = 2/2, which is not the multiplicative identity 1/1. Hence, the first opportunity to define a useful concept of division is for the rational numbers.
Definition: The reciprocal operation is defined as an indication of the multiplicative-inverse of a given number (a/b);
that is 1/(a/b) = b/a.
Theorem: The product of a number a/b and its reciprocal is the multiplicative-identity; that is (a/b) * 1/(a/b) = 1/(a/b) *
(a/b) = ((a b) / (a b)) = 1/1 = 1.
Theorem: The solution of the equation a/b = (x/y) * (c/d) is (x/y) = (a/b) * 1/(c/d).
Corollary: The same x/y solves the equation a/b = (c/d) * (x/y).
Definition: Hence, we define division as (a/b) / (c/d) = x/y.
We define the real numbers as the compactification of the rational numbers. There are several ways to do so. The aforementioned reference employs Dedekind cuts.
Theorem: The rational-numbers are everywhere dense in the real-numbers.
Proof: Consider a sequence of decimal-numbers, with progressively more places to the right of the decimal-point. Each such number is a
rational-number; but, they may converge to any given real-number.
Corollary: The algebraic-numbers are everywhere dense in the real numbers.
Proof: The algebraic-numbers are s super-set of the rational-numbers; but, a sub-set of the real-numbers.
Theorem: The real numbers are totally ordered. However, we cannot extend the concept of ordering to anything more complicated -- vectors,
complex numbers, or matrices, for instance.
The obvious extension of addition and multiplication to the real numbers yields another field.
Definition: We call any real-number of the form (a, 1) "integer-like".
Definition: We call any real-number which is a root of a polynomial with integer-like coefficients "algebraic".
Theorem: The root of the polynomial x^2 - 2 = 0 is irrational.
We classify the real numbers as follows:
This classification obviously is a partitioning; because it satisfies the two requirements that the classes be mutually exclusive and, taken
together, exhaustive.
All that remains is to establish that each class is not null. For each of the first three classes, we have provided an example. The
fourth class requires more work.
Definition: The cardinality of a set is defined as the
collection of sets which may be placed in one-to-one correspondence to it.
Definition: The cardinality of the natural-numbers is denumerable, often called "countable". Thus, we
say that the natural numbers are denumerable. This cardinality is denoted by the capital Aleph, with a subscript of zero. It is read
"Aleph null". Aleph is the first letter of the Hebrew alphabet.
Theorem: Consider the ordered pairs (a, b) with a and b drawn from a denumerable set. These ordered pairs are
denumerable.
Corollary: The ordered pairs (a, b, c, ....) are denumerable, even if the dimensionality itself is denumerable.
Proof by Mathematical Induction.
Lemma: The interval [0, 1) of real-numbers is not denumerable.
Proof be contradiction: Assume that it is denumerable. By the definition of being denumerable, the numbers in this interval may be
placed in one-to-one correspondence with the natural-numbers. Do so and write them in ternary-notation. Go down the diagonal, from the
top-left, and change each zero-digit to a one, each one-digit to a zero, and each two-digit to -- your choice -- either a zero or a one. Read
the number along that diagonal. It is not in the original list. We have obtained a contradiction; hence, QED. A rhetorical
question: Why a binary base would not work? Because. these digits might each have been a zero originally and would have to have been
changed to a one. Then, a repeating one -- 0.111.... -- is equal to 1.0, which would be in the original list. Thus, we would not have
obtained a contradiction.
Theorem: The real-numbers are not denumerable.
Corollary: The transcendental-numbers are not denumerable.
Definition: The interval [0, 1) of real-numbers is said to have a cardinality of the power of the
continuum. This cardinality is denoted by the lower-case Greek omega.
Cantor originated the study of cardinality. His textbook remains the best introduction to the subject.
The complex numbers may be introduced by any of these equivalent methods, among others.
Since, as will be shown, these methods are isomorphic, there is no logical distinction among them. Thus, there cannot be any logical preference for any one of them. However, a particular method may be more convenient in a given context. Each method provides its own insight and pedagogical value.
For the following three theorems, we will employ the second of the foregoing methods. The proofs are obvious and hence will be omitted.
Lemma: The complex number (0,0) is the additive identity.
Theorem: Complex addition is an Abelian (i.e., commutative) group.
Lemma: The complex number (1, 0) is the multiplicative identity.
