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Numbers and Their Application - Lesson 13

It's Been Real

Lesson Overview

Reals

There are numbers on the number line which are not rational.

We already showed that the [square root of two] was irrational. We also stated that the rationals were dense—between each rational number was another rational number. However, apparently they are not continuous or complete. Somehow if we only had rational numbers on our number line, we would skip over the [square root of two] even though any decimal approximation, such as 1.414, 1.4142, ... is on our number line!

The Real Numbers are all the numbers on the number line.

Physicists like to say that they work with continuous functions with continuous derivatives (slopes), whereas mathematicians spend a lot of time worrying about whether or not a function or its derivatives are continuous. You will explore this concept further in Algebra II and Calculus. Suffice it to say now that if you can plot the function without picking up your pencil, it is continuous. A number line is such a plot.

Real Numbers are either rational or irrational.
All rational and all irrational numbers are real numbers.

The rational and irrational numbers are disjoints sets which together make up the real numbers.

The symbol denotes the set of real numbers.

      .

John Derbyshire in Prime Obsession, page 170, offers the mnemonic: Nine Zulu Queens Rule China to help remember how these nested Russian dolls are arranged.

The real numbers are nondenumerable (uncountable).

Proof by contradition:
Assume that the real numbers are denumerable (meaning, they have one-to-one correspondence to natural numbers). Then there exists a pairing of each number such that neither set has any elements left over. The following notation indicates one such pairing where the a's, b's, c's, etc. represent digits and the subscripts indicate the location to the right of the decimal point: 1<=>.a1a2a3..., 2<=>.b1b2b3..., 3<=>.c1c2c3..., etc. But we will now show that there is at least one real number which is not included in this pairing. Let N = .n1n2n3... where the n's represent any digits such that: n1 is not equal to a1, n2 is not equal to b2, n3 is not equal to c3, etc. Thus N is a real number and is different from each of the real numbers in the one-to-one correspondence. Thus the set of real numbers is non-denumerable. This proof goes back to Georg Cantor in 1874.

The Field Axioms

We introduced the group axioms in number lesson 7. Another interesting mathematical object is a ring. They have two operators usually called addition (+) and multiplication (× or • or just juxtapositioned (from Latin: to be placed side by side)). Since × and x can so easily be confused, • is often preferred. A ring is an abelian group under addition, where abelian means it is commutative (see the axiom below), and comes from a famous Norwegian mathematician named Niels Henrik Abel (1802–1829). (Abel is generally pronounced with a long e sound and accented second syllable.) A ring must also be closed under multiplication, and must also be associative (for an associative ring). There is also an axiom to interrelate addition and multiplication (see the distributive property below). The rings of interest to us have a unit element which will serve as our multiplicative identity (1), and are commutative under multiplication. A field is just another mathematical object with more structure than a ring.

If the elements different from 0 in a commutative ring with unit element form
an abelean group under multiplication, the ring is called a field.

Zero must be excluded because it does not have a multiplicitive inverse—division by zero is not allowed. The only fields we will be concerned with are the binaries (0,1), the rational numbers, the real numbers, and in lesson 15, the complex numbers. The eleven field axioms are listed below and are true for any real numbers, represented below by x, y, and z.

Closure under addition: real numbers are closed under addition.

That is, adding any pair of real numbers will result in a unique real number. 1 + 1 = 2. Always. This also means we stay inside the set.

Closure under multiplication: real numbers are closed under multiplication.

Multiplying any real number pair together will result in a unique real number. 2 × 2 = 4 and never 5.

Additive Commutativity: x + y = y + x.

Order does not matter. You can add a column of numbers from the top or from the bottom.

Multiplicative Commutativity: x • y = y • x.

The root word commute is commonly used to describe exchanging places, like going forth and back between home and work.

Additive Associativity: (x + y) + z = x + (y + z).

Multiplicative Associativity: (xy)z = x(yz).

Distributivity: Multiplication distributes over addition. x(y + z) = xy + xz

Additive Identity Element: The additive identity is a unique element,
which can be added to any element without altering it.
The additive identity is zero (0).   x + 0 = x.

We have both a left and right additive identity element and they are the same: x + 0 = x = 0 + x.

Multiplicative Identity Element: The multiplicative identity is unique; it is one (1).   x•1 = x.

We also have both a left and right multiplicative identity element and they are the same: x•1 = x = 1•x.

Additive Inverses: For every real number there exists a unique inverse, such that when added together, the result is the additive identity (0). The additive inverse is the opposite (negative) of the given real number,   x + (-x) = 0.

Multiplicative Inverses: For every real number not equal to zero there exists a unique inverse, such that when multiplied together, the result is the multiplicative identity.   x • x-1 = 1.

x-1 is a general designation for an inverse, but here denotes the multiplicative inverse or reciprocal (1/x).

Reflexive, Symmetric, Transitive, Trichotomy

In addition to the field axioms, real numbers satisfy additional important axioms or properties.

