Brunei Math Club

 The mathematical constant \(e\), sometimes called Euler's number or Napier's constant, is an irrational number that appears everywhere in mathematics and other sciences and has a value of \(e=2.71828\cdots\). Now we see how this constant is defined and how it is related to complex numbers. We assume you know some calculus. Definition (Natural logarithm) The natural logarithm function \(\log: (0, \infty) \to \mathbb{R}\) is defined by \[\log x = \int_1^{x}\frac{1}{t}dt.\] From the definition, we can immediately derive a few important properties of \(\log\). \(\log(1) = 0\). \(\log\) is a strictly increasing function as \(1/t > 0\) for all \(t > 0\). This means, for any \(x_1, x_2 \in (0,\infty)\), \[x_1 < x_2 \implies   \log x_1 < \log x_2.\] \(\log\) is a continuous function. This means that for any \(a \in (0, \infty)\), we have \[\lim_{x\to a}\log x = \log a.\] Lemma (The logarithm of a product is the sum of logarithms) For \(y_1, y_2 > 0, y_1, y_2\in\mathbb{

We define two trigonometric functions, namely the sine and cosine functions, based on the geometric interpretation of complex numbers. Using these functions, we define an alternative representation of complex numbers called the polar form . Definition (Sine and cosine) Let \(u \in \mathbb{C}\) such that \(|u| = 1\) and \(\theta \in \arg(u)\). We define \(\sin: \mathbb{R} \to \mathbb{R}\) and \(\cos: \mathbb{R} \to \mathbb{R}\) by  \[\begin{eqnarray} \cos(\theta) &=& \Re(u),\\ \sin(\theta) &=& \Im(u). \end{eqnarray}\] Thus, we have \(u = \cos\theta + i\sin\theta\). Example . In particular, we have \[\begin{eqnarray} \cos(0) &=& 1,\\ \cos(\pi/2) &=& 0,\\ \cos(\pi) &=& -1,\\ \sin(0) &=& 0,\\ \sin(\pi/2) &=& 1,\\ \sin(\pi) &=& 0. \end{eqnarray}\] Also, note that \[\begin{eqnarray} \cos(-\theta) &=& \cos\theta,\\ \sin(-\theta) &=& -\sin\theta. \end{eqn

The addition and multiplication of complex numbers have interesting geometric interpretations in terms of translation, rotation, and scaling on the complex plane. Addition is translation Let \(z_1 = a_1 + ib_1\) and \(z_2 = a_2 + ib_2\) where \(a_i, b_i \in \mathbb{R}\) for \(i = 1, 2\). Then, \(z_1 + z_2 = (a_1 + a_2) + i(b_1+b_2)\). Therefore, on the complex plane, adding \(z_1\) to \(z_2\) to obtain \(z_1 + z_2\) corresponds to translating the point \((a_1, b_1)\) by the vector \((a_2, b_2)\) to obtain the point \((a_1+a_2, b_1+b_2)\). Thus, the four points, \(0, z_1, z_1+z_2\), and \(z_2\) comprise a parallelogram (Figure 2). Figure 2. Adding two complex numbers. Multiplication by a real number is scaling Let \(c \in \mathbb{R}\) and \(z = a + ib\in \mathbb{C}\) with \(a, b\in\mathbb{R}\). Then, \(cz = ca + i(cb)\), which corresponds to the point \((ca, cb)\) on the complex plane. Meanwhile, we have \(|cz| = |c|\cdot|z|\), so the modulus is scaled by \(|c|\). If \(c > 0\), \(cz\

 From the definition of complex numbers, we can plot any complex number \(z = a + ib\in \mathbb{C}\) with \(a, b \in \mathbb{R}\) as a point \((a,b)\) in a two-dimensional plane. Here the ``\(x\)'' axis represents the real part of complex numbers, whereas the ``\(y\)'' axis represents the imaginary part. When we use this ``\(x-y\)'' plane to plot complex numbers, this plane is called the complex (number) plane, also known as the Argand diagram (Figure 1). Figure 1. The complex plane or Argand diagram. Definition (Modulus) Let \(z = a + ib \in \mathbb{C}\) with \(a, b \in \mathbb{R}\). The modulus of \(z\) is defined by \[|z| = \sqrt{a^2 + b^2}.\] That is, the modulus \(|z|\) is the distance between \(z\) and the origin in the complex plane (Figure 1). In comparison, the modulus of a real number \(x\in\mathbb{R}\) is defined (usually) as \[|x| = \left\{ \begin{array}{cc} x & \text{(if $x \geq 0$)},\\ -x & \text{(otherwise)}. \end{array}\ri