Lemma: Given a complex number (a, b). Its inverse is (a / (a^2 + b^2), - b / (a^2 + b^2)).
Theorem: Complex multiplication is an Abelian (i.e., commutative) group.
Theorem: The complex numbers constitute a field.
Theorem: The five methods cited above are isomorphic.
Proof: The proof of the isomorphism 1 <==> 2, that of 2 <==> 3, and that of 2 <==> 5 is obvious. The proof of the isomorphism 2 <==> 4 is provided in the Grundlagen reference.
Hence, regardless of the method employed to obtain them, the complex numbers are yet another field.
Theorem: The complex numbers (x, 0) are isomorphic to the real numbers x.
Proof is obvious.
Pray, observe that we did not say that (x, 0) is a real number.
Since in the field of real numbers the square-root of minus one does not exist, it would be illogical do define i as the square-root of minus one. Observe that the #1 alternative, above, sidesteps this logical hurdle by evaluating the complex numbers modulo the given defining equation. Once we have obtained complex numbers, however, we observe that the square-root of minus one becomes the set {+ i, - i} or any of its isomorphs.
The same method may be applied to the integers, to obtain the Gaussian integers.
Theorem: The Gaussian integers constitute an Abelian group under addition.
Definition: A matrix is a rectangular array of elements drawn from a division-ring, most often either the real or complex
numbers. However, since -- as will be shown presently -- the matrices themselves constitute a division-ring, a matrix may be over another set
of matrices. The Mathematical induction implicit in this recursive definition means that it suffices to prove a given theorem regarding
matrices for a 2 by 2 matrix.
Definition: A zero matrix is a matrix consisting of zero-matrices or the matrix (0).
Definition: An identity matrix is a zero-matrix with its principal diagonal replaced by identity-matrices or the matrix (1).
Definition: The determinant of a matrix is defined as the sum of the products of elements taken once from each column or
row. For a 2 by 2 matrix (a, b; c, d) its determinant is det = a d - b c.
Definition: A matrix is said to be singular iff its determinant is zero.
We define addition of matrices as the addition of corresponding elements. Thus c = a + b has the elements c(i, j) = a(i, j) + b(i, j). Obviously, the matrices a and b have to be compatible for addition; that is, the i and j have to have the same domains. Otherwise, we pad with the zero-matrices, as required.
We define multiplication of matrices, employing the Einstein convention, that c = a * b has the elements c(i, k) = a(i, j) b(j, k), with an implied summation over the domain of j. Obviously, the matrices a and b have to be compatible for multiplication; that is, their j have to have the same domain. Otherwise, we pad with the identity-matrices, as required.
Lemma: Matrices constitute an Abelian group under addition. We define subtraction in an analogous manner to that
employed for the integers.
Lemma: Matrices -- with the singular matrices excluded -- constitute a non-commutative group under multiplication. We
define division in an analogous manner to that employed for the rational numbers. In particular, the reciprocal (usually called inverse) of a
2 by 2 matrix (a, b; c, d) is inv = (d, -b; -c, a) / det(a, b; c, d).
Theorem: Matrices constitute a division-ring.
The study of matrices is called Linear Algebra. I provide a discussion and a formal presentation, which includes a shareware library.
A group may be axiomatized in four equivalent ways, by means of these four axioms:
From the foregoing first four axioms, it may be shown that the identity is both a left and a right identity and that the inverse for any given
element is both its left and right inverse.
Sometimes, we forego certain of the group axioms; but, in such a case we have to take a stronger form of the remaining axioms. In particular,
we will require both a left and a right version of the second (identity) and the third (inverse) axioms. Then, we call it a semi-group or
groupoid. We might give up the third (inverse) axiom xor the first (closure) axiom.
The order of a group is defined as its cardinality. Why a different name? Groups had order long before Cantor invented the concept of cardinality.
The infinite group of the integers under addition.
