THE HYPERREAL NUMBERS

1
THE HYPERREAL NUMBERS

• We want to introduce two new types of numbers,
  namely, Infinite and infinitesimal numbers.
• What we should do here is an extension from R,
  the R, the hyper-real numbers.
• Specifically, R should be a subset of R and
  still preserve all the properties we know and
  next point is to make sure that our new field R
  is ordered. That is, for every a, b in R, one
  and only one of the followings hold
• ab, altb, or agtb.

2
Construction of Hyperreal numbers

• The first point in our agenda is to establish
  some measure of equality and order between two
  hyper-real numbers. When comparing two hyper-real
  numbers, a and b, we can form three disjoint
  sets the agreement set (set of indices of
  corresponding equal terms of the sequences), and
  a-greater set (set of indices of corresponding
  terms greater in a than b), and the b-greater
  set (whatever is left-over.).
• We want to define a definition of largeness which
  will allow us to choose exactly one of these
  three terms, so that we can choose one and only
  one large set.

3
Let us first consider when we want to say that
two infinite sequence are in fact the same
hyper-real number. Since we want a measure of
equality, we want an equivalence relation.
(Reflexivity, Symmetry, transitivity must
hold.) The agreement set of a sequence
with itself will be the set of natural numbers.
This is our first requirement of largeness 1-
)The set of natural numbers N1,2,3,.. is
large. This ensures the reflexivity. Symmetry
will be satisfied trivially, but transitivity
requires more thought.
4
Now suppose that we know that ab, bc. This
means that the agreement set between a and b,
Eab, is large, and so is the agreement set Ebc.
The intersection of Eab and Ebc is a subset of
Eac. We want Eac to be large, so that we may
say that ab and bc implies ac. This is our
next requirement of largeness. 2- ) If two
subsets of N are large, then all supersets of
their intersection are also large. In
particular, this condition entails that if A and
B are large, then so is their intersection.
Also, any superset of a large set is large. 3-
)The empty set is not large.
5

• 4- ) Set B is large if and only if its complement
  is not large.
• It is now apparent that we will need some sort of
  rigorous criterion to be able to filter out which
  subsets of N are large and which are not.
• A filter F is a set that only includes all large
  sets, where large sets are only as those
  described in condition 1 and 2 above. If a filter
  includes all large sets as defined by conditions
  1-4 above, It is called an ultra filter.

6

• 5- ) A finite set is not large.
• An ultra-filter that also satisfies condition 5
  is enough to define an order and equivalence
  relation between any two hyper real numbers.
• A non-principal ultra-filter F is a set of
  subsets of the natural numbers that satisfies the
  following
• 1. F contains the set of natural numbers N.
• 2. If F contains the subsets B and C, then it
  also contains their intersection.
• 3. If F contains the subset B, then it contains
  all of the supersets of B as well.
• 4. F does not contain a subset B if and only if
  it contains its complement.

7

• 5. F does not contain any finite subsets of N.
• With these remarks, the most important task of
  our construction is complete.
• /\
• HYPER-REAL ARITHMETIC
• \/
• Let F be a fixed non-principal ultra-filter on N.
  Define the following three relations module F,
  for any infinite sequences of real numbers a and
  b.
• ab if the agreement set between a and b is in
  F,

8

• agtb if the a-greater set is in F,
• altb if the b-greater set is in F.
• From now on, saying that a subset of N is large
  means that it is included in our ultra-filter F.
  Define the following equivalence class on an
  infinite real sequence r
• C rall real infinite sequences s such that
  sr,

9

• and a hyper-real number r is the equivalence
  class C r. The set of hyper-real numbers is
  denoted by R.
• Hyper-real arithmetic is all done term-by-term.
  To add two hyper-real numbers, we add all of the
  corresponding entries, to multiply them, we
  multiply corresponding terms. The set of real
  numbers R is a subset of R, and a member r of R
  is the equivalence class identified by the
  constant sequence r.
• (I. e, r(r, r, r, r, r, r, ..)).

10

• Let us give a few examples, where the choice of F
  is irrelevant, since I will only use the fact
  that it contains all cofinite sets.
• The hyper-reals we will use in these examples
• 0(0, 0, 0,.)
• 1 (1, 1, 1, ..)
• 2(2, 2, 2, .)
• a (1, ½, 1/3, ..)

11

• b (1, 2, 3,.,n,.)
• c (2, 4, 6, .,2n,..)
• d (0, 0, 0, .....fifty more zero, 54, 55, 56)
• f (3, 6, 9, ..,3n,.)
• Now, we can see the following order and
  equivalence relations among them.
• 0lt1, db, fgtcgtbgta.

12

• Examples of addition, subtraction,
  multiplication, division
• a0(10, ½0, .1/n0,.)
• a
• fx1(3x1, 6x1, ,3nx1,..)
• f
• b c (12, 24, 36,,n2n,..)
• (3, 6, 9, )
• f

13
References

• 1-Lectures on the Hyperreals, by robert Goldblatt
• 2-Nonstandard Analysis Theory and Application by
  Leif O. Arkeryd.
• 3-Non-standard Analysis by Abraham Robinson.
• 4-The math forum- Ask Dr. Math A Mathematical
  Essay Non-standard Analysis and the Hyper-real
  numbers.