Which of these is not a property of ordered field?

If x is infinitesimal its inverse is also infinite

If x is infinitesimal and r is rational their products are also infinite

Which of these statements is the Axiom of Archimedes?

Let x be any element of K. Then there exists a natural number n such that n > x

Let x be any element of K. Then there exists a rational number r such that r > x

Let x be any element of K. Then there exists a natural number n such that n = x

Let x be any element of K. Then there exists a natural number n such that n < x

Which of the following is not equivalent characterizations of Archimedean fields?

Every nonempty open interval of K contains a natural number

The natural numbers are cofinal in K

For any x in K the set of integers greater than x has a least element

A Quiz About Axiom Archimedes

1.

Who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes on Spheres and Cylinders?
- A.
  
  Holms
- B.
  
  Otto Stolz
- C.
  
  Ostrowski
- D.
  
  Archimedes
Correct Answer
B. Otto Stolz

Explanation
Otto Stolz gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes on Spheres and Cylinders.

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2.

Which of these is not a property of ordered field?
- A.
  
  It contains irrational number
- B.
  
  If x is infinitesimal its inverse is also infinite
- C.
  
  It contains rational number
- D.
  
  If x is infinitesimal and r is rational their products are also infinite
Correct Answer
A. It contains irrational number

Explanation
An ordered field is a mathematical structure that satisfies certain properties, such as the existence of rational numbers and the preservation of order under addition and multiplication. The statement "It contains irrational numbers" is not a property of an ordered field because an ordered field can contain both rational and irrational numbers. Therefore, the correct answer is that an ordered field does not necessarily contain irrational numbers.

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3.

When a pair of non-zero elements is infinitesimal with respect to the other, what is it called?
- A.
  
  Strowski theorem
- B.
  
  Archimedean
- C.
  
  Non-Archimedean
- D.
  
  Heron's theorem
Correct Answer
C. Non-Archimedean

Explanation
A pair of non-zero elements is said to be infinitesimal with respect to the other when the magnitude of these elements is extremely small compared to the magnitude of the other elements. This concept is known as non-Archimedean. Non-Archimedean numbers are used in mathematics to describe situations where infinitely small quantities are considered. Strowski theorem, Archimedean, and Heron's theorem are unrelated to the concept of infinitesimal elements.

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4.

In which of the following is the Archimedean principle very useful?
- A.
  
  Cinematography
- B.
  
  Local field
- C.
  
  Mechanics
- D.
  
  Medicine
Correct Answer
B. Local field

Explanation
The Archimedean principle is very useful in the field of local field. This principle states that when an object is immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces. In local field, such as hydrology or geophysics, this principle is often applied to calculate the buoyant force acting on objects submerged in water or other fluids. By understanding the Archimedean principle, scientists and engineers can accurately determine the behavior and stability of objects in local fields, making it an essential concept in this field.

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5.

When was the Archimidean property named The Axiom Archimedes?
- A.
  
  1970s
- B.
  
  1980s
- C.
  
  1880s
- D.
  
  1870s
Correct Answer
C. 1880s
6.

Which of these statements is the Axiom of Archimedes?
- A.
  
  Let x be any element of K. Then there exists a rational number r such that r > x
- B.
  
  Let x be any element of K. Then there exists a natural number n such that n = x
- C.
  
  Let x be any element of K. Then there exists a natural number n such that n < x
- D.
  
  Let x be any element of K. Then there exists a natural number n such that n > x
Correct Answer
D. Let x be any element of K. Then there exists a natural number n such that n > x

Explanation
This statement is the Axiom of Archimedes because it states that for any element x in the set K, there exists a natural number n which is greater than x. This axiom is fundamental in the field of real numbers and is used to prove various properties and theorems. It essentially states that the set of natural numbers is unbounded, meaning that there is no natural number that is greater than all other natural numbers.

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7.

What does K represent in Axiom of Archimedes?
- A.
  
  Ordered field
- B.
  
  Strowski determinant
- C.
  
  Axiom constant
- D.
  
  Archimedes constant
Correct Answer
A. Ordered field

Explanation
K represents an ordered field in the Axiom of Archimedes. The Axiom of Archimedes states that for any positive real number x, there exists a positive integer n such that nx is greater than any given real number. An ordered field is a mathematical structure that satisfies certain properties, including the existence of a total order relation and the ability to perform addition, subtraction, multiplication, and division. Therefore, K in the Axiom of Archimedes represents an ordered field where this property holds true.

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8.

Which structure holds the Archimedean principle?
- A.
  
  Index structure
- B.
  
  Strowski Structure
- C.
  
  Algebraic structure
- D.
  
  Rational number structures
Correct Answer
C. Algebraic structure

Explanation
An algebraic structure is a mathematical structure that consists of a set of elements and a set of operations defined on those elements. The Archimedean principle states that for any two positive real numbers, there exists a positive integer multiple of one number that is greater than the other number. This principle is applicable in various branches of mathematics and physics. Therefore, it is reasonable to say that the Archimedean principle is held within an algebraic structure.

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9.

Which property must be non-Archimedean in the context of ordered field?
- A.
  
  Irrational function
- B.
  
  Rational function
- C.
  
  Real number function
- D.
  
  Strowski constant
Correct Answer
B. Rational function

Explanation
In the context of an ordered field, the property that must be non-Archimedean is a rational function. A rational function is defined as the ratio of two polynomial functions, where the degree of the numerator is less than or equal to the degree of the denominator. This property is non-Archimedean because it does not satisfy the Archimedean property, which states that for any two positive numbers, there exists a positive integer multiple of the first number that is greater than the second number.

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10.

Which of the following is not equivalent characterizations of Archimedean fields?
- A.
  
  The natural numbers are cofinal in K
- B.
  
  For any x in K the set of integers greater than x has a least element
- C.
  
  Zero is the infimum of a set
- D.
  
  Every nonempty open interval of K contains a natural number
Correct Answer
D. Every nonempty open interval of K contains a natural number

Explanation
Archimedean fields are characterized by the property that the natural numbers are cofinal in the field, meaning that for any element x in the field, there exists a natural number n such that n is greater than x. Additionally, Archimedean fields have the property that for any element x in the field, the set of integers greater than x has a least element. This means that the set of integers greater than x is bounded below and has a minimum element. Another equivalent characterization of Archimedean fields is that zero is the infimum (greatest lower bound) of a set. However, the statement that every nonempty open interval of K contains a natural number is not an equivalent characterization of Archimedean fields.

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