Study sheet: Real Numbers and Their Properties

📋 Course Outline

1. Real Number Operations
2. Powers and Algebraic Identities
3. Methods of Mathematical Proof
4. Order and Inequalities
5. Intervals of Real Numbers
6. Absolute Value and Distance
7. Roots and Rational Powers

📖 1. Real Number Operations

🔑 Key Concepts & Definitions

• Rational number : can be written as $\frac{a}{b}$, where a and b are integers and $b\ne0$
• Irrational number : An irrational number cannot be written as $\frac{a}{b}$ with integers a and b and $b\ne0$, and therefore cannot be written as either a terminating or a repeating decimal.

📝 Essential Points

• Real numbers consist of rational numbers and irrational numbers and can be represented by points on the real-number axis.

• Subtraction is defined by $a-b=a+(-b)$, and division by a nonzero number is defined by $\frac{a}{b}=a\cdot\frac{1}{b}$ with $b\ne0$.

• For real numbers, $ab=0$ if and only if $a=0$ or $b=0$, and consequently $ab\ne0$ if and only if $a\ne0$ and $b\ne0$.

💡 Memory Hook

Rational numbers have fractional or repeating-decimal forms, whereas irrational numbers do not.

📖 2. Powers and Algebraic Identities

🔑 Key Concepts & Definitions

• Identity : an equality containing variables that is true for every allowed value of those variables

★ Must-know

📐 Formula — The notable identities include $\left(a+b\right)^2=a^2+2ab+b^2$, $a^2-b^2=(a+b)(a-b)$, and $a^3-b^3=(a-b)(a^2+ab+b^2)$.

📌 If the powers are defined, the laws of integer exponents apply to products, quotients, and powers of powers.

Further detail

📐 Formula — The square of a sum of three numbers is $\left(a+b+c\right)^2=a^2+b^2+c^2+2ab+2bc+2ca$.

💡 Memory Hook

Power rules lead to identities, then to factorization.

📖 3. Methods of Mathematical Proof

📝 Essential Points

• A direct proof starts from the hypothesis and uses successive valid transformations until the required conclusion is obtained.

📌 To disprove a universal claim, it is sufficient to provide one counterexample for which the claim is false.

• A proof by contradiction assumes that the desired conclusion is false, derives a contradiction with a known hypothesis or fact, and concludes that the assumption was false.

💡 Memory Hook

Direct proof → equivalent transformations → counterexample → contradiction.

📖 4. Order and Inequalities

🔑 Key Concepts & Definitions

• Order relation : For real numbers, $a>b$ means that $a-b>0$; geometrically, a is to the right of b on the real-number axis.

📝 Essential Points

📌 Adding the same real number to both sides of an inequality preserves its direction, while multiplying both sides by a positive number preserves the direction and multiplying by a negative number reverses it.

📌 For positive numbers and a positive integer n, $a>b$ if and only if $a^n>b^n$.

📐 Formula — For every real number a, $a^2\ge0$, with equality only when $a=0$.

💡 Memory Hook

Multiplying an inequality by a positive number preserves its direction, whereas a negative number reverses it.

📖 5. Intervals of Real Numbers

🔑 Key Concepts & Definitions

• Closed interval : The closed interval $[a,b]$ is the set of real numbers x satisfying $a\le x\le b$ and contains both endpoints.
• Open interval : The open interval $(a,b)$ is the set of real numbers x satisfying $a<x<b$ and excludes both endpoints.

📝 Essential Points

📌 The half-open intervals $[a,b)$ and $(a,b]$ include exactly one endpoint, while $[a,+\infty)$ and $(-\infty,a]$ describe one-sided unbounded sets.

💡 Memory Hook

Closed intervals include both endpoints, whereas open intervals exclude them.

📖 6. Absolute Value and Distance

🔑 Key Concepts & Definitions

• Absolute value : The absolute value $|a|$ of a real number is its distance from zero on the real-number axis, so $|a|=a$ for $a\ge0$ and $|a|=-a$ for $a<0$.
• Distance : The distance between real numbers a and b is $d(a,b)=|a-b|$ and is symmetric: $d(a,b)=d(b,a)$.

📝 Essential Points

📐 Formula — Absolute value satisfies $|ab|=|a||b|$, $\left|\frac{a}{b}\right|=\frac{|a|}{|b|}$ for $b\ne0$, and $|a+b|\le|a|+|b|$.

📌 For $\rho>0$, $|x-x_0|<\rho$ is equivalent to $x_0-\rho<x<x_0+\rho$, while $|x-x_0|>\rho$ is equivalent to $x<x_0-\rho$ or $x>x_0+\rho$.

💡 Memory Hook

Absolute value is the distance from a point on the number line to the origin.

📖 7. Roots and Rational Powers

🔑 Key Concepts & Definitions

• Square root : For $a\ge0$, the square root $\sqrt a$ is the nonnegative number whose square is a, so it is the nonnegative solution of $x^2=a$.
• Nth root : For $a\ge0$ and a positive integer n, the nth root $\sqrt[n]{a}$ is the nonnegative number whose nth power equals a.
• Rational exponent : For $a>0$, an integer m, and a positive integer n, a power with rational exponent is defined by $a^{m/n}=\sqrt[n]{a^m}$.

📝 Essential Points

📌 For nonnegative numbers, roots satisfy $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$ and, when $b\ne0$, $\sqrt[n]{\frac ab}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$.

💡 Memory Hook

Root definition → root properties → rational exponents.

📊 Synthesis Tables

Representations of Real Numbers

Class	Fractional form	Decimal form
Rational	$\frac{a}{b}$ with $b\ne0$	Terminating or repeating decimal
Irrational	No such fractional form	Neither terminating nor repeating decimal