Chapter 1 - Real Numbers, Euclidean Spaces, and Complex Numbers

1.1 The Field of Real Numbers: The Foundation of Analysis

We begin our journey into analysis by understanding the real numbers, denoted by $\mathbb{R}$. You’ve likely worked with them for years, but here we’ll look at their fundamental structure, the very bedrock upon which calculus and analysis are built.

Imagine the number line – that’s our intuitive picture of the reals. It’s continuous, unlike the integers ($\mathbb{Z}$) or rational numbers ($\mathbb{Q}$) which have “gaps.” The real numbers fill in all those gaps.

We start even simpler, with the natural numbers $\mathbb{N}=\{0,1,2,3,...\}$. These are the counting numbers. With natural numbers, we can add and multiply, but we quickly run into problems. What’s the solution to $x+1=0$? Not a natural number!

To solve this, we expand to the integers $\mathbb{Z}=\{...,-2,-1,0,1,2,...\}$. Now we have negatives and zero. But we still can’t solve $2x=1$.

So, we expand again to the rational numbers $\mathbb{Q}=\{\frac{p}{q}\mid p,q\in \mathbb{Z},q\ne 0\}$. Fractions! Now we can divide (except by zero). It seems like we have a pretty good system, right?

But think about geometry. Consider a right-angled triangle with both shorter sides of length 1. What’s the length of the hypotenuse? By the Pythagorean theorem, it’s $\sqrt{1^2+1^2}=\sqrt{2}$. Is $\sqrt{2}$ a rational number?

Proof that $\sqrt{2}$ is irrational

Assume, for contradiction, that $\sqrt{2}$ is rational. Then we can write $\sqrt{2}=\frac{p}{q}$, where $p$ and $q$ are integers with no common factors other than 1 (i.e., the fraction is in lowest terms), and $q\ne 0$.

Squaring both sides gives $2=\frac{p^2}{q^2}$, so $2q^2=p^2$. This means $p^2$ is even. If $p^2$ is even, then $p$ must also be even (because if $p$ were odd, $p^2$ would be odd).

Since $p$ is even, we can write $p=2k$ for some integer $k$. Substituting this into $2q^2=p^2$ gives $2q^2=(2k{)}^2=4k^2$, so $q^2=2k^2$. This means $q^2$ is even, and therefore $q$ must also be even.

But this is a contradiction! We assumed that $p$ and $q$ had no common factors other than 1, but we’ve shown that both $p$ and $q$ are even, meaning they share a common factor of 2. Therefore, our initial assumption that $\sqrt{2}$ is rational must be false. Thus, $\sqrt{2}$ is irrational.

$\sqrt{2}$ is not rational! And what about $\pi$, the ratio of a circle’s circumference to its diameter? It turns out $\pi$ is also irrational, and even more than that, it’s transcendental, meaning it’s not a root of any polynomial equation with rational coefficients.

Lindemann's Theorem (1882)

There is no equation of the form $x^n+a_{n-1}{x}^{n-1}+...+a_0=0$ with rational coefficients $a_i\in \mathbb{Q}$, such that $x=\pi$ is a solution.

This is where the real numbers $\mathbb{R}$ come in. They are a “completion” of the rational numbers, filling in all the gaps and including numbers like $\sqrt{2}$ and $\pi$. We’ll take the real numbers as given and focus on their fundamental properties.

The Real Numbers as a Field

The real numbers $\mathbb{R}$ with addition (+) and multiplication ($\cdot$) form a commutative, ordered field that is order-complete. Let’s break down what this means:

Field Axioms (for Addition and Multiplication):

These are the rules you’re used to working with:

Axioms of Addition:

A1: Associativity of Addition

For all $x,y,z\in \mathbb{R}$, $x+(y+z)=(x+y)+z$.

A2: Existence of Additive Identity (Zero)

For all $x\in \mathbb{R}$, there exists $0\in \mathbb{R}$ such that $x+0=x$.

A3: Existence of Additive Inverse (Negation)

For every $x\in \mathbb{R}$, there exists $-x\in \mathbb{R}$ such that $x+(-x)=0$.

A4: Commutativity of Addition

For all $x,z\in \mathbb{R}$, $x+z=z+x$.

