{"id":125,"date":"2006-08-21T21:17:07","date_gmt":"2006-08-21T21:17:07","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/08\/21\/a-glance-at-hyperreal-numbers\/"},"modified":"2006-08-21T21:17:07","modified_gmt":"2006-08-21T21:17:07","slug":"a-glance-at-hyperreal-numbers","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/08\/21\/a-glance-at-hyperreal-numbers\/","title":{"rendered":"A Glance at Hyperreal Numbers"},"content":{"rendered":"

Since we talked about the surreals, I thought it would be interesting to take a *very* brief look at an alternative system that also provides a way of looking at infinites and infinitessimals: the *hyperreal* numbers.
\nThe hyperreal numbers are not a construction like the surreals; instead they’re defined by axiom. The basic idea is that for the normal real numbers, there are a set of basic statements that we can make – statements of first order logic; and there is a basic structure of the set: it’s an *ordered field*.
\nHyperreals add the “number” ω, the size of the set of natural numbers, so that you can construct numbers using ω, like ω+1, 1\/ω, etc; but it constrains it by axiom so that the set of hyperreals is *still* an ordered field; and all statements that are true in first-order predicate logic over the reals are true in first-order predicate logic over the hyperreals.
\nFor notation, we write the real field ℜ, and the hyperreal field ℜ*.
\nThe Structure of Reals and Hyperreals: What is an ordered field?
\n——————————————————————-
\nIf you’ve been a long-time reader of GM\/BM, you’ll remember the discussion of group theory. If not, you might want to take a look at it; there’s a link in the sidebar.
\nA field is a commutative ring, where the multiplicative identity and the additive identity are not equal; where all numbers have an additive inverse, and all numbers except 0 have a multiplicative inverse.
\nOf course, for most people, that statement is completely worthless.
\nIn abstract algebra, we study things about the basic structure of the sets where algebra works. The most basic structure is a *group*. A group is basically a set of values with a single operation, “×”, called the *group operator*. The “×” operation is *closed* over the set, meaning that for any values x and y in the set, x × y produces a value that is also in the set. The group operator also must be associative – that is, for any values x, y, and z, x&times(y×z) = (x×y)×z. The set contains an *identity element* for the group, generally written “1”, which has the property that for every value x in the group, “x×1=x”. And finally, for any value x in the set, there must be a value x-1<\/sup> such that x×x-1<\/sup>=1. We often write a group as (G,×) where G is the set of values, and × is the group operator.
\nSo, for example, the integers with the “+” operation form a group, (Z,+). The real numbers *almost* form a group with multiplication, except that “0” has no inverse. If you take the real numbers without 0, then you get a group.
\nIf the group operator is also commutative (x=y if\/f y=x), then it’s called an *abelian group*. Addition with “+” is an abelian group.
\nA *ring* (R,+,×) is a set with two operations. (R,+) must be an abelian group; (R-{0},×) needs to be a group. If × is commutative (meaning (R-{0},×) is abelian), then the group is called a *commutative* group.
\nA *field* (F,+,×) is a commutative ring with two operators “+” and “×”; where the identity value for “+” is written 0, and the identity for “×” is written 1; all values have additive inverses, all values except 0 have multiplicative inverses; and 0 ≠ 1. A *subfield* (S,+,×) of a field (F,+,×) is a field with the same operations as F, and where its set of values is a subset of the values of F.
\n*Finally*, an *ordered* field is a field with a total order “≤”: for any two values x and y, either “x ≤ y” or “y ≤ x”, and if x ≤ y ∧ y ≤ x then x=y. The total order must also respect the two operations: if a ≤ b, then a + x ≤ b + x; and if 0 ≤ a and 0 ≤ b then 0 ≤ a×b.
\nThe real numbers are the canonical example of an ordered field.
\n*(The definitions above were corrected to remove several errors pointed out in the comments by readers “Dave Glasser” and “billb”. As usual, thanks for the corrections!)*
\nOne of the things we need to ensure for the hyperreal numbers to work is that they form an ordered field; and that the real numbers are an ordered subfield of the hyperreals.
\nThe Transfer Principle
\n————————
\nTo do the axiomatic definition of the hyperreal numbers, we need something called *the transfer principle*. I’m not going to go into the full details of the transfer principle, because it’s not a simple thing to fully define it, and prove that it works. But the intuitive idea of it isn’t hard.
\nWhat the transfer principle says is: *For any **first order** statement L that’s true for the ordered field of real numbers, L is also true for the ordered field of hyperreal numbers*.
\nSo for example: ∀ x ∈ ℜ, &exists; y ∈ ℜ : x ≤ y. Therefore, for any hyperreal number x ∈ ℜ*, &exists y ∈ ℜ* : x ≤ y.
\nDefining the Hyperreal Numbers
\n——————————–
\nTo define the hyperreal numbers so that they do form an ordered field with the right property, we need to two things:
\n1. Define at least one hyperreal number that is not a real number.
\n2. Show that the *transfer principle* applies.
\nSo we define ω as a hyperreal number such that ∀ r ∈ ℜ, r < ω.
\nWhat we *should* do next is prove that the transfer principle applies. But that’s well beyond the scope of this post.
\nWhat we end up with is very similar to what we have with the surreal numbers. We have infinitely large numbers. And because of the transfer principle, since there’s a successor for any real number, that means that there’s a successor for ω, so there is an ω+1. Since multiplication works (by transfer), there is a number 2×ω. Since the hyperreals are a field, ω has a multiplicative inverse, the infinitessimal 1\/ω, and an additive inverse, -ω.
\nThere is, of course, a catch. Not quite everything can transfer from ℜ to ℜ*. We are constrained to *first order* statements. What that means is that we are limited to simple direct statements; we can’t make statements that are quantified over other statements.
\nSo for example, we can say that for any real number N, the series 1,1+1,1+1+1,1+1+1,1+1+1+1,… will eventually reach a point where every element after that point will be larger than N.
\nBut that’s not a first order statement. The *series* 1, 1+1, 1+1+1, … is a *second order* statement: it isn’t talking about a simple single statement. It’s talking about a *series* of statements. So the transfer principle fails.
\nThat does end up being a fairly serious limit. There are a lot of things that you can’t say using first-order statements. But in exchange for that limitation, you get the ability to talk about infinitely large and infinitely small values, which can make some problems *much* easier to understand.<\/p>\n","protected":false},"excerpt":{"rendered":"

Since we talked about the surreals, I thought it would be interesting to take a *very* brief look at an alternative system that also provides a way of looking at infinites and infinitessimals: the *hyperreal* numbers. The hyperreal numbers are not a construction like the surreals; instead they’re defined by axiom. The basic idea is […]<\/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":[43],"tags":[],"class_list":["post-125","post","type-post","status-publish","format-standard","hentry","category-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-21","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/125","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=125"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/125\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=125"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=125"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}