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Lecture 1 - Real Number
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Overview
This lecture introduces the concept of real numbers, which are fundamental to calculus. It begins by defining natural numbers, integers, and rational numbers (fractions). The lecture then demonstrates that not all numbers are rational, using the example of the square root of 2, and introduces irrational numbers. The real number line is presented as a continuous line where every point corresponds to a real number, filling the gaps left by rational numbers. The lecture highlights the completeness axiom (or supremum axiom) as a crucial property of real numbers, which guarantees that every non-empty, bounded subset of real numbers has a least upper bound. This axiom leads to the Archimedean property, which is then applied to prove the denseness of both rational and irrational numbers within the real number system.
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Chapters
- Calculus studies functions defined on the set of real numbers (R).
- Key analytical properties of functions studied in calculus include continuity, differentiability, and integrability.
- Understanding the properties of real numbers is essential before studying these functions.
- Familiar number sets include natural numbers (N), integers (Z), and rational numbers (Q, expressed as fractions m/n where n ≠ 0).
This chapter sets the stage by explaining why understanding the nature of real numbers is a prerequisite for studying calculus and its core concepts.
Examples of rational numbers are 1/2, 2/3, and 3/4.
- The diagonal of a square with side length 1 has a length of √2.
- √2 cannot be expressed as a fraction m/n of two integers, meaning it is not a rational number.
- Proof by contradiction shows that assuming √2 is rational leads to the conclusion that it must have a common factor, contradicting the initial assumption.
- This demonstrates that numbers exist beyond the set of rational numbers, necessitating the introduction of irrational numbers.
This section proves the existence of numbers that cannot be represented as simple fractions, expanding our understanding of the number system beyond rationals.
The proof that √2 is irrational, by assuming √2 = m/n and deriving a contradiction.
- The real number line provides a geometric representation where each point corresponds to a unique real number.
- Rational numbers occupy specific points on the line, while irrational numbers fill the 'gaps'.
- The set of real numbers is defined by its 'completeness' property, ensuring no gaps exist.
- The completeness axiom (or supremum axiom) states that every non-empty, bounded subset of real numbers has a least upper bound (supremum).
This chapter introduces the visual model of the real number line and the critical completeness axiom, which distinguishes real numbers from rational numbers and is foundational for calculus.
Consider a set A of rational numbers r where r² < 2. The completeness axiom guarantees that this set has a largest value (which is √2, an irrational number).
- The Archimedean property, derived from the completeness axiom, states that for any positive real number x and any real number y, there exists a natural number n such that nx > y.
- This property implies that you can always find a multiple of a positive number that exceeds any other real number.
- It can be proven using a proof by contradiction, leveraging the supremum property.
- A direct consequence is that for any positive epsilon, there exists an n such that 1/n < epsilon.
The Archimedean property is a powerful tool that allows us to manipulate inequalities and limits, essential for understanding convergence and behavior of sequences and functions.
Given a positive number like 0.001 (epsilon), the Archimedean property guarantees we can find a natural number n (e.g., n=1001) such that 1/n is smaller than 0.001.
- The denseness of rational numbers means that between any two distinct real numbers, there exists a rational number.
- This is proven using the Archimedean property to find integers m and n such that x < m/n < y.
- Similarly, the denseness of irrational numbers means that between any two distinct real numbers, there also exists an irrational number.
- This property is demonstrated by first finding a rational number between x and y, and then constructing an irrational number using that rational number and a known irrational like √2.
These denseness properties confirm that rational and irrational numbers are intricately interwoven on the real line, with no 'space' between them, which is crucial for continuous functions.
Between any two real numbers, say 3.14 and 3.15, you can find a rational number (like 3.141) and an irrational number (like 3.14159265...). For the irrational, one might construct it as r + √2/n where r is a rational and n is chosen appropriately.
- Real numbers correspond one-to-one with points on the real line.
- The completeness (supremum) axiom is the most fundamental property, ensuring no gaps in the real number line.
- The Archimedean property, derived from completeness, is vital for limit processes.
- Both rational and irrational numbers are dense on the real line, meaning they are found everywhere between any two distinct real numbers.
This summary reinforces the core ideas about the structure and properties of real numbers that will be frequently used in the study of calculus.
Key takeaways
- The set of real numbers includes rational numbers (fractions) and irrational numbers (like √2), which cannot be expressed as fractions.
- The real number line is a continuous, unbroken line where every point represents a real number.
- The completeness axiom is the defining characteristic of real numbers, ensuring that there are no 'gaps' and that every non-empty, bounded subset has a least upper bound.
- The Archimedean property, a consequence of completeness, allows us to find multiples of a positive number that exceed any given real number.
- Rational and irrational numbers are densely packed on the real number line, meaning between any two distinct real numbers, you can always find both a rational and an irrational number.
Key terms
Real Numbers (R)Natural Numbers (N)Integers (Z)Rational Numbers (Q)Irrational NumbersCompleteness Axiom (Supremum Axiom)Upper BoundLeast Upper Bound (Supremum)Archimedean PropertyDenseness of Rational NumbersDenseness of Irrational Numbers
Test your understanding
- Why is it necessary to understand the properties of real numbers before studying calculus?
- How does the proof that √2 is irrational demonstrate the existence of numbers beyond the rational set?
- What is the significance of the completeness axiom for the structure of the real number line?
- How does the Archimedean property relate to the concept of limits or infinite processes in mathematics?
- What does it mean for rational and irrational numbers to be 'dense' on the real number line, and why is this property important?