The cyclic family of groups of any natural-number order. We present the Caylay table for the first five members of this family. We display them with addition as the group operator. (It is conventional to label the operator "addition" for an Abelian; that is, commutative, group.)
| + | 0 |
| 0 | 0 |
| + | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 1 | 0 |
| + | 0 | 1 | 2 |
| 0 | 1 | 2 | 0 |
| 1 | 2 | 0 | 1 |
| 2 | 0 | 1 | 2 |
| + | 0 | 1 | 2 | 3 |
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 | 3 | 0 | 1 | 2 |
| + | 0 | 1 | 2 | 3 | 4 |
| 0 | 0 | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 | 0 |
| 2 | 2 | 3 | 4 | 0 | 1 |
| 3 | 3 | 4 | 0 | 1 | 2 |
| 4 | 4 | 0 | 1 | 2 | 3 |
Another example of an order-four group, often called the four-group. We display it with multiplication as the group operator. (It is conventional to label the operator "multiplication" for a group which is not commutative.)
| x | 1 | -1 | i | -i |
| 1 | 1 | -1 | i | -i |
| -1 | -1 | 1 | -i | i |
| i | i | -i | -1 | 1 |
| -i | -i | i | 1 | -1 |
These are the only groups of order five or less.
Definition. A subset S of G is said to be a subgroup of G if S is a group under the same operation
as that of G.
Theorem. A subset S of a group G is a subgroup; if for any elements a and b of S the sum of a and the additive inverse
of b is an element of S, namely, a + (-b) is in S.
Theorem: Transitivity of the subgroup property. If G2 is a subgroup of G1 and G1 is a subgroup of G0; then, G2 is
a subgroup of G0. Written in symbols: If G2 <= G1 <= G0 ; then, G2 <= G0.
Definition: For any given element g of G, the set C of sums g + s, where s is in S, is a right-coset of
S. We observe that for an Abelian (that is, commutative) group, the concept of a left-coset coincides with that of a right-coset.
Hence, then we speak of a coset.
Theorem: For any given subgroup S, the group G is partitioned into a collection of right-cosets.
Corollary: The order (that is, cardinality) of each right-coset is the same.
Theorem (LaGrange): The order of a subgroups S divides the order of
the group G.
Theorem: Every group G has these two subgroups: {e} (the identity element) and G (the whole group).
Definition: These two subgroups comprise the trivial subgroups.
Definition: A subgroup N of a group G is said to be a normal-subgroup of G if a) each right-coset of N also
is a left-coset of N and b) each left-coset of N also is a right-coset of N.
Theorem: For any group G, each of the trivial subgroups is a normal-subgroup of G.
Theorem: Any subgroup S of an Abelian (that is, commutative) group G is a normal-subgroup of G.
Definition: A group G is said to be simple; if its only normal-subgroups are the trivial ones.
Hey! It was not my idea to call them simple. :-)
Theorem: If an Abelian (that is, commutative) group G has a non-trivial subgroup; the group G is not simple.
Definition: A group G is said to be cyclic; if there is an element c which generates all of the elements of
G, by successive addition.
Theorem: Any cyclic group G is Abelian (that is, commutative).
Theorem: For any natural number n, there exists a unique cyclic group G of order n.
Theorem: In a cyclic group G of prime order, any element g generates all of the elements of G, by successive addition.
Lemma: A cyclic group G of prime order has only the trivial subgroups.
Proof by LaGrange theorem.
Theorem: Any cyclic group G of prime order is simple.
Definition: A ring has two operators. It is an Abelian (that is, commutative) group under addition. But, under multiplication, it only satisfies certain of the group axioms. For multiplication, we have to exclude certain elements -- at the very least the additive identity; but, sometimes some other specified elements. We need to couple the multiplication to the addition by means of the distributive axiom of multiplication over addition
Which axioms of multiplication do we desire? Even the terminology varies among authors. Progressively greater restriction --
Definition: A ring satisfies the first (closure) and fourth (associative) axioms under multiplication.
Definition: A ring-with-identity satisfies the second (identity -- both left and right) axiom as well.
Definition: A commutative ring with identity satisfies the fifth (commutative) axiom as well.
Definition: A division-ring (sometimes called a non-commutative field) satisfies all of the group axioms under
multiplication.
Definition: A field is commutative under multiplication.
Hence, a field is a division-ring, a division-ring is a ring-with-identity, and a ring-with-identity is a ring. Thus, a field is a ring-with-identity and also a ring. Likewise a division-ring is a ring.
The additive identity we designate by the glyph 0 and call "zero". The multiplicative identity we designate by the glyph 1 and call "one". Furthermore, we identify the multiplicative identity with the natural-number one.
Since the preceding set of axioms did not tell us anything about how multiplication handles the zero element, we have to prove some theorems.
Theorem: For any element a, we have 0 a = 0 and a 0 = 0.
