Real number - Basic Foundations

Understanding Real Numbers What Are Real Numbers? Real numbers form the foundation of calculus and mathematical analysis. A real number is any number that measures a continuous one-dimensional quantity, such as length, temperature, time, or position on a line. What makes real numbers "continuous"? Continuity means that between any two different real numbers, no matter how close together, you can always find another real number. The numbers never have gaps—they form a seamless, unbroken collection. This distinguishes real numbers from, say, integers, which do have gaps between them. Every real number can be represented as an (almost) unique infinite decimal expansion. For example, $\frac{1}{2} = 0.5000\ldots$ or $\pi = 3.14159265\ldots$. We use the special symbol $\mathbb{R}$ to denote the set of all real numbers. Subsets of Real Numbers Not all real numbers are the same. Within $\mathbb{R}$, we can identify several important subsets based on their properties. Rational numbers are real numbers that can be expressed as a fraction of two integers. In other words, a number $r$ is rational if we can write $r = \frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. Examples include $-5 = \frac{-5}{1}$, $0$, and $\frac{4}{3}$. Rational numbers have decimal expansions that either terminate (like $0.5$) or eventually repeat (like $0.333\ldots = \frac{1}{3}$). Irrational numbers are real numbers that are not rational—they cannot be written as a fraction of two integers. Their decimal expansions never terminate and never repeat. Common examples include $\sqrt{2} = 1.41421356\ldots$ and $\pi = 3.14159265\ldots$. Together, rational and irrational numbers partition all real numbers into two disjoint groups. <extrainfo> There are also more specialized classifications. Algebraic numbers are real numbers that serve as solutions to polynomial equations with rational coefficients. For instance, $\sqrt{2}$ is algebraic because it solves $x^2 - 2 = 0$. All rational numbers are algebraic, and some irrational numbers (like $\sqrt{2}$) are also algebraic. Transcendental numbers are real numbers that are not algebraic. Famous examples include $\pi$ and $e = 2.7182\ldots$. These numbers arise naturally in mathematics but have more exotic properties than algebraic numbers. </extrainfo> Visualizing Real Numbers: The Number Line A helpful way to understand real numbers is to visualize them geometrically. The number line (or real line) is a straight line where each point corresponds to exactly one real number, and each real number corresponds to exactly one point. On the number line, integers $\ldots, -2, -1, 0, 1, 2, \ldots$ appear as equally spaced points. Rational numbers fit between these integers—for example, $\frac{1}{2}$ lies exactly halfway between $0$ and $1$. Irrational numbers also occupy specific points on the line, though they cannot be expressed as simple fractions. This geometric view reminds us that real numbers form a unified, continuous system with no "holes" or missing points. Key Vocabulary: Supremum and Infimum When working with sets of real numbers, we often need to identify their boundary points. The supremum (plural: suprema) of a set $S$, denoted $\sup S$, is the least upper bound of that set. In other words, it is: An upper bound: every element of $S$ is less than or equal to $\sup S$ The smallest such bound: no smaller number has this property For example, if $S = \{1, 2, 3\}$, then $\sup S = 3$. If $S = (0, 1)$ (the open interval from $0$ to $1$, not including the endpoints), then $\sup S = 1$, even though $1$ is not actually in $S$. The infimum of a set $S$, denoted $\inf S$, is the greatest lower bound—the largest number such that every element of $S$ is greater than or equal to it. For $S = (0, 1)$, we have $\inf S = 0$. For $S = \{1, 2, 3\}$, we have $\inf S = 1$. These concepts are essential because they allow us to describe the "boundaries" of sets precisely, even when those boundaries aren't actually in the set itself. Absolute Value The absolute value of a real number $x$, denoted $|x|$, measures the distance from $x$ to $0$ on the number line. Since distance is always non-negative: If $x \geq 0$, then $|x| = x$ If $x < 0$, then $|x| = -x$ (which is positive) For example, $|3| = 3$ and $|-5| = 5$. The absolute value always gives a non-negative result. Absolute value is useful when we care about "how far away" a number is from zero, regardless of direction. The Real Numbers as an Ordered Field At a deeper level, the real numbers have a special algebraic structure. They form an ordered field, meaning they support three key features: Addition and multiplication: You can add and multiply any two real numbers to get another real number A total order: Any two real numbers can be compared; either $a < b$, $a = b$, or $a > b$ Compatibility between operations and order: The order respects the algebraic operations This compatibility is formalized by key axioms: Order is preserved by addition: If $a < b$ and $c < d$, then $a + c < b + d$. In simpler terms, you can add the same number to both sides of an inequality without changing its direction. Positive numbers multiply productively: If $0 < a$ and $0 < b$, then $0 < ab$. This means the product of two positive numbers is always positive. These properties ensure that algebraic manipulations respect the ordering of real numbers—a crucial feature for calculus and analysis. Treating Integers and Rationals as Real Numbers A beautiful feature of the real number system is that it extends naturally from simpler number systems. There exist injective homomorphisms (structure-preserving embeddings) from: Natural numbers $\mathbb{N}$ into integers $\mathbb{Z}$ Integers $\mathbb{Z}$ into rational numbers $\mathbb{Q}$ Rational numbers $\mathbb{Q}$ into real numbers $\mathbb{R}$ These embeddings preserve arithmetic operations and order. Because of these embeddings, we can treat natural numbers, integers, and rational numbers as if they are subsets of the real numbers, without making a formal distinction. For example, we write "$5 \in \mathbb{R}$" even though $5$ started as a natural number, because the embedding justifies this identification. Notation You'll Encounter Throughout your study, you'll see: $\mathbb{R}$: the set of all real numbers $\mathbb{R}^n$: the set of all $n$-tuples (ordered lists of $n$ numbers) of real numbers. For instance, $\mathbb{R}^2$ represents all ordered pairs $(x, y)$ where $x$ and $y$ are real, which corresponds to the coordinate plane The term "real" as an adjective (e.g., "real matrix," "real polynomial") indicates that the entries or coefficients are real numbers, as opposed to complex numbers