Notes of Calculus

Chapter 1 : Real and Complex Systems

Author: Kenneth, S.K. Cheng

Table of Contents

• Notes of Calculus
  • Chapter 1 : Real and Complex Systems
    • Author: Kenneth, S.K. Cheng
  • Table of Contents
    • 1.1 Introduction
    • 1.2 The Field Axioms
    • 1.3 Order Axioms
    • 1.4 Intervals
    • 1.5 Integers
    • 1.6 Rational Numbers
    • 1.7 Irrational Numbers
    • 1.8 Real Numbers
    • 1.9 Complex Numbers
    • Exercises
    • References and Further Reading

1.1 Introduction

Throughout our studying in secondary school, we have been exposed to the concept of functions. In this chapter, we will be looking at functions in a more mathematical way. We will be looking at the definition of functions, the domain and range of functions, and the different types of functions.

1.2 The Field Axioms

We have encounted many domains in mathematics, such as the set of integers which we denote as $\mathbb{Z}$, the set of real numbers which we denote as $\mathbb{R}$, and the set of complex numbers which we denote as $\mathbb{C}$. These sets are known as fields. Fields $\mathbb{F}$ are sets that satisfy the following axioms:

1. Closure under addition: For all $a, b \in \mathbb{F}$, $a + b \in \mathbb{F}$.
2. Closure under multiplication: For all $a, b \in \mathbb{F}$, $ab \in \mathbb{F}$.
3. Commutativity of addition: For all $a, b \in \mathbb{F}$, $a + b = b + a$.
4. Commutativity of multiplication: For all $a, b \in \mathbb{F}$, $ab = ba$.
5. Associativity of addition: For all $a, b, c \in \mathbb{F}$, $(a + b) + c = a + (b + c)$.
6. Associativity of multiplication: For all $a, b, c \in \mathbb{F}$, $(ab)c = a(bc)$.
7. Existence of additive identity: There exists an element $0 \in \mathbb{F}$ such that for all $a \in \mathbb{F}$, $a + 0 = a$.
8. Existence of multiplicative identity: There exists an element $1 \in \mathbb{F}$ such that for all $a \in \mathbb{F}$, $a1 = a$.
9. Existence of additive inverse: For all $a \in \mathbb{F}$, there exists an element $-a \in \mathbb{F}$ such that $a + (-a) = 0$.
10. Existence of multiplicative inverse: For all $a \in \mathbb{F}$, there exists an element $a^{-1} \in \mathbb{F}$ such that $a \cdot a^{-1} = 1$.
11. Distributive property: For all $a, b, c \in \mathbb{F}$, $a(b + c) = ab + ac$.

1.3 Order Axioms

When we learn about inequalities, we often use the order axioms. However, we just simply accept them without knowing why they are true. The order axioms are as follows:

1. Trichotomy: For all $a,b \in \mathbb{F}$, exactly one of the following holds: $a > b$, $a = b$, or $a < b$.
2. Transitivity: For all $a,b,c \in \mathbb{F}$, if $a > b$ and $b > c$, then $a > c$.
3. Addition of inequalities: For all $a,b,c \in \mathbb{F}$, if $a > b$, then $a + c > b + c$.
4. Multiplication of inequalities: For all $a,b,c \in \mathbb{F}$, if $a > b$ and $c > 0$, then $ac > bc$.
5. Order of reciprocals: For all $a,b \in \mathbb{F}$, if $a > b > 0$, then $\frac{1}{a} < \frac{1}{b}$.

Remark: The symbol $\leq$ denotes "less than or equal to", and the symbol $\geq$ denotes "greater than or equal to".

1.4 Intervals

In mathematics, we often use intervals to describe a set of numbers. There are four types of intervals:

1. Open interval: The open interval $(a,b)$ is the set of all $x \in \mathbb{R}$ such that $a < x < b$.
2. Closed interval: The closed interval $[a,b]$ is the set of all $x \in \mathbb{R}$ such that $a \leq x \leq b$.
3. Half-open interval: The half-open interval $[a,b)$ is the set of all $x \in \mathbb{R}$ such that $a \leq x < b$.
4. Half-closed interval: The half-closed interval $(a,b]$ is the set of all $x \in \mathbb{R}$ such that $a < x \leq b$.

Apart from these, we also have infinite intervals which are used for describing unbounded sets of numbers. There are also five types of infinite intervals:

1. Infinite open interval: The infinite open interval $(-\infty, a)$ is the set of all $x \in \mathbb{R}$ such that $x < a$.
2. Infinite closed interval: The infinite closed interval $(-\infty, a]$ is the set of all $x \in \mathbb{R}$ such that $x \leq a$.
3. Infinite open interval: The infinite open interval $(a, \infty)$ is the set of all $x \in \mathbb{R}$ such that $x > a$.
4. Infinite closed interval: The infinite closed interval $[a, \infty)$ is the set of all $x \in \mathbb{R}$ such that $x \geq a$.
5. Infinite interval: The infinite interval $(-\infty, \infty)$ is the set of all $x \in \mathbb{R}$.
6. Empty set: The empty set $\emptyset$ is the set of all $x \in \mathbb{R}$ such that there are no elements in the set.

1.5 Integers

We learn about integers in secondary school. Integers are numbers that can be written without a fractional component. Integers can be positive, negative, or zero. The set of integers is denoted as $\mathbb{Z}$. The set of integers is a subset of the set of real numbers $\mathbb{R}$.