Axioms of Multiplication:

M1: Associativity of Multiplication

For all $x,y,z\in \mathbb{R}$, $x\cdot (y\cdot z)=(x\cdot y)\cdot z$.

M2: Existence of Multiplicative Identity (One)

For all $x\in \mathbb{R}$, there exists $1\in \mathbb{R}$ such that $x\cdot 1=x$.

M3: Existence of Multiplicative Inverse (Reciprocal)

For every $x\in \mathbb{R},x\ne 0$, there exists $x^{-1}\in \mathbb{R}$ such that $x\cdot x^{-1}=1$.

M4: Commutativity of Multiplication

For all $x,z\in \mathbb{R}$, $x\cdot z=z\cdot x$.

Distributive Axiom (linking Addition and Multiplication):

D: Distributivity

For all $x,y,z\in \mathbb{R}$, $x\cdot (y+z)=x\cdot y+x\cdot z$.

These axioms ensure that basic algebraic manipulations work as expected in $\mathbb{R}$.

Order Axioms (introducing the order relation $\le$):

These axioms define how real numbers are ordered on the number line:

O1: Reflexivity

For all $x\in \mathbb{R}$, $x\le x$.

O2: Transitivity

If $x\le y$ and $y\le z$, then $x\le z$.

O3: Antisymmetry

If $x\le y$ and $y\le x$, then $x=y$.

O4: Totality (Completeness of Order)

For all $x,y\in \mathbb{R}$, either $x\le y$ or $y\le x$.

Compatibility Axioms (linking Order and Field Operations):

These ensure the order is consistent with addition and multiplication:

K1: Compatibility with Addition

If $x\le y$, then for all $z\in \mathbb{R}$, $x+z\le y+z$.

K2: Compatibility with Multiplication by Non-negative Numbers

If $x\ge 0$ and $y\ge 0$, then $x\cdot y\ge 0$.

Order-Completeness (The crucial property distinguishing $\mathbb{R}$ from $\mathbb{Q}$):

This is the most subtle and important property. It guarantees that there are no “gaps” in the real number line.

V: Order-Completeness Axiom

Let $A$ and $B$ be non-empty subsets of $\mathbb{R}$ such that for every $a\in A$ and every $b\in B$, $a\le b$. Then there exists a number $c\in \mathbb{R}$ such that for all $a\in A$, $a\le c$ and for all $b\in B$, $c\le b$.

Imagine set $A$ being all numbers less than $\sqrt{2}$ and set $B$ being all numbers greater than $\sqrt{2}$. In the rationals, there’s no rational number ‘c’ that fits between them. Order-completeness guarantees such a number exists in the reals – in this case, it’s $\sqrt{2}$ itself.

Important Consequences of the Axioms

From these basic axioms, we can derive familiar properties of real numbers. Here are a few important corollaries:

Corollary 1.1.6: Basic Properties of Real Numbers

(1) Uniqueness of Additive and Multiplicative Inverses: The additive inverse $(-x)$ and multiplicative inverse $(x^{-1})$ are unique. (2) Zero Property of Multiplication: For any $x\in \mathbb{R}$, $0\cdot x=0$. (3) Multiplication by -1: For any $x\in \mathbb{R}$, $(-1)\cdot x=-x$, and in particular, $(-1{)}^2=1$. (4) Relationship between $\ge 0$ and $\le 0$: $y\ge 0$ if and only if $-y\le 0$. (5) Squares are Non-negative: For any $y\in \mathbb{R}$, $y^2\ge 0$, and in particular, $1=1\cdot 1\ge 0$. (6) Adding Inequalities: If $x\le y$ and $u\le v$, then $x+u\le y+v$. (7) Multiplying Inequalities (with non-negative numbers): If $0\le x\le y$ and $0\le u\le v$, then $x\cdot u\le y\cdot v$.

Proof of Corollary 1.1.6 (Selection of Proofs)

(2) Proof that $0\cdot x=0$: We start with the additive identity: $0+0=0$ (A2). Multiply both sides by $x$: $(0+0)\cdot x=0\cdot x$. Using the distributive axiom (D): $0\cdot x+0\cdot x=0\cdot x$. Add the additive inverse of $(0\cdot x)$ to both sides (A3): $(0\cdot x+0\cdot x)+(-(0\cdot x))=0\cdot x+(-(0\cdot x))=0$. Using associativity (A1): $0\cdot x+(0\cdot x+(-(0\cdot x)))=0$. Using the additive inverse property (A3): $0\cdot x+0=0$. Using the additive identity property (A2): $0\cdot x=0$.