Proof: Since 0 is the additive identity, for any element b, we have b = 0 + b. Post-multiply by the a, to obtain b a = (0 + b) a. Employ the second distributive axiom (right distribution) to obtain = 0 a + b a. By transitivity of the equality, we obtain b a = 0 a + b a. Hence, from the uniqueness of addition, we obtain 0 = 0 a. QED. An analogous proof for the right-multiplication by zero.
Theorem: There is no divisor of zero. If a b = 0; then either a or b is zero.
Proof by contra-positive: Assume that neither a nor b is zero. By the previous theorem, we have that a 0 = 0. Hence, by transitivity, we obtain a b = a 0. By the uniqueness of the multiplicative group, we obtain b = 0. This is a contradiction. QED.
Theorem: An element a of the ring commutes with its additive inverse (- a); that is, we have a (- a) = (- a) a.
Proof: Start with the additive identity zero, to obtain 0 = a 0 = a (a + (- a)) = a a + a (- a) and 0 = 0 a = (a + (- a)) a = a a + (- a) a . By transitivity of the equality, we have a a + a (- a) = a a + (- a) a . Hence, from the uniqueness of addition, we obtain a (- a) = (- a) a. QED.
Please see the reference URL for an excellent exposition.
The usual glyph for the equivalence relation is a pair of tildes, one on top of the other. For lack of such a symbol, I will employ a single tilde.
The equivalence relation is axiomatized by means of these three axioms:
Theorem: The equality relation -- symbolized by the = glyph -- is an equivalence relation.
The usual glyph for the congruence relation is a set of three parallel lines above each other.
There has to be an underlying group for a congruence relation to make sense. In this description, we will employ addition as the group
operator.
We say that a is left-congruent to b, modulo c, if either of the equalities holds. The term inside of the parentheses may consist of zero or
more elements.
Likewise, we say that a is right-congruent to b, modulo c, if either of the equalities holds.
Please observe that we have employed neither division nor even multiplication in these definitions.
Theorem: If the group is Abelian; then these two concepts coincide.
Theorem: The congruence relation is an equivalence relation.
Theorem: If a = b; then a is both left-congruent and right-congruent to b.
Theorem: In a ring, a is congruent to b, modulo c, iff there exists an integer n such that a - b = n c.
Theorem: In a ring, if a is congruent to x, modulo c, and b is congruent to y, modulo c; then (a * b) is congruent to (x *
y), modulo c.
Theorem: In a division-ring, there are no divisors of zero. That is, if a b = 0; then either a or b must be zero.
Theorem: For a prime c, the equivalence classes of the congruencies of the integers, modulo c, constitute a field of order
c. These fields are named after their discovered, Galois. We write such a field as GF(p), where p is any prime.
Examples of the first several Galois fields, one for each prime. The addition table is that for the corresponding cyclic group, as shown previously. Hence, we demonstrate only the multiplication table.
| * | 1 |
| 1 | 1 |
| * | 1 | 2 |
| 1 | 1 | 2 |
| 2 | 2 | 1 |
| * | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 |
| 2 | 1 | 0 | 2 |
| 3 | 3 | 2 | 1 |
| * | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 1 | 3 |
| 3 | 3 | 1 | 4 | 2 |
| 4 | 4 | 3 | 2 | 1 |
| * | 1 | 2 | 3 | 4 | 5 |
| 1 | 1 | 2 | 3 | 4 | 5 |
| 2 | 2 | 4 | 0 | 2 | 1 |
| 3 | 3 | 0 | 3 | 0 | 3 |
| 4 | 4 | 2 | 0 | 4 | 2 |
| 5 | 5 | 4 | 3 | 2 | 1 |
| * | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 4 | 6 | 1 | 3 | 5 |
| 3 | 3 | 6 | 2 | 5 | 1 | 4 |
| 4 | 4 | 1 | 5 | 2 | 6 | 3 |
| 5 | 5 | 3 | 1 | 6 | 4 | 2 |
| 6 | 6 | 5 | 4 | 3 | 2 | 1 |
Each of these tables proves that the corresponding Galois field does not have any divisors of zero. Of course, we already knew this property in general -- it follows from the fact that any field -- with the exclusion of the zero element -- is a group under multiplication.