Definition: The set of integers $\mathbb{Z}$ is defined as $\mathbb{Z} = \{0, \pm 1, \pm 2, \pm 3, \ldots\}$.

1.6 Rational Numbers

Rational numbers, one may think of them as extension of integers. Rational numbers are numbers that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. The set of rational numbers is denoted as $\mathbb{Q}$. The set of rational numbers is a subset of the set of real numbers $\mathbb{R}$.

Definition: The set of rational numbers $\mathbb{Q}$ is defined as $\mathbb{Q} = \left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0\right\}$.

1.7 Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. The set of irrational numbers is denoted as $\mathbb{I}$. The set of irrational numbers is a subset of the set of real numbers $\mathbb{R}$.

Definition: The set of irrational numbers $\mathbb{I}$ is defined as $\mathbb{I} = \mathbb{R} \setminus \mathbb{Q}$.

Some famous irrational numbers include $\pi$ and $\sqrt{2}$.

1.8 Real Numbers

Real numbers are numbers that can be represented on the number line. Real numbers include integers, rational numbers, and irrational numbers. The set of real numbers is denoted as $\mathbb{R}$.

Definition: The set of real numbers $\mathbb{R}$ is defined as $\mathbb{R} = \mathbb{Z} \cup \mathbb{Q} \cup \mathbb{I}$.

For example, the number $0.5$ is a real number because it can be represented on the number line.

Next we extend the concept of real numbers by adjoining two new elements, $\infty$ and $-\infty$. These two elements are used to represent infinity. The set of extended real numbers is denoted as $\mathbb{R}^*$.

Definition: The set of extended real numbers $\mathbb{R}^*$ is defined as $\mathbb{R}^* = \mathbb{R} \cup \{-\infty, \infty\}$.

1.9 Complex Numbers

Complex numbers might be not familiar to you. It actually follows from the axioms governing the relation $<$ that the square of any real number is nonnegative. This means that the equation $x^2 = -1$ has no solution in the set of real numbers. To solve this problem, we introduce a new number $i$ such that $i^2 = -1$. The set of complex numbers is denoted as $\mathbb{C}$.

Definition: Complex number is an ordered pair of real numbers $(a,b)$, where $a$ is the real part and $b$ is the imaginary part. The set of complex numbers $\mathbb{C}$ is defined as $\mathbb{C} = \{(a,b) \mid a,b \in \mathbb{R}\}$. For such $x \in \mathbb{C}$, we have $x = a + bi$.

Remark: The real part of a complex number $z = a + bi$ is denoted as $\Re(z) = a$, and the imaginary part of a complex number $z = a + bi$ is denoted as $\Im(z) = b$. One may wonder what is the meaning of $i$. It is defined as $i = \sqrt{-1}$, we called it the imaginary unit.

Theorem: For all $z, w \in \mathbb{C}$, we have:

1. Addition: $z + w = (a + c) + (b + d)i$.
2. Multiplication: $zw = (ac - bd) + (ad + bc)i$.
3. Conjugate: The conjugate of $z = a + bi$ is $\bar{z} = a - bi$.
4. Modulus: The modulus of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$.
5. Division: $\frac{z}{w} = \frac{z\bar{w}}{|w|^2}$.
6. Polar form: $z = r(\cos \theta + i \sin \theta)$. (It consists knowledge of trigonometry.)
7. De Moivre's Theorem: $(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$. (I still remember this theorem is needed for the joing admission exam in Macau...)
8. Roots of unity: The $n$th roots of unity are given by $z_k = \cos \frac{2\pi k}{n} + i \sin \frac{2\pi k}{n}$ for $k = 0, 1, 2, \ldots, n-1$.
9. Euler's Formula: $e^{i\theta} = \cos \theta + i \sin \theta$.

Remark: Since we have $\mathbb{C}$ is a field, we can perform addition, subtraction, multiplication, and division on complex numbers.

By the polar form of complex numbers, we can represent complex numbers in the form of $z = r(\cos \theta + i \sin \theta)$. This form is useful for calculating the powers of complex numbers. The angle $\theta$ is known as the argument of the complex number $z$. The modulus $r$ is the distance of the complex number $z$ from the origin.

Exercises

1. Express the following complex numbers in the form of $a + bi$:
   1. $(1+i)^3$
   2. $\frac{2+3i}{3-4i}$
   3. $i^5+i^{16}$
2. In each case, determine all real $x$ and $y$ which satisfy the given relation.
   1. $x+iy = |x-iy|.$
   2. $x+iy = (x-iy)^2.$
3. If $z = x + iy$ and $\overline{z} = x - iy$ is the conjugate of $z$, prove the following:
   1. $\overline{z + w} = \overline{z} + \overline{w}$.
   2. $\overline{zw} = \overline{z} \cdot \overline{w}$.
   3. $\overline{\overline{z}} = z$.
   4. $z+\overline{z} = 2x$.

References and Further Reading

1. Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley.
2. Zorich, V. A. (2004). Mathematical Analysis I. Springer.
3. Lang, S. (1986). A First Course in Calculus. Springer.
4. Hass J., Heil C., Weir M. (2013). Thomas' Calculus. Pearson. (I think this should be the easiest book for you to understand the concepts of calculus.)
5. Silverman, R.A. Modern Calculus and Analytic Geometry. Springer. (This book is also good for you to understand the concepts of calculus.)