(3) Proof that $(-1)\cdot x=-x$: Consider $x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x$ (Distributive Law D). Since $1+(-1)=0$ (Additive Inverse A3), we have $(1+(-1))\cdot x=0\cdot x=0$ (from part (2)). So, $x+(-1)\cdot x=0$. This means $(-1)\cdot x$ is the additive inverse of $x$, which is $-x$ by uniqueness of inverses.

Corollary 1.1.7: Archimedean Principle

For any $x\in \mathbb{R}$ with $x>0$ and any $y\in \mathbb{R}$, there exists a natural number $n\in \mathbb{N}$ such that $y\le n\cdot x$.

The Archimedean principle basically says that you can always find a multiple of a positive number $x$ that is greater than any number $y$. There are no “infinitely large” real numbers compared to the natural numbers.

Theorem 1.1.8: Existence of Square Roots

For every $t\ge 0,t\in \mathbb{R}$, the equation $x^2=t$ has a solution in $\mathbb{R}$.

This is a significant result. It shows that we’ve successfully extended the rational numbers to include solutions to equations like $x^2=2$. For $t\ge 0$, there is a unique non-negative solution, denoted by $\sqrt{t}$.

Proof of Theorem 1.1.8 (Existence of Square Roots)

For $t=0$, the solution is clearly $x=0$. Let’s consider $t>0$. We define two sets: $A=\{y\in \mathbb{R}\mid y\ge 0,y^2\le t\}$ $B=\{z\in \mathbb{R}\mid z>0,z^2\ge t\}$

We want to use the order-completeness axiom (V). First, we need to show that sets $A$ and $B$ are non-empty and satisfy the condition for axiom V.

V(i): Non-empty sets: $A\ne \varnothing$ because $0\in A$. To show $B\ne \varnothing$, by the Archimedean Principle (Corollary 1.1.7), there exists $n\in \mathbb{N}$ such that $n\ge t$ and $n\ge 1$. Then $n^2\ge t\cdot 1=t$, so $n\in B$.

V(ii): Condition for Axiom V: For any $y\in A$ and $z\in B$, we have $y^2\le t\le z^2$. Since $y,z>0$, this implies $y\le z$.

Now, by the order-completeness axiom (V), there exists $c\in \mathbb{R}$ such that for all $y\in A,y\le c$ and for all $z\in B,c\le z$.

We will show that $c^2=t$. We know $y^2\le t\le z^2$ for all $y\in A,z\in B$. By properties of order, this implies $y^2\le c^2\le z^2$.

Proof by Contradiction that $c^2\ge t$: Assume $c^2<t$. Then $c\in A$. Since $t-c^2>0$ and $2c+1\ge 1>0$, by the Archimedean principle, there exists $n\in \mathbb{N},n\ge 1$ such that $2c+1\le n(t-c^2)$. Consider $c+\frac{1}{n}$. Then $(c+\frac{1}{n}{)}^2=c^2+2c\cdot \frac{1}{n}+\frac{1}{n^2}\le c^2+\frac{2c+1}{n}\le c^2+(t-c^2)=t$. This means $(c+\frac{1}{n})\in A$. But $c+\frac{1}{n}>c$, which contradicts the fact that $c$ is an upper bound for $A$. Thus $c^2\ge t$.

Proof by Contradiction that $c^2\le t$: Assume $c^2>t$. Then $c\in B$. Since $c^2-t>0$ and $2c+1\ge 1>0$, by Archimedean principle, there exists $n\in \mathbb{N},n\ge 1$ such that $2c+1\le n(c^2-t)$. Consider $c-\frac{1}{n}$. If $c-\frac{1}{n}>0$, then $(c-\frac{1}{n}{)}^2=c^2-2c\cdot \frac{1}{n}+\frac{1}{n^2}\ge c^2-\frac{2c+1}{n}\ge c^2-(c^2-t)=t$. Thus $(c-\frac{1}{n})\in B$. But $c-\frac{1}{n}<c$, contradicting the fact that $c$ is a lower bound for $B$. Thus $c^2\le t$.

