# Additive & Multiplicative Properties of Ordered Field

• MHB
• A.Magnus
In summary, the chapter section on Ordered Field discusses the additive and multiplicative properties, which are about closure and the equality-substitution-property. These properties ensure that the set has defined operations of addition and multiplication that yield results within the set, and that the equality-substitution-property holds true. These properties are important in defining an Ordered Field and can be illustrated with simple examples.
A.Magnus
I am reading a chapter section on Ordered Field that starts off with the additive and multiplicative properties:

$\mathscr A_1$: If $x, y \in \mathbb R, x + y \in \mathbb R$ and if $x = w$ and $y = z$, then $x + y = w + z$.

$\mathscr M_1$: If $x, y \in \mathbb R, xy \in \mathbb R$ and if $x = w$ and $y = z$, then $xy = wz$.

To my untrained eyes, they do not mean anything at all. Could somebody therefore give an intuitive significance of the two properties, perhaps with examples - please. Are they about closure?

Thank you for your time and gracious helps. ~MA

MaryAnn said:
I am reading a chapter section on Ordered Field that starts off with the additive and multiplicative properties:

To my untrained eyes, they do not mean anything at all. Could somebody therefore give an intuitive significance of the two properties, perhaps with examples - please. Are they about closure?

Thank you for your time and gracious helps. ~MA

Hey MaryAnn,

Yes, they are about closure. And they include the substitution-property of the equality-relation although I'm not exactly sure why. I haven't seen that before. Usually the substitution-property is part of the separate definition of the equality-relation and can then just be used everywhere, including as it is used here in these properties about closure.

We're starting with a set that may contain literally anything, could be symbols that resemble numbers, could be apples and pears.
Then we add the required structure to it to be able to call it an Ordered Field.
Typically we define for instance that $1+1=2$ and that $2\times 2=4$.
If the set contains apples and pears, we have to define what the result is if we add an apple and a pear together.
That is, if we want to call it an Ordered Field.

Closure means that it doesn't suffice to just say that $1+1=2$.
Instead we have to define addition for every combination of 2 elements that are in the set.
And the result must be in the set as well.

Suppose we have the 2-element set $\{0, 1\}$.
What does it take to be able to call it and Ordered Field?
Well, first off we need to define addition, and ensure that it is closed under addition.
For instance:
$$\begin{array}{c|cc}+&0&1 \\ \hline 0 & 1 & 1 \\ 1 & 0 & 0 \end{array}$$

There you go, now for any $x$ and $y$ in the set, the addition $x+y$ is defined and is included in the set.

As for the equality-substitution-property, the only way that it can fail is if $0=1$.
But then our set would only contain 1 element, which contradicts that we started with a 2-element set.
Therefore the equality-substitution-property is also satisfied.

Next we define multiplication, say with the table:
$$\begin{array}{c|cc}\times &0&1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 1 \end{array}$$
Now for any $x$ and $y$ in the set, the multiplication $x\times y$ is defined and is included in the set.
And the equality-substitution-property is satisified for the same reason as before.

Thank you for your helps! ~MA

## 1. What are the additive and multiplicative properties of an ordered field?

The additive properties of an ordered field refer to the properties of addition, such as commutativity, associativity, and the existence of a unique identity element (zero). The multiplicative properties refer to the properties of multiplication, such as commutativity, associativity, and the existence of a unique identity element (one).

## 2. How do the additive and multiplicative properties affect the order of elements in an ordered field?

The additive and multiplicative properties play a crucial role in determining the order of elements in an ordered field. These properties dictate that if a < b, then a + c < b + c and ac < bc, where c is any element in the field. This means that the order of elements is preserved under addition and multiplication.

## 3. Are the additive and multiplicative properties the same for all ordered fields?

No, the additive and multiplicative properties can vary depending on the specific ordered field. For example, in the field of real numbers, both addition and multiplication are commutative, but in the field of matrices, addition is commutative, but multiplication is not.

## 4. How do the additive and multiplicative properties of an ordered field relate to the distributive property?

The distributive property is a combination of the additive and multiplicative properties. It states that a(b + c) = ab + ac, where a, b, and c are elements of the ordered field. This property holds true for all ordered fields, and it is a fundamental property used in algebraic manipulations.

## 5. Can the additive and multiplicative properties of an ordered field be used to simplify equations?

Yes, the additive and multiplicative properties can be used to simplify equations in an ordered field. These properties allow us to manipulate equations and rearrange terms to make solving for an unknown variable easier. This is particularly useful in solving linear equations and systems of equations.

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