Problem 6 - A Primer of Real Analysis

Question

Show that \(\mathbb{R}\) is an ordered field, that is, verify the following: a. For any \(a, b \in \mathbb{R},\) one and only one of the following must hold: (i) \(ab\) b. If \(a, b, c \in \mathbb{R}\) with \(a0\) and \(b>0,\) then \(a b>0\).

Step-by-Step Solution

Answer

\(\mathbb{R}\) is an ordered field as it satisfies trichotomy, transitivity, and both addition and multiplication properties for order.

Step 1: Trichotomy Law

To verify part (a), for any real numbers \(a\) and \(b\), exactly one of the following must be true: (i) \(a < b\), (ii) \(a = b\), or (iii) \(a > b\). In the field of real numbers, each number can be compared to another to determine their order relation uniquely. The properties of the real numbers ensure that these operations are well-defined.

Step 2: Transitivity

For part (b), if \(a < b\) and \(b < c\), then it must follow that \(a < c\). This is the transitive property of inequality in real numbers. If \(a\) is less than \(b\), and \(b\) is less than \(c\), it logically follows that \(a\) is less than \(c\). This property ensures consistent ordering in sequences of inequalities.

Step 3: First Addition Property

Part (c) requires showing if \(a < b\), then \(a + c < b + c\) for any real number \(c\). The addition of real numbers preserves the inequality because the same quantity is added to both sides of the inequality. If \(c\) were added to values where \(a < b\), \(a + c\) must remain less than \(b + c\).

Step 4: Multiplication Property

Lastly, for part (d), if \(a > 0\) and \(b > 0\), then \(a \cdot b > 0\). In the ordered field of real numbers, the product of two positive numbers remains positive. This is fundamental to the properties of positivity in the multiplication of real numbers.

Key Concepts

Real Numbers

Real numbers are an essential concept in mathematics, representing a continuous and ordered set of numbers. They include various types of numbers such as:

• Natural numbers (e.g., 1, 2, 3...)
• Integers (e.g., -1, 0, 1...)
• Rational numbers (fractions like 1/2, 3/4)
• Irrational numbers (e.g., π, √2)

The real number system is vast and infinitely dense, meaning between any two real numbers, there exists another real number.
This incredible diversity allows real numbers to model continuous quantities in the real world, such as distance, time, and temperature.
Moreover, real numbers form an ordered field, meaning they adhere to both the field properties (like addition and multiplication) and order relations.

Trichotomy Property

The trichotomy property is a distinct feature of real numbers within an ordered field. It states that for any two real numbers, say \(a\) and \(b\), one, and only one, of the following three relations holds:

• \(a < b\)
• \(a = b\)
• \(a > b\)

This property ensures that real numbers are completely ordered, providing a clear basis for comparison between any two numbers.
In essence, the trichotomy property enables us to definitively determine the relationship between two numbers, ruling out any ambiguity.
This clear distinctivity is foundational for further properties such as the transitive property and rules of inequalities.

Transitive Property

Transitivity is another critical property of inequalities in real numbers. It claims that if one number is less than a second number, and this second number is less than a third number, then the first number must be less than the third.
In symbolic form, if \(a < b\) and \(b < c\), then \(a < c\). Transitivity ensures that inequalities follow a logical progression and maintain a consistent order.
This property prevents contradictions in ordered sequences and allows us to make logical deductions in mathematical reasoning.
For example, if John is shorter than Mary, and Mary is shorter than Paul, then John must logically be shorter than Paul. This rule applies perfectly within the world of ordered mathematics.

Inequalities

Inequalities are a way of expressing that one quantity is less than or greater than another. They play a significant role in mathematics, economics, physics, and many other fields.
In mathematics, they often involve real numbers and are governed by several key properties:

• If \(a < b\), then adding the same number to both sides maintains the inequality: \(a + c < b + c\).
• Similarly, for multiplication, if \(a > 0\) and \(b > 0\), then \(a \cdot b > 0\). This guarantees positive products when dealing with positive real numbers.

Understanding these rules allows us to solve equations and inequalities effectively, supporting problem-solving in diverse mathematical contexts.
By working with inequalities consistently and accurately, we can understand and explore complex relationships in quantitative data.