Since $c^2\ge t$ and $c^2\le t$, we conclude $c^2=t$.

This completes our introduction to the real numbers as a field. They are the foundation for everything that follows in analysis.

1.2 The Euclidean Space: Moving Beyond the Number Line

We’ve spent some time understanding the real numbers, which we can visualize as a one-dimensional number line. But mathematics and the physical world are often multi-dimensional. Euclidean space generalizes the real number line to higher dimensions.

${\mathbb{R}}^n$: Coordinates in $n$ Dimensions

For any positive integer $n$, we define ${\mathbb{R}}^n$ as the set of all ordered $n$-tuples of real numbers:

${\mathbb{R}}^n=\{(x_1,x_2,\dots ,x_n)\mid x_j\in \mathbb{R}\text{ for }j=1,2,\dots ,n\}$

• ${\mathbb{R}}^1$ is just the real number line $\mathbb{R}$.
• ${\mathbb{R}}^2$ is the familiar 2-dimensional Euclidean plane (the $xy$-plane).
• ${\mathbb{R}}^3$ is 3-dimensional Euclidean space (the $xyz$-space).
• And so on for higher dimensions.

Each element in ${\mathbb{R}}^n$ is a vector or a point in $n$-dimensional space. We can perform vector operations in ${\mathbb{R}}^n$:

• Vector Addition: If $x=(x_1,\dots ,x_n)$ and $y=(y_1,\dots ,y_n)$, their sum is: $x+y=(x_1+y_1,x_2+y_2,\dots ,x_n+y_n)$ Just add component-wise.
• Scalar Multiplication: If $x=(x_1,\dots ,x_n)$ and $\alpha \in \mathbb{R}$ is a scalar, then: $\alpha \cdot x=(\alpha x_1,\alpha x_2,\dots ,\alpha x_n)$ Multiply each component by the scalar.

With these operations, ${\mathbb{R}}^n$ becomes a vector space over $\mathbb{R}$. This means it satisfies the axioms of a vector space (closure under addition and scalar multiplication, associativity, commutativity, distributivity, existence of zero vector and additive inverses, etc.), which you might have encountered in linear algebra.

The Scalar Product: Measuring Angles and Lengths

To introduce geometry and notions of length and angle in ${\mathbb{R}}^n$, we define the scalar product (or dot product) of two vectors $x=(x_1,\dots ,x_n)$ and $y=(y_1,\dots ,y_n)$:

$⟨x,y⟩={\sum }_{j=1}^n{x}_j{y}_j=x_1{y}_1+x_2{y}_2+\cdots +x_n{y}_n$

The scalar product takes two vectors and produces a real number (a scalar). It has important properties:

Properties of the Scalar Product

1. Symmetry: $⟨x,y⟩=⟨y,x⟩$ for all $x,y\in {\mathbb{R}}^n$.
2. Bilinearity: For all ${\alpha }_1,{\alpha }_2\in \mathbb{R}$ and $x_1,x_2,y\in {\mathbb{R}}^n$: $⟨{\alpha }_1{x}_1+{\alpha }_2{x}_2,y⟩={\alpha }_1⟨x_1,y⟩+{\alpha }_2⟨x_2,y⟩$
3. Positive Definiteness: $⟨x,x⟩={\sum }_{j=1}^n{x}_j^2\ge 0$ for all $x\in {\mathbb{R}}^n$, and $⟨x,x⟩=0$ if and only if $x=(0,0,\dots ,0)$ (the zero vector).

The scalar product is a generalization of the dot product you might have seen in 2D or 3D vector calculus.

The Norm (Euclidean Norm): Measuring Length

Using the scalar product, we can define the norm (or Euclidean norm or length) of a vector $x\in {\mathbb{R}}^n$:

$\Vert x\Vert =\sqrt{⟨x,x⟩}=\sqrt{{\sum }_{j=1}^n{x}_j^2}=\sqrt{x_1^2+x_2^2+\cdots +x_n^2}$

The norm $\Vert x\Vert$ represents the length or magnitude of the vector $x$. In ${\mathbb{R}}^2$, this is just the length of the vector in the plane, given by the Pythagorean theorem.

