{"id":301,"date":"2007-02-05T08:30:00","date_gmt":"2007-02-05T08:30:00","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/02\/05\/basics-real-numbers\/"},"modified":"2007-02-05T08:30:00","modified_gmt":"2007-02-05T08:30:00","slug":"basics-real-numbers","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/02\/05\/basics-real-numbers\/","title":{"rendered":"Basics: Real Numbers"},"content":{"rendered":"

What are the real numbers?<\/p>\n

Before I go into detail, I need to say up front that I hate<\/em> the term
\nreal number<\/em>. It implies that other kinds of numbers are not<\/em> real,
\nwhich is silly, annoying, and frustrating. But we’re pretty much stuck with it.<\/p>\n

There are a couple of ways of describing the real numbers. I’m going to take you through a couple of them: first, an informal intuitive description; then an axiomatic<\/em> definition, and finally, a constructive<\/em> definition.<\/p>\n

<\/p>\n

The Reals, Informally<\/h3>\n

The informal, intuitive description is the basic number line. Think about
\na line, that goes on forever in both directions. There’s one spot on it labeled 0,
\nand on either side of zero it’s labeled like a ruler. Every point on that line is
\na number. And given any two points on the line, there’s an infinite number of points
\nbetween them. The numbers that make up that line are the real numbers.<\/p>\n


\n\"number-line.jpg\"<\/p>\n

The intuition of the number line is a good one – most of the important properties
\nof the real numbers can be described in very nice, intuitive ways using the number
\nline. The ideas of addition, subtraction, ordering, and contintuity are all very
\nclear in the intuition of the number line; multiplication seems tricky, but that can be explained in terms of the number line (you can look at my posts about slide rules<\/a> to get an idea of how.)<\/p>\n

Assuming you’re not an insane math geek, you probably want to stop reading right here; I’m going to move on to more formal definitions of the reals, which most people probably don’t want to bother with. \ud83d\ude42<\/p>\n

The Reals, Axiomatically<\/h3>\n

The axiomatic<\/em> definition is, in many ways, quite similar to the definition
\nof the reals in terms of the number line – it just does it in a very formal way. An
\nAxiomatic<\/em> definition doesn’t tell you how to get the real numbers – it just
\ndescribes them in terms of a set of rules in terms of simple set theory and logic.<\/p>\n

The reals are defined by a tuple: (R<\/b>,+,0,×,1,≤), where R<\/b> is an infinite set; “+” and × are binary operators on members of R<\/b>;
\n“0” and “1” are special distinguished elements of R<\/b>; and ≤ is a binary
\nrelation over members of R<\/b>. The elements of the tuple must satisfy the following
\naxioms:<\/p>\n

    \n
  • (R<\/b>,+,×) are a field. What this means is:\n
      \n
    1. “+” and “×” are closed, total, and onto in R<\/b>.<\/li>\n
    2. “+” and “×” are commutative: a+b=b+a, a×b=b×a.<\/li>\n
    3. “×” is distributive with respect to each “+”: (3+4)×5 = 3×5 + 4×5.<\/li>\n
    4. 0 is the only identity value for “+”: For all a, a+0=a.<\/li>\n
    5. For every member i∈R<\/b>, there is exactly one<\/em>
      \nvalue -i. called the additive inverse<\/em> of i, so that
      \ni+-i=0, and for all i≠0, i≠-i.<\/li>\n
    6. 1 is the only identity value for “×”; for all a, a×1=a.<\/li>\n
    7. For every member i∈R<\/b> except 0<\/em>, there is
      \nexactly one<\/em>
      \nvalue i-1<\/sup>, called the multiplicative inverse<\/em> of i,\t\t such that i×i-1<\/sup>=1. For all i≠1, i≠i-1<\/sup>.<\/li>\n<\/ol>\n
    8. (R<\/b>,≤) is a total order:\n
        \n
      1. For all members a,b∈ R, a≤b or b≤a.<\/li>\n
      2. “≤” is transitive: if a≤b and b≤c then a≤c.<\/li>\n
      3. “≤” is antisymmetric: if a≤b, and a≠b, then ¬(b≤a)<\/li>\n<\/ol>\n<\/li>\n
      4. “≤” is compatible with “+” and “×”:\n
          \n
        1. If i≤j then (i+1)≤(j+1)<\/li>\n
        2. If i≤j, then ∀x≥0, (i×x)≤(j×x)<\/li>\n
        3. If i≤j, then ∀x≤0, (j×x)≤(i×x)<\/li>\n<\/ol>\n<\/li>\n
        4. For every subset S⊂R<\/b> where S≠∅, if S has an upper bound, then
          \nit has a least upper bound<\/em> l such that for any x∈R<\/b> that
          \nis an upper bound for S, l≤x.<\/li>\n<\/ul>\n