For a fixed prime p, take as the ground-field, the Galois field GF(p). Consider the set of monic polynomials x^k + a x^(k-1) + ... d, of
degree at most m. Write each as a vector (0, 0, 0, ..., 0, 1, a, ..., d). Take the metric function g as the
degree of this polynomial. This is an m-dimensional vector space. Hence, there are p^m such vectors. They comprise the Galois
field GF(p^m). Furthermore, it is an Euclidean domain. Thus,
we may find the greatest common divisor of a pair a and b of polynomials and we may find s and t in GF(p) such that s a + t b = GCD(a, b). Do
not take the expression p^m literally, however. For instance, 2^3=8; but GF(2^3) is not GF(8) -- there is no such thing as GF(8).
Theorem: For any prime p and natural number m, GF(p^m) is a Galois field. It may be represented as either
Corollary: The GF(p^m) is an Eucliean domain, with the degree of the polynomial (or corresponding position in the vector)
as the metric.
My study of Galois fields continues.
We summarize the algebraic properties of the entities which we have described, so far, in this table
| natural | integers | fractions | rational | algebraic | real | complex | Gaussian integers | matrix | ||
| distributive | mult over add * | x | x | x | x | x | x | x | x | x |
| addition | Abelian group | x | x | x | x | x | x | x | x | |
| closed | x | x | x | x | x | x | x | x | x | |
| identity * | x | x | x | x | x | x | x | x | ||
| inverse * | x | x | x | x | x | x | x | x | ||
| associative | x | x | x | x | x | x | x | x | x | |
| commutative | x | x | x | x | x | x | x | x | x | |
| subtraction | closed | x | x | x | x | x | x | x | x | |
| multiplication | group | x | x | x | x | x | ||||
| closed | x | x | x | x | x | x | x | x | x | |
| identity * | x | x | x | x | x | x | x | x | x | |
| inverse * | x | x | x | x | x | |||||
| associative | x | x | x | x | x | x | x | x | x | |
| commutative | x | x | x | x | x | x | x | x | ||
| division | closed | x | x | x | x |
* As a property, each of these is both left and right.
Consider the field F = {a, b, c, ....}. Then the set of vectors V = {w, x, y, z, ....} may be axiomatized as follows:
With this definition of vectors, a set of matrices (over any field -- that is, the elements of each matrix are drawn from some field) is a vector space. Another example is the set of n-tuples of the ground-field F. In this latter case, we define the inner-product (also known as dot-product) -- symbolized by the binary infix operator . --, as the sum of the products of the corresponding components of a pair of vectors. In general, we axiomatized as the inner-product as follows:
Definition: A subset S of the vector space V is said to span V; if any v in V can be expressed as a
linear combination of vectors s in S. Namely, v is the sum of terms of the form (f s), where f is in F.
Definition: A subset S of the vector space V is said to be linearly dependent; if a linear combination of
vectors s in S, not all of the coefficients f of which are zero, sums to the zero vector.
Definition: A subset S of the vector space V is said to be linearly independent; if it is not linearly
dependent.
Definition: A spanning set B is said to be a basis of the vector space V; if the elements of B are linearly
independent.
Theorem: A vector v of V has a unique expression in terms of a given basis B of V.
Theorem: Each basis of V has the same cardinality, which we call the
dimension of V.
Definitions: A geometry is said to be positive; if for every vector v in v, we have v . v >= 0.
Otherwise, it is a pseudo-geometry. If in a positive-geometry; the only vector for which v . v = 0 is the null-vector, the
geometry is said to be positive-definite. Both the special and general relativity employ pseudo-geometries. However,
here, we confine ourselves to a positive-definite geometry.
Axiom: A norm has to satisfy the triangle-inequality, namely, for any two vectors u and v in V, ||u + v||
<= ||u|| + ||v||.
Definition: We define our norm as, for any vector v in v, the norm of v is ||v|| = sqrt(v . v).
The Gramm-Schmidt ortho-normalization algorithm. Given a basis B = {b0, b1, b2, ..., b(n-1)} of the vector space V.
The corresponding ortho-normal basis C = {c0, c1, c2, ..., c(n-1)} may be constructed as follows:
Theorem: None of the ci is null. Proof: Otherwise, the basis B would be linearly dependent.
Now, for any vector v in V, we may express it uniquely in terms of an ortho-normal basis C = {c0, c1, c2, ... c(n-1)} as a sum of terms of the form
(fi, ci), Then, the inner-product becomes the sum pf the products of the corresponding coefficients fi of the two vectors.
Theorem: This inner-product satisfies the aforementioned axioms for an inner-product.