Theorem 1.2.2: Properties of the Euclidean Norm

1. Non-negativity: $\Vert x\Vert \ge 0$ for all $x\in {\mathbb{R}}^n$, and $\Vert x\Vert =0$ if and only if $x=0$.
2. Homogeneity: For any scalar $\alpha \in \mathbb{R}$ and vector $x\in {\mathbb{R}}^n$, $\Vert \alpha \cdot x\Vert =|\alpha |\Vert x\Vert$. Scaling a vector by $\alpha$ scales its length by $|\alpha |$.
3. Triangle Inequality: For all $x,y\in {\mathbb{R}}^n$, $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$. The length of the sum of two vectors is less than or equal to the sum of their lengths.

Proof of Triangle Inequality (Theorem 1.2.2 (3))

$\Vert x+y{\Vert }^2=⟨x+y,x+y⟩=⟨x,x⟩+2⟨x,y⟩+⟨y,y⟩=\Vert x{\Vert }^2+2⟨x,y⟩+\Vert y{\Vert }^2$

By the Cauchy-Schwarz Inequality (Theorem 1.2.1 in the script, stated but proof omitted in this summary, standard result in linear algebra): $|⟨x,y⟩|\le \Vert x\Vert \Vert y\Vert$.

Thus, $2⟨x,y⟩\le 2|⟨x,y⟩|\le 2\Vert x\Vert \Vert y\Vert$.

Therefore, $\Vert x+y{\Vert }^2\le \Vert x{\Vert }^2+2\Vert x\Vert \Vert y\Vert +\Vert y{\Vert }^2=(\Vert x\Vert +\Vert y\Vert {)}^2$. Taking the square root of both sides (since norms are non-negative): $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$.

Euclidean space ${\mathbb{R}}^n$ with the norm and scalar product provides a geometric setting for analysis in higher dimensions.

1.3 Complex Numbers: Expanding Numbers to a Plane

We started by extending the natural numbers to integers, then to rationals, and finally to reals to solve more and more equations. Now we take another leap and extend the real numbers to the complex numbers $\mathbb{C}$.

Motivation: Solving $x^2+1=0$

The real numbers $\mathbb{R}$ are “order-complete” and very powerful, but they still have limitations. For example, the simple equation $x^2+1=0$ has no solution in real numbers. We need to introduce a new kind of number to solve this.

Defining Complex Numbers as Ordered Pairs

We define complex numbers $\mathbb{C}$ not just as an extension of the real line, but as a two-dimensional plane:

$\mathbb{C}=\{(x,y)\mid x,y\in \mathbb{R}\}$

Each complex number is an ordered pair $(x,y)$ of real numbers. We define addition component-wise, just like vector addition in ${\mathbb{R}}^2$:

• Addition: $(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$

But multiplication is defined in a special way:

• Multiplication: $(x_1,y_1)\cdot (x_2,y_2)=(x_1{x}_2-y_1{y}_2,x_1{y}_2+x_2{y}_1)$

This definition might seem strange at first, but it’s precisely what allows us to solve $x^2+1=0$.

The Imaginary Unit $i$

Let’s consider the complex number $(0,1)$. Let’s square it using our multiplication rule:

$(0,1)\cdot (0,1)=(0\cdot 0-1\cdot 1,0\cdot 1+1\cdot 0)=(-1,0)$

The complex number $(-1,0)$ is identified with the real number $-1$. So, if we define $i=(0,1)$, then $i^2=-1$. We have found a solution to $x^2+1=0$! The number $i$ is called the imaginary unit.

We can write any complex number $(x,y)$ in the standard form:

$z=x+yi$

where $x$ is the real part of $z$ (Re $z$) and $y$ is the imaginary part of $z$ (Im $z$). We identify real numbers $x$ with complex numbers of the form $x+0i$, effectively embedding $\mathbb{R}$ within $\mathbb{C}$.

Theorem 1.3.1: Complex Numbers as a Field

The set of complex numbers $\mathbb{C}$ with addition and multiplication as defined forms a commutative field with additive identity $(0,0)$ (or $0$) and multiplicative identity $(1,0)$ (or $1$).

