# Are Non-Ordered Numbers More Than Complex Numbers?

• MHB
• highmath
In summary, complex numbers are not the only numbers that are not ordered. Other examples include quaternion numbers, octonion numbers, Gaussian integers, and integers modulo n. Infinitesimal numbers are also not ordered, and there is a difference between one infinitesimal number and another. The term "infinitesimally" should be replaced with "infinitesimal" when talking about these numbers.
highmath
1. The complex number are not ordered. Which else number are not ordered?
2. Are the infinitesimally numbers are ordered numbers? It there a difference between infinitesimally number to another infinitesimally number?

highmath said:
1. The complex number are not ordered. Which else number are not ordered?
2. Are the infinitesimally numbers are ordered numbers? It there a difference between infinitesimally number to another infinitesimally number?
I'm not going to write it out again. Please see your question here. If you don't understand the response please say so on this or the other site.

-Dan

highmath said:
1. The complex number are not ordered. Which else number are not ordered?

Other examples: Quaternion numbers ($\mathbb H$), Octonion numbers, Gaussian integers ($\mathbb Z(i)$), integers modulo $n$ ($\mathbb Z_n=\mathbb Z/n\mathbb Z$ or $F_n$).

highmath said:
2. Are the infinitesimally numbers are ordered numbers? It there a difference between infinitesimally number to another infinitesimally number?

Yes.
Let $\varepsilon$ be a positive infinitesimal, and let $\omega=\frac 1\varepsilon$ be the corresponding infinity.
Then:
$$2<2+\varepsilon<2+2\varepsilon<\frac\omega 2<\omega-\varepsilon<\omega<\omega+1+\varepsilon$$

By the way "infinitesimally", ending in "ly", is an adverb and cannot modify a noun like "numbers". The corresponding adjective is "infintesmal" You want to talk about "infinitesimal numbers" not "infinitesimally numbers".

## 1. Are non-ordered numbers the same as complex numbers?

No, non-ordered numbers and complex numbers are different mathematical concepts. Non-ordered numbers refer to numbers that cannot be arranged in a specific order, such as rational and irrational numbers. Complex numbers, on the other hand, are numbers that have both a real and imaginary component, expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

## 2. How do non-ordered numbers and complex numbers differ in terms of representation?

Non-ordered numbers can be represented on a number line, while complex numbers are typically represented on a complex plane, with the real component on the x-axis and the imaginary component on the y-axis.

## 3. Can non-ordered numbers be classified as a subset of complex numbers?

No, non-ordered numbers and complex numbers are two distinct sets of numbers. While some non-ordered numbers, such as real numbers, can be expressed as complex numbers, not all non-ordered numbers can be represented as complex numbers.

## 4. What are some examples of non-ordered numbers and complex numbers?

Examples of non-ordered numbers include irrational numbers like π and √2, and rational numbers like 1/2 and 3/4. Examples of complex numbers include 2 + 3i, -4 + 5i, and 0 + 2i.

## 5. In what situations would we use non-ordered numbers versus complex numbers?

Non-ordered numbers are useful in situations where precise measurements or calculations are needed, such as in physics and engineering. Complex numbers are often used in electrical engineering, signal processing, and quantum mechanics.

• Linear and Abstract Algebra
Replies
1
Views
826
• Linear and Abstract Algebra
Replies
1
Views
996
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
11
Views
2K
• General Math
Replies
108
Views
3K
• Linear and Abstract Algebra
Replies
6
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
14
Views
1K
• Calculus
Replies
24
Views
3K
• Calculus
Replies
0
Views
2K