Corollary: The value of the inner-product of two vectors is the same whether computed directly or from any basis, including
an ortho-normal basis.
Definition: A subset S of the vector space V is said to be s subspace if it is closed under vector
addition.
Definition: Two subspaces S1 and S2 of the vector space V are said to be orthogonal; if for any s1 of S1
and any s2 of S2, it is true that their inner-product s1 . s2 = 0.
Theorem: The sum of their dimensions is less than or equal to that of the vector space V.
Definition: If we have equality; the subspaces are said to be dual.
Theorem: Given a subspace S of the vector space V, a dual subspace exists and is unique.
In passing, we observe that the scalars (real or complex numbers) are zero-rank tensors, the vectors are first-rank tensors, and the matrices are second-rank tensors. There is no special name for any third or higher rank tensors.
Next, we will describe quaternions, which were discovered/invented by Hamilton. They are eminently suited to the study of Electro-Magneto Dynamics.
We adjoin the i, j, k to the field of real numbers. Hamilton provided the summary i^2 = j^2 = k^2 = i j k = - 1. The Q8 group consists of the eight elements {1, i, j, k, -1, -i, -j, -k). Its whole multiplication table is
| * | 1 | i | j | k | -1 | -i | -j | -k |
| 1 | 1 | i | j | k | -1 | -i | -j | -k |
| i | i | -1 | k | -j | -i | 1 | -k | j |
| j | j | -k | -1 | i | -j | -k | 1 | -i |
| k | k | j | -i | -1 | -k | -j | i | 1 |
| -1 | -1 | -i | -j | -k | 1 | i | j | k |
| -i | -i | 1 | -k | j | i | -1 | k | -j |
| -j | -j | k | 1 | -i | j | -k | -1 | i |
| -k | -k | -j | i | 1 | k | j | -i | -1 |
I hope that I did not make any mistakes in filling-out the foregoing table. :-)
The quaternion also may be written as an ordered quadruple (a, b, c, d). It is isomorphic to the complex matrix (a + i d, b + i c; - b + i c, a - i d), which is isomorphic to the real matrix (a, b, d, c; b; - b, a, c, -d; - d, - c, a, b; - c, d, - b, a). I suggest that you write these two matrices in their square form -- it will be much clearer to comprehend them.
More often, the multiplication table is written as
| * | 1 | i | j | k |
| 1 | 1 | i | j | k |
| i | i | - 1 | k | - j |
| j | j | - k | - 1 | i |
| k | k | j | - i | - 1 |
Did you notice the difference? Here we have adjoined (not "added") the three symbols {i, j, k} to the field of real numbers and defined their multiplication as shown. Hence, the minus sign is not part of the symbol.
Finally, we observe that the quaternion whose first element is zero is equivalent to a three-dimensional vector. The product of two such quaternions yields minus the inner (dot) product in the first element and the cross-product in the following three elements. Thus quaternions are equivalent to three-dimensional vectors. Willard Gibbs discovered/devised vectors as a simplified version of quaternions. The successive generalizations of real, complex, quaternion cannot go any further. The vectors, on the other hand, generalize to any dimensional space. Except that the cross-product of vectors is peculiar to three-dimensional space.
The quaternion (0, 0, 0, 0) is the additive identity. The quaternion (-a, -b, -c, -d) is the additive inverse of the quaternion (a, b, c, d). The quaternion (1, 0, 0, 0) is the multiplicative identity. The quaternion (1/a, 0, 0, 0) is the multiplicative inverse of the quaternion (a, 0, 0, 0). We define the quaternion (a, -b, -c, -d) as the conjugate of the quaternion (a, b, c, d). Given a quaternion q = (a, b, c, d), the product q qconj yields the square of the magnitude in the first element; the remaining elements are zero. Hence the product of qconj by the inverse of (q qconj) is the multiplicative inverse of the given quaternion. We observe that the product q qconj commutes; that is, qconj q = q qconj. We also observe that the product of a quaternion with zero in the last three elements -- (a, 0, 0, 0) -- commutes with any other quaternion. Thus, quaternions constitute a division ring.
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A glossary of terms follows, in a logical order
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Well. There you are. A comprehensive summary of the number system! Any comments or criticisms? Any corrections or additions??
Copyright (c) 2002, 3, 4, 5 by R.I. 'Scibor-Marchocki. Last revised on Monday 18-th July 2005. Broken link removed on Thursday 25-th August 2005. mailto:webmaster@rism.com