This means that complex numbers, like real numbers, satisfy all the field axioms (commutativity, associativity, distributivity, inverses, identities). $\mathbb{C}$ is a richer number system than $\mathbb{R}$ because it is algebraically closed (Fundamental Theorem of Algebra - any polynomial equation with complex coefficients has complex roots).

Complex Conjugate and Norm

For a complex number $z=x+yi$, we define:

• Complex Conjugate: $\bar{z}=x-yi$ (reflection across the real axis)
• Norm (Modulus): $|z|=\sqrt{z\bar{z}}=\sqrt{x^2+y^2}$ (distance from the origin in the complex plane)

The norm of a complex number is always a non-negative real number.

Theorem 1.3.2: Properties of Complex Conjugate and Norm

1. Conjugation is an Automorphism: $\overline{(z_1+z_2)}={\bar{z}}_1+{\bar{z}}_2$, $\overline{(z_1{z}_2)}={\bar{z}}_1{\bar{z}}_2$.
2. Norm Squared: $z\bar{z}=x^2+y^2=|z{|}^2$.

Multiplicative Inverse in $\mathbb{C}$:

For any non-zero complex number $z\ne 0$, its multiplicative inverse is given by:

$z^{-1}=\frac{\bar{z}}{|z{|}^2}$

This formula explicitly shows how to find the reciprocal of a complex number using its conjugate and norm.

Polar Form of Complex Numbers:

Just like points in the plane can be represented by Cartesian coordinates $(x,y)$ or polar coordinates $(r,\theta )$, complex numbers can be represented in Cartesian form ($z=x+yi$) or polar form.

In polar form, a complex number $z$ is represented by its modulus $r=|z|$ (distance from origin) and its argument $\theta =\arg (z)$ (angle with the positive real axis).

$z=r(\cos \theta +i\sin \theta )$

Using Euler’s formula $e^{i\theta }=\cos \theta +i\sin \theta$, we can write the polar form compactly as:

$z=r{e}^{i\theta }$

The polar form is particularly useful for multiplication and powers of complex numbers.

Theorem 1.3.3: De Moivre's Theorem (Powers of Complex Numbers in Polar Form)

If $z=r(\cos \theta +i\sin \theta )=r{e}^{i\theta }$, then for any integer $n$:

$z^n=r^n(\cos (n\theta )+i\sin (n\theta ))=r^n{e}^{in\theta }$

Roots of Unity: Symmetries in the Complex Plane

Every non-zero complex number has exactly $n$ distinct $n$-th roots. These roots are evenly spaced around a circle in the complex plane. For the $n$-th roots of unity (solutions to $z^n=1$), these roots are given by:

$z_j=\cos \left(\frac{2\pi j}{n}\right)+i\sin \left(\frac{2\pi j}{n}\right)=e^{i(2\pi j/n)},j=1,2,\dots ,n$

They form the vertices of a regular $n$-gon inscribed in the unit circle in the complex plane.

Fundamental Theorem of Algebra: Completeness of Complex Numbers

A truly remarkable result about complex numbers is the Fundamental Theorem of Algebra:

Theorem 1.3.4: Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has at least one complex root. Consequently, a polynomial of degree $n$ with complex coefficients has exactly $n$ complex roots (counting multiplicities).

This theorem states that the complex numbers are algebraically closed. Any polynomial equation you can write down with complex coefficients will have all its roots within the complex numbers. This is a property that the real numbers do not possess (e.g., $x^2+1=0$ has no real roots).

Complex numbers are not just an abstract mathematical curiosity. They are essential in many areas of mathematics, physics, engineering, and computer science, providing a powerful and complete number system for solving a wide range of problems.

This concludes Chapter 1. We’ve built our number systems from natural numbers to integers, rationals, reals, and finally complex numbers, exploring their properties and motivations. This foundation is crucial for delving into the core concepts of analysis in the chapters to come.

Next chapter: Chapter 2 - Sequences and Series, Approaching Infinity, Divergence, Infinity, Algebra of Limits, Monotone Sequences, Weierstrass, Cauchy, Series, Summing Infinitely, Special Series and Operations