1.
Who gave the axiom of Archimedes its name?
Correct Answer
A. Otto Stolz
Explanation
Otto Stolz is credited with giving the axiom of Archimedes its name.
2.
Which structure holds the principle?
Correct Answer
B. Algebraic structure
Explanation
An algebraic structure holds the principle because it is a mathematical structure that consists of a set of elements along with operations that can be performed on those elements. These operations follow certain rules and properties, such as closure, associativity, and distributivity. Algebraic structures are used to study and analyze various mathematical concepts, including equations, functions, and transformations. They provide a framework for understanding and manipulating mathematical objects, making them essential in many branches of mathematics and its applications.
3.
What is the property of the element?
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.
4.
What do we call a structure in which a pair of non-zero elements is infinitesimal with respect to the other?
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.
5.
What do we call a structure in which neither of the non-zero elements is infinitesimal with respect to the other?
Correct Answer
B. 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.
6.
The theory plays an important role in which of the following?
Correct Answer
C. Local field
Explanation
The theory plays an important role in the local field.
7.
Which property must be Archimedean in the context of ordered field?
Correct Answer
B. 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.
8.
Which property must be non-Archimedean in the context of ordered field?
Correct Answer
D. 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.
9.
In which decade was the concept named?
The...
Correct Answer
B. 1880s
Explanation
The concept named was introduced in the 1880s.
10.
For linearly ordered group G, what do x and y denote?
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.