          That’s an extremely concise version of the axiomatic definition of reals. It describes what properties the real numbers must<\/em> have, in terms of statements
          \nthat could<\/em> be written out in first-order predicate logic form. An actual set of values that match that description is called a model<\/em> for the definition;
          \nyou can show that there are<\/em> models that match the definition, and that
          \nall of the models that match the definition are equivalent.<\/p>\n

          The Reals, Constructively<\/h3>\n

          Finally, the constructive<\/em> definition. A constructive definitions is a
          \nprocedure for creating the set of real numbers.<\/p>\n

          We’ll start with the set of natural numbers. All of the natural numbers
          \nare real numbers, with exactly the properties that they had as natural numbers.<\/p>\n

          Then we’ll add rational<\/em> numbers. A positive<\/em> rational number is defined by a pair<\/em> of non-zero<\/em> natural numbers called a ratio<\/em>. A ratio n\/d<\/em> represents a real number which, when multiplied by “d”, gives the value “n”. The set of ratios numbers constructed this way ends up with lots of duplicates – 1\/2, 2\/4, 3\/6, etc; so we ‘ll define the rationals as a set of equivalence classes<\/em> over the ratios. To define ratio equality, we need to have multiplicative inverses<\/em>: given a ratio a\/b, the its inverse (a\/b)-1<\/sup> is b\/a. Using this, we can say that two ratios n\/d and m\/e are equivalent if n\/d × (m\/e)-1<\/sup> = 1. Using this, we can also see that every natural number except zero<\/em> has a multiplicative inverse – N-1<\/sup>= 1\/N, because N×1\/N = 1.<\/p>\n

          So each of the equivalence classes of the rationals is a real number. Now we’ve got the positive rationals. Next, we add the additive inverses<\/em> of all of the reals we’ve accumulated so far: for every positive real number N, there is exactly one negative<\/em> real number -N, such that N+-N=0.<\/p>\n

          That gives us the complete set of rationals. For convenience, we’ll use Q<\/b> to represent the set of rational numbers. Now we’re kind of stuck. We know that there are<\/em> irrational numbers – we can define them axiomatically, and they fit the axiomatic definition of reals. So we need to be able to construct them. But how?<\/p>\n

          There are a bunch of tricks. The one I’m going to use is based on something called Dedekind cuts<\/em>. Dedekind cuts basically says that you can define a real number r as a pair (A,B) of sets: A is the set of rational numbers smaller<\/em> than r; B is the set of real numbers larger than<\/em> r. Because of the nature of the rationals, these two sets have really peculiar properties. The set A is a set
          \ncontaining values smaller than<\/em> some number r; but there is no largest value<\/em> or A. B is similar – there is no smallest<\/em> value in B. r
          \nis the number in the gap<\/em> between the two sets in the cut.<\/p>\n

          How does that get us the irrational numbers? Here’s a simple example: let’s define
          \nthe square root of two using a Dedekind cut: <\/p>\n

            \n
          • A = { r : r×r < 2 or r < 0}<\/li>\n
          • B = { r : r×r > 2 and r > 0}<\/li>\n<\/ul>\n

            So we can say that the set of real numbers is: the set of numbers that can be defined using Dedekind cuts of the rationals. <\/p>\n

            We know that addition, multiplication, and comparisons work nicely on the rationals – they form a field, and they are totally ordered. Just to give you a sense of how we can show that the cuts also fit into that, we can show the definitions of “=”, “+”, and “≤” in terms of cuts.<\/p>\n

            \n
            Addition<\/dt>\n
            The sum X+Y of two cuts X=(XL<\/sub>,XR<\/sub>) and Y=(YL<\/sub>,YR<\/sub>) = (ZL<\/sub>,ZR<\/sub>) where ZR<\/sub> = { x+y | x ∈ XR<\/sub> and y ∈ YR<\/sub>}.<\/dd>\n
            Equality<\/dt>\n
            Two cuts X=(XL<\/sub>,XR<\/sub>) and Y=(YL<\/sub>,YR<\/sub>) are equal if and only if XL<\/sub> ⊆ YL<\/sub> and XR<\/sub> ⊆ YR<\/sub>.<\/dd>\n
            Ordering<\/dt>\n
            Given two cuts X=(XL<\/sub>,XR<\/sub>) and Y=(YL<\/sub>,YR<\/sub>), X ≤ Y if and only if XL<\/sub> ⊆ YL<\/sub> and YR<\/sub> ⊆ XR<\/sub>.<\/dd>\n<\/dl>\n","protected":false},"excerpt":{"rendered":"

            What are the real numbers? Before I go into detail, I need to say up front that I hate the term real number. It implies that other kinds of numbers are not real, which is silly, annoying, and frustrating. But we’re pretty much stuck with it. There are a couple of ways of describing the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[74,43],"tags":[],"class_list":["post-301","post","type-post","status-publish","format-standard","hentry","category-basics","category-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-4R","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/301","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=301"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/301\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=301"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}