Which property must be Archimedean in the context of ordered field?

Field of the trigonometric functions

Which property must be non-Archimedean in the context of ordered field?

Field of the trigonometric functions

What Do You Know About Axiom Of Archimedes?

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1/10 Questions

Which structure holds the principle?
- Linear structure
- Algebraic structure
- Trigonometric structure
- Exponential structure

About This Quiz

Thanks to the Axiom V of Archimedes’ On the Sphere and Cylinder, the axiom of Archimedes (or Archimedean axiom, Archimedes' axiom, Archimedes' lemma, or the continuity axiom) is one of the most widespread concepts in mathematical analysis and the field of abstract algebra.
It states that one magnitude can find a multiple of either of two magnitudes which will exceed See morethe other, given the magnitudes have a ratio.

What Do You Know About Axiom Of Archimedes? - Quiz

2.

What is the property of the element?
- Infinitely small
- Large
- Equal
- Small
Correct Answer

A. Infinitely small

Explanation

The property of the element is described as "infinitely small." This implies that the element is extremely tiny or minuscule in size.

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3.

Which property must be Archimedean in the context of ordered field?
- Field of the trigonometric functions
- Field of real numbers
- Field of integers
- Field of rational functions
Correct Answer

A. Field of real numbers

Explanation

The property that must be Archimedean in the context of an ordered field is the field of real numbers. The Archimedean property states that for any two positive numbers a and b, there exists a positive integer n such that na > b. This property is important in the real numbers because it allows for the concept of limits and the ability to compare and order real numbers. The other options, such as the field of trigonometric functions, integers, and rational functions, do not necessarily have this property.

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4.

In which decade was the concept named? The...
- 1870s
- 1880s
- 1890s
- 1900s
Correct Answer

A. 1880s

Explanation

The concept named was introduced in the 1880s.

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5.

What do we call a structure in which neither of the non-zero elements is infinitesimal with respect to the other?
- Non-Archimedean
- Archimedean
- Linear-Archimedean
- Algebraic-Archimedean
Correct Answer

A. Archimedean

Explanation

An Archimedean structure is one in which neither of the non-zero elements is infinitesimal with respect to the other. In other words, there is no element that is infinitely smaller or larger than another element in the structure. This concept is important in mathematics and physics, as it allows for the comparison and measurement of quantities.

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6.

Who gave the axiom of Archimedes its name?
- Otto Stolz
- Reynald Harvey
- Paul Autz
- Charles Puma
Correct Answer

A. Otto Stolz

Explanation

Otto Stolz is credited with giving the axiom of Archimedes its name.

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7.

What do we call a structure in which a pair of non-zero elements is infinitesimal with respect to the other?
- Non-Archimedean
- Archimedean
- Linear-Archimedean
- Algebraic-Archimedean
Correct Answer

A. Non-Archimedean

Explanation

A non-Archimedean structure refers to a mathematical system where there exist pairs of non-zero elements that are considered infinitesimal in comparison to the other elements. In such a structure, the concept of size or magnitude is not solely determined by the absolute value of the elements, but rather by their relative relationship to each other. This is in contrast to an Archimedean structure, where the concept of size is determined solely by the absolute value of the elements. The terms "Linear-Archimedean" and "Algebraic-Archimedean" are not relevant to the concept described in the question.

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8.

The theory plays an important role in which of the following?
- Global field
- Exponential functions
- Local field
- Trigonometric functions
Correct Answer

A. Local field

Explanation

The theory plays an important role in the local field.

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9.

For linearly ordered group G, what do x and y denote?
- Positive elements
- Negative elements
- Integers
- Real numbers
Correct Answer

A. Positive elements

Explanation

The terms "x" and "y" in a linearly ordered group G denote positive elements. A linearly ordered group is a mathematical structure where the elements can be arranged in a linear order, and in this case, the elements represented by "x" and "y" are specifically the positive elements of the group.

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10.

Which property must be non-Archimedean in the context of ordered field?
- Field of the trigonometric functions
- Field of real numbers
- Field of integers
- Field of rational functions
Correct Answer

A. Field of rational functions

Explanation

The property that must be non-Archimedean in the context of an ordered field is the field of rational functions. An ordered field is said to be non-Archimedean if there exists an element that is greater than all the natural numbers. In the field of rational functions, there are elements that can be infinitely large, such as 1/x, where x approaches 0. This violates the Archimedean property, making the field of rational functions non-Archimedean.

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