Real number Study Guide

📖 Core Concepts Real Number ($\mathbb{R}$) – a continuous one‑dimensional quantity; any two values can differ by an arbitrarily small amount. Subsets – Rational $ \in \mathbb{Q}$: fraction of two integers (e.g., $-5$, $\tfrac{4}{3}$). Irrational: not rational. Algebraic: root of a polynomial with rational coefficients (includes all rationals, e.g. $\sqrt2$). Transcendental: not algebraic (e.g. $\pi$, $e$). Ordered Field – $\mathbb{R}$ supports $+$, $\times$, and a total order $\le$ compatible with these operations. Dedekind Completeness – every non‑empty bounded‑above subset of $\mathbb{R}$ has a least upper bound (supremum) in $\mathbb{R}$. Metric Completeness – every Cauchy sequence of real numbers converges to a real limit. Supremum / Infimum – $\sup S$ = least upper bound; $\inf S$ = greatest lower bound. Archimedean Property – for any $x\in\mathbb{R}$ there exists $n\in\mathbb{Z}$ with $n>x$. Decimal (or base‑$b$) Expansion – $x=\sum{i=-\infty}^{k} di b^{\,i}$; terminating expansions have a twin representation ending in infinite $b-1$’s. Cardinality of the Continuum – $|\mathbb{R}|=\mathfrak c$, strictly larger than $\aleph0$ (the size of $\mathbb{N}$). --- 📌 Must Remember $\mathbb{R}$ = unique Dedekind‑complete ordered field (up to isomorphism). Least Upper Bound Property distinguishes $\mathbb{R}$ from $\mathbb{Q}$. Metric ⇔ Dedekind completeness (together with order field axioms). Archimedean: no infinitesimal real numbers; $\forall x\;\exists n\in\mathbb{Z}: n>x$. Decimal non‑uniqueness: $0.5000\ldots = 0.4999\ldots$ (terminating vs. repeating $b-1$). Density: $\mathbb{Q}$ is dense in $\mathbb{R}$ (every open interval contains a rational). Uncountability: Cantor’s diagonal argument → $\mathbb{R}$ cannot be listed. Computable reals: countable; “almost all’’ reals are non‑computable. --- 🔄 Key Processes Finding a Supremum Verify set $S$ is non‑empty and bounded above. Show any upper bound $U$ satisfies $U\ge\sup S$. Use completeness: $\sup S$ exists in $\mathbb{R}$. Proving Cauchy ⇒ Convergent (Metric completeness) Given $(xn)$, for each $\varepsilon>0$ find $N$ with $|xn-xm|<\varepsilon$ for $n,m\ge N$. Invoke completeness ⇒ $\exists L\in\mathbb{R}$ with $xn\to L$. Constructing a Real via Decimal Truncations For each $n$, take finite sum $Tn=\sum{i=-n}^{k} di10^{\,i}$. Set $x=\sup\{Tn\}$. Dedekind Cut Construction Split $\mathbb{Q}$ into $L$ (no greatest element) and $U$ (no least element) with $L<U$. The cut $(L,U)$ defines a real number. Cauchy‑Sequence Construction Form equivalence classes of rational Cauchy sequences where termwise difference $\to0$. Each class = a real number. --- 🔍 Key Comparisons Rational vs. Irrational – rational = fraction of integers; irrational cannot be expressed that way. Algebraic vs. Transcendental – algebraic solves a rational‑coeff polynomial; transcendental does not (e.g., $\sqrt2$ vs. $\pi$). Dedekind Cut vs. Cauchy Sequence – both give the same $\mathbb{R}$; cuts emphasize order, sequences emphasize metric. Terminating Decimal vs. Repeating $9$’s – $0.5000\ldots$ equals $0.4999\ldots$; only terminating fractions have two representations. $\mathbb{R}$ vs. $\mathbb{Q}$ – $\mathbb{R}$ is complete, uncountable, has supremum property; $\mathbb{Q}$ lacks these. --- ⚠️ Common Misunderstandings “Every bounded set has a maximum.” Only supremum is guaranteed; maximum exists only if the supremum belongs to the set. “Cauchy ⇒ Bounded” is true, but the converse is not true; bounded sequences need not be Cauchy. “$0.999\ldots$ is “less than” $1$.” By definition $0.999\ldots = 1$; the infinite tail of $9$’s represents the same limit. “Real numbers can be ordered like complex numbers.” $\mathbb{C}$ cannot be given a total order compatible with field operations. --- 🧠 Mental Models / Intuition Number line as a “filled‑in” line – think of $\mathbb{Q}$ as the “dots” and the gaps filled by irrationals to make a solid line (completeness). Supremum as “the door just beyond the set” – the smallest point you can step to without entering the set. Cauchy sequences as “zoom‑in” – terms get arbitrarily close, like a microscope sharpening on a point. Decimal expansions as infinite Lego towers – each digit adds a smaller block; truncations are partial builds whose limit is the full tower. --- 🚩 Exceptions & Edge Cases Terminating decimals – have two decimal expansions (e.g., $0.2500\ldots = 0.2499\ldots$). Supremum not in set – e.g., $S=(0,1)$ has $\sup S = 1 \notin S$. Archimedean fails in non‑standard models – hyperreal fields contain infinitesimals; not relevant for $\mathbb{R}$ but good to note. Metric completeness does not hold in $\mathbb{Q}$ – Cauchy sequences like the decimal expansion of $\sqrt2$ have no rational limit. --- 📍 When to Use Which Supremum/Infimum – use when a problem asks for “least upper bound” or “greatest lower bound”. Cauchy sequence test – apply to prove convergence without finding the limit explicitly (e.g., series convergence). Dedekind cut reasoning – handy for order‑based arguments (e.g., proving existence of irrational numbers). Decimal expansion – useful for constructing explicit approximations or proving density of rationals. Cardinality arguments – invoke when comparing sizes of infinite sets (e.g., prove “most” reals are non‑computable). --- 👀 Patterns to Recognize Bounded + Non‑empty ⇒ Look for supremum (especially in optimization problems). Sequence with decreasing differences ⇒ Likely Cauchy → can claim convergence in $\mathbb{R}$. Infinite repeating $9$’s at the end of a decimal → the number equals the next terminating decimal. Any statement involving “for every $\varepsilon>0$ … there exists $N$ …” → a hallmark of limits, Cauchy, or continuity. “No greatest element” in a set of rationals → signals a cut representing an irrational. --- 🗂️ Exam Traps Choosing “maximum” instead of “supremum” – the set may lack a maximum; answer will be wrong if you claim one exists. Assuming all Cauchy sequences converge in $\mathbb{Q}$ – they converge in $\mathbb{R}$, not necessarily in $\mathbb{Q}$. Confusing density with completeness – $\mathbb{Q}$ is dense but not complete; a dense set need not be complete. Misreading “terminating decimal” – forgetting the alternate $...999$ representation can lead to off‑by‑one errors. Overlooking the Archimedean property – some “infinitesimal” answers are invalid for real numbers. ---