Arithmetic Study Guide

📖 Core Concepts Arithmetic – study of numbers & operations (add, subtract, multiply, divide; plus exponentiation, roots, logarithms). Number sets – Natural $\mathbb N$: 1,2,3,… Whole $\mathbb W$: $\mathbb N$ plus 0 Integers $\mathbb Z$: …‑2,‑1,0,1,2,… Rationals $\mathbb Q$: fractions $\frac{p}{q}$, $q\neq0$ Reals $\mathbb R$: all rationals + irrationals Complex $\mathbb C$: $a+bi$, $i^2=-1$ Identity elements – additive identity = 0, multiplicative identity = 1. Inverse elements – additive inverse of $a$ is $-a$; multiplicative inverse (reciprocal) of $a\neq0$ is $\frac1a$. Commutativity – order doesn’t matter for $+$ and $\times$. Associativity – grouping doesn’t matter for $+$ and $\times$. Exponentiation – $b^{e}$; neutral exponent $b^{1}=b$. Not commutative/associative. Logarithm – inverse of exponentiation: $\log{b}(x)=y \iff b^{y}=x$. 📌 Must Remember $\displaystyle a+0=a,\qquad a\times1=a$ (identity laws). $\displaystyle a+(-a)=0,\qquad a\times\frac1a=1$ (inverse laws). Fraction addition (common denominator): $\displaystyle \frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$. Fraction addition (different denominators): $\displaystyle \frac{a}{c}+\frac{b}{d}= \frac{ad+bc}{cd}$. Fraction multiplication: $\displaystyle \frac{a}{b}\times\frac{c}{d}= \frac{ac}{bd}$. Fraction division: $\displaystyle \frac{a}{b}\div\frac{c}{d}= \frac{a}{b}\times\frac{d}{c}= \frac{ad}{bc}$. Decimal ↔ rational – any terminating or repeating decimal is rational; e.g., $0.\overline{3}=1/3$, $0.75=75/100$. Floating‑point non‑associativity – order of addition can change the rounded result. Significant digits – only non‑leading zeros and all digits after a decimal point count; propagate uncertainty by adding absolute (add/sub) or relative (mult/div) uncertainties. 🔄 Key Processes Adding fractions with different denominators Find LCD: $ \text{LCD}=c d$ (or least common multiple). Convert: $\frac{a}{c}\to\frac{a d}{c d}$, $\frac{b}{d}\to\frac{b c}{c d}$. Add numerators, keep LCD. Multiplying/Dividing fractions Multiply numerators & denominators directly. For division, flip the divisor (reciprocal) then multiply. Exponentiation by squaring (integer exponent) If $e$ even: $b^{e}=(b^{e/2})^{2}$. If $e$ odd: $b^{e}=b\times b^{e-1}$. Repeat recursively – reduces multiplications from $e$ to $\log2 e$. Rounding vs. Truncation Truncate: drop digits beyond desired place (no change to last kept digit). Round: look at first discarded digit; if ≥5, increase last kept digit by 1. 🔍 Key Comparisons Addition vs. Subtraction Addition combines; subtraction undoes addition: $(a+b)-b=a$. Multiplication vs. Division Multiplication repeats addition; division undoes multiplication: $(a\times b)\div b = a$. Rational vs. Real arithmetic Rational: closed under $+,-,\times,\div$ (except division by 0). Real: exponentiation only closed for positive bases; logarithms require positive arguments & base $\neq1$. Floating‑point addition vs. exact addition Exact: associative; Floating‑point: rounding introduces non‑associativity. Decimal vs. Binary numeral systems Base‑10 uses powers of 10; Base‑2 uses powers of 2 (used in computer arithmetic). ⚠️ Common Misunderstandings “0 is a natural number.” – In the outline, natural numbers start at 1; whole numbers include 0. “All exponentiation is associative.” – It is not; $(a^{b})^{c}\neq a^{(b^{c})}$ in general. “Rounding always makes a number larger.” – Only when the first discarded digit ≥5; otherwise the number stays the same or becomes smaller after truncation. “Floating‑point addition can be reordered without effect.” – Because of rounding, reordering can change the result. 🧠 Mental Models / Intuition Identity & Inverse – Think of a “do‑nothing” button (0 for addition, 1 for multiplication) and a “undo” button (negative sign or reciprocal). Positional value – Each digit is a “stack of blocks” whose size is the base‑power; moving a digit left multiplies its contribution by the base. Fraction operations – Aligning denominators is like converting two recipes to the same measuring cup size before mixing. Floating‑point error – Imagine rounding each step of a long calculation on a cheap calculator; small errors accumulate and order matters. 🚩 Exceptions & Edge Cases Integer division – Not closed: $7\div2 = 3.5$ (non‑integer). Exponentiation base ≤0 – Not closed for real numbers; only positive bases guarantee real results. Logarithm base = 1 – Undefined (no exponent yields a different number). Division by zero – Always undefined, even in rational or real arithmetic. 📍 When to Use Which Add/subtract numbers → use plain addition/subtraction if same denominator; otherwise find common denominator. Multiply/divide fractions → multiply directly; for division, flip the divisor first. Compute large integer powers → use exponentiation by squaring to minimise multiplications. Represent a decimal exactly → convert to a fraction with denominator $10^{k}$ (terminating) or use repeating‑decimal fraction form. Estimate with limited precision → truncate for a quick lower bound, round for the nearest estimate. Perform arithmetic on computers → prefer binary arithmetic; be aware of floating‑point non‑associativity and rounding errors. 👀 Patterns to Recognize Repeating decimal → rational – any bar over digits signals a fraction (e.g., $0.\overline{7}=7/9$). Common denominator pattern – when adding/subtracting fractions, denominators become the product $cd$ unless a smaller LCD exists. Exponentiation pattern – powers of 2 in binary correspond to left‑shifts; squaring repeatedly doubles the exponent. Floating‑point error pattern – large magnitude differences between operands cause the smaller one to be lost (“catastrophic cancellation”). 🗂️ Exam Traps Choosing the wrong denominator – Adding $\frac{1}{2}+\frac{1}{3}$ as $\frac{1+1}{2+3}$ (incorrect). Correct: $\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$. Assuming commutativity of subtraction/division – $(a-b)\neq(b-a)$, $(a\div b)\neq(b\div a)$. Treating $0$ as a natural number – Depends on convention; the outline defines naturals as starting at 1. Applying logarithm rules with base 1 or negative arguments – $\log{1}(x)$ undefined; $\log{b}(x)$ requires $b>0$, $b\neq1$, $x>0$. Floating‑point associativity – Computing $(a+b)+c$ vs. $a+(b+c)$ can give different results; exam may test which order yields a more accurate sum. --- Use this guide for quick recall before the exam – focus on the bullets, practice the step‑by‑step processes, and watch out for the listed traps!