Reflexive Property: If x is a real number, then x = x.

Operations which are reflexive look the same in a mirror. This axiom establishes that a variable stands for the same number wherever it appears in an expression. Order is not reflexive: 5 < 5 is a counterexample.

Symmetry: If x = y, then y = x.

Notice that symmetry is true for only the "=" sign. Order relationships, such as < and >, cannot have the numbers rearranged without changing the meaning. For example, 4 < 5 is not the same as 5 < 4.

If x = y and y = z, then x = z.
Transitivity: If x < y and y < z, then x < z.
If x > y and y > z, then x > z.

The prefix trans- means across like rapid transit quickly takes you across a city. An easy way to remember which of these three properties is which is to note that the initial letters RST are in alphabetic order and corresponds to 123 or the number of variables which appear in the description!

Trichotomy: If x and y are two real numbers, then exactly one of the following must be true:
y < x,     y > x,   or   y = x.

Trichotomy means to section or cut into three pieces. Please note it is three pieces not two because the reals are continuous (not just dense).

Orders of Infinity

George Cantor introduced transfinite numbers back in the 1870's as a way to deal with the fact that not all infinite sets are equivalent. The cardinality of the integers, rational numbers, even algebraic numbers is designated the first order of infinity and assigned the name aleph null (0) where aleph () is the first Hebrew letter. However, the cardinality of the real numbers or such important subsets as the transcendentals or irrationals is beyond that of a countable infinity. This cardinality became known as the cardinality of the continuum and was designated by c. By forming power sets (the set of all subsets of a given set), Cantor was able to form higher order infinities. These became known as 0, 1, 2, .... where 20 =1. Cantor believed this first aleph (1) was the cardinality of the continuum and was sometimes able and sometimes not able to prove it. This may well have been a contributing factor to his mental instabilities. This hypothesis (20 =1=c) became known as the Continuum Hypothesis (CH) (see this link). This power set relationship was later generalized to apply to any successive pair of alephs and became known as the generalized continuum hypothesis. Only much later was it shown that CH is independent of the usual axioms of set theory and was thus unproveable (Kurt Gödel, 1937 and Paul Cohen, 1963). The method used by Cohen became known as forcing.

While we are on the topic, another axiom, the axiom of choice (AC) suffered a similar fate, being proved independent of the rest of mathematics (Gödel, 1940 and Cohen, 1963). However, unlike CH, it is still routinely, but not universally, used in the development of mathematics. See this link for further details. One last related topic is Gödel's Incompleteness Theorem, 1931, which showed that there were things within any formal system which were neither provable nor not provable. These recent developments make one question the very merits of establishing a rigorous foundation for mathematics.

The Axioms of Set Theory

Following are the axioms of set theory generally used in mathematics. They were designed by Ernst Zermelo, et al at the beginning of the 20th century. This minimal set of assumptions leads to a consistent body of mathematical knowledge, including the natural, real, and complex numbers along with their properties and arithmetic. Along with other axioms, the areas of geometry, algebra, topology, etc. can also be formed. Georg Cantor developed set theory but implicitly assumed many of these.

  1. Existence: There exists at least one set. (The empty set can be chosen. The set containing the empty set would then be constructed...)
  2. Extension: Two sets are equal iff they have the same elements.
  3. Specification: To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This axiom leads to Russell's paradox.
  4. Pairing: For any two sets there exists a set to which they both belong.
  5. Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection.
  6. Powers: For each set there exists a collection of sets that contains among its elements all the subsets of the given set.
  7. Infinity: There exists a set containing 0 and containing the successor of each of its elements.
  8. Choice: For every set A there is a choice function, f, such that for any non-empty subset B of A, f(B) is a member of B.

Surreal Numbers

John Conway invented surreal numbers in recent years. These numbers have multiple infinities and many other unusual but useful properties. Donald Knuth wrote a novellete to help explain these numbers even before the technical paper was published.

Continuity

Our macroscopic existence means that most of our physical observations are continuous. Thus most physical phenomina is modelled by continuous functions with continuous derivatives (slopes). Some cutting edge models attempting to unify gravity with quantum mechanics while retaining general relativity (as in loop quantum gravity, unlike string or M-theory) treat space as quantized. However, the mathematical treatment of functions is riddled with concerns about continuity. Discontinuities fall into two catagories: removable and nonremovable. We stated before that continuous functions can be drawn without having to lift your pencil from the paper. For removable discontinuities one must only avoid an occasional point whereas nonremovable discontinuities involve moving your pencil up or down. The function x/x would have a removeable discontinuity at x=0, whereas |x|/x would have a nonremoveable discontinuity. The definition of continuity is wrapped up with the concept of limit and will not be discussed further here.

Paradoxes

We already encountered various paradoxes in lesson 2 and lesson 5. More needs to be put here, but hasn't been yet! Some of the most ancient paradoxes are: Dichotomy, Achilles, Arrow, and Stadium.

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