MthT 491 Distributive Properties and Negative Numbers

To emphasize the important role of the distributive property in dealing with positive and negative numbers, we construct a system of Numbers, [weird] numbers, which

•

satisfies all the properties of an ordered field except for the distributive property:

P9

For all a, b, c,

a times (b + c) = (a times b) + (a times c) = a times b + a times c.

•

the product of two [weird] negative numbers is a [weird] negative number.

For the time being we will denote the numbers we are using to by Numbers. We shall list the primitive properties - that is, develop a minimal list of properties from which results can be deduced.

We shall assume there is a set Numbers, with binary operations + (plus, addition) and · (times, multiplication) defined.

We start with a Commutative Group, (G, +) - a set of numbers G, with a binary operation + (plus, addition), which satisfies

Properties of +

P1

For all a, b, c, in G,

a + (b + c) = (a + b) + c

P2

There is a number 0 in G such that for all a,

a + 0 = 0 + a = a.

P3

For all a, there is a number −a such that

a + (−a) = (−a) + a = 0.

P4

For all a, b,

a + b = b + a.

Examples include

•

Z, the set of all integers.

•

R, the set of real numbers.

•

Q, the set of rational numbers.

•

C, the set of complex numbers.

•

Z + iZ, the set of "complex integers."

Temporarily, we will assume

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G is nontrivial in the sense that there is an element U ∈ G, U ≠ 0.

We now define weird multiplication, ∗, on G by

For all a, b ∈ G,

a ∗b ≡ a + b − U.

Properties of ∗

P5

For all a, b, c,

a ∗(b ∗c) = (a ∗b) ∗c

Proof.

a ∗(b ∗c) = a + (b + c − U) − U = … = (((a + b) − U) + c) −U = ((a ∗b) + c) − U = (a ∗b) ∗c.

P6

There is a number 1 ≠ 0 such that for all a,

a ∗1 = 1 ∗a = a.

Proof.

Let 1 ≡ U.

1 ∗a = U + a − U = a = a ∗U.

P7

For all a ≠ 0, there is a number a⁻¹ such that

a ∗(a−1) = (a−1) ∗a = 0.

Proof.

For any a, let

a−1 ≡ −a + U + U, a ∗a−1 = a + (−a + U + U) − U = U

P8

For all a, b,

a ∗b = b ∗a.

N.B. With the multiplication ∗,

0 ∗0 = 0 + 0 − U = − U ≠ 0. (− U)∗(− U) = − U − U − U.

If U ≠ −U,

− (0 ∗0) = −(− U) = U ≠ (−0) ∗0 = 0 * 0 = −U.

The structure (G, +, ∗) satisfies all the properties of a field , except the glue which relates multiplication and addition, the distributive property :

Property of · with +

P9

For all a, b, c,

a ·(b + c) = (a ·b) + (a·c) = a ·b + a·c.

Positive Numbers and Order

Within our set of numbers, we say that a collection of numbers, P, is a positive set, or a set of positive numbers if P satisfies P10 - P12:

P10

For every a, one and only one of the following holds:

(i)

a = 0,

(ii)

a is in the collection P,

(iii)

−a is in the collection P.

P11

If a and b are in the collection P, then a + b is in the collection P.

P12

If a and b are in the collection P, then the product of a and b is in the collection P.

If P is a given positive set, we define inequalities or P-inequalities by:

a < b (a < P b) iff b −a ∈ P.

Weird Example (Z, +, ∗)

As an example, we consider (Z, +, ∗), U = 1, the usual "1". The system

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Satisfies P1 - P8.

•

Does not satisfy P9. Give a counterexample!

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0 ∗0 ≠ 0.

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There are nonzero a and b such that a ∗b = 0. Give examples.

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The set can be ordered in such a way that "1" is not positive.

In our example (Z, + , ∗), we take as a weird positive set

P∗ = { −1, −2, …},

the usual set of negative integers. The weird negative integers are

N∗ = { 1, 2, …},

the usual set of positive integers.

We have P10 (trichotomy).

Now verify P11 and P12. A typical element of P_∗ is of the form −a, with a a usual positive integer. If (−a), (− b), are in P_∗, then

(−a) + (− b) = − (a + b) ∈ P∗, (−a) ∗(− b) = − (a + b) − 1 = −(a + b + 1) ∈ P∗.

Here the weird product of two weird negative integers is always weird negative : For a and b weird negative , i.e., usual positive integers

a ∗b = a + b − 1

is a usual positive integer, i.e., weird negative .

More Examples

We consider the even and odd integers. We know that

odd+ even = odd, even+ even = even, odd·odd = odd, odd·even = even.

Thus the role of zero for addition is played by even .

We construct the addition table:

+ (plus)         odd         even         odd         even                 even

The usual multiplication table is:

· (times)         odd         even         odd                         even

The weird multiplication table is:

∗ (weird)         odd         even         odd         even         odd         even         odd         even

The role of 1 for weird multiplication is played by odd .

Note that

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