The sets \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) behave in a similar manner. They all have addition, subtraction, multiplication, and division that satisfy certain properties. This common behavior is summarised in the notion of fields. Thus, \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) are examples of fields. Definition (Field) A set \(F\) is called a  field  if it is endowed with binary operations \(+\) and \(*\) and satisfies the following axioms. \(F\) is closed under \(+\) (closure): \(\forall a, b \in F\), \(a + b \in F\). \(+\) is commutative: \(\forall a, b\in F\), \(a + b = b + a\). \(+\) is associative: \(\forall a, b, c\in F\), \(a + (b + c) = (a + b) + c\). There is an additive identity element \(0\) with respect to \(+\): \(\exists 0 \in F\) such that \(\forall a\in F\), \(a + 0 = a\). There is an inverse of each element with respect to \(+\): \(\forall a\in F,\)\(\exists -a \in F\) such that \(a + (-a) = 0\). \(F\) is closed under \(*\) (closure): \(\foral

We define the set of complex numbers, \(\mathbb{C}\), as the set of pairs of real numbers, \(\mathbb{R}^2\), equipped with addition and multiplication. However, our original motive for complex numbers was to solve the quadratic equation in general. How do they relate to each other?  Theorem (square root of a complex number) Suppose that \(a, b\in \mathbb{R}\). The square roots of \(a + ib\) are \(\pm(x + iy)\) where \[\begin{eqnarray} x &=& \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}},\\ y &=& \text{sign}(b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}} \end{eqnarray}\] where \[\text{sign}(b) = \left\{ \begin{array}{cl} -1 & \text{(if $b < 0$)},\\ 1 & \text{(otherwise)}. \end{array}\right.\] Remark . As useful as it is, memorizing this formula is not recommended. Why? Because this "formula" is so complicated that you will most likely forget it! Try to derive the square root of \(a+ib\) every time by solving \[(x + iy)^2 = a+ ib\] for \(x\) and \(y\). □ Proof . The

 For any pair of real numbers, \(a, b\in\mathbb{R}\), we can define a complex number \(a + bi\) where \(i\) is the imaginary unit (\(i = \sqrt{-1}\)). However, the imaginary unit is not a real number. What do we exactly mean by the expression like \(a + bi\)? We answer this question by constructing a system of complex numbers from the set of pairs of real numbers. Recall \(\mathbb{R}^2\), the set of ordered pairs of real numbers: \[\mathbb{R}^2 = \{(a, b) \mid a, b\in\mathbb{R}\}.\] Each element of \(\mathbb{R}^2\) is something like \((a,b)\) where \(a, b\in \mathbb{R}\). Let us define addition in \(\mathbb{R}^2\) by \[(a, b) + (c, d) = (a + c, b + d)\] where \((a,b), (c, d) \in\mathbb{R}^2\). Note that \(+\) on the left-hand side is the addition in \(\mathbb{R}^2\) being defined, and \(+\) on the right-hand side is the addition of real numbers (\(\mathbb{R}\)), which is already defined. Thus defined addition is commutative and associative (verify this): \[(a, b) + (c, d) = (c, d) + (

We review how complex numbers are introduced, starting from a simple linear equation. We also review the derivation of the quadratic formula. Consider the following equation: \[3x = 12.\] The coefficient of \(x\) is 3, and the right-hand side is 12. 3 and 12 are natural numbers: \(3, 12 \in \mathbb{N}\). Solving this for \(x\), we have \[x = 4 \in \mathbb{N}.\] So, the solution is obtained within \(\mathbb{N}\). Next, consider \[2x + 4 = 0.\] Again we have \(2, 4 \in \mathbb{N}\). But the solution \[x = -2 \in \mathbb{Z}\] is not found in \(\mathbb{N}\), but in \(\mathbb{Z}\). Next, consider \[2x - 3 = 0\] where \(2, -3 \in \mathbb{Z}\). The solution is \[x = \frac{3}{2} \in \mathbb{Q}\] which is not in \(\mathbb{Z}\). In general, consider \[ax + b = 0, ~ a, b \in \mathbb{Q}, a \neq 0.\] Then, the solution is always found within \(\mathbb{Q}\): \[x = -\frac{b}{a} \in \mathbb{Q}.\] As long as all the coefficients are in \(\mathbb{Q}\) and the equations are linear , we can always find th

The cardinalities (numbers of elements) of the sets \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\) are all infinite. Which one is "larger" or "smaller"? It turns out that the sets \(\mathbb{N}\), \(\mathbb{Z}\), and \(\mathbb{Q}\) all have the "same" number of elements and are called countable , whereas \(\mathbb{R}\) contains a far greater number of elements and is called uncountable . But how can we "count" infinities? The sets \(\{a, b, c, d\}\) and \(\{1, 2, 3, 4\}\) have no elements in common, but they have the same number of elements (i.e., 4). Note that there are bijections between them. For example, (but not limited to) \[\begin{align} 1 &\mapsto \text{a},\\ 2 &\mapsto \text{b},\\ 3 & \mapsto \text{c},\\ 4 & \mapsto \text{d}. \end{align}\]  Quiz.  How many bijections are possible from \(\{1, 2, 3, 4\}\) to \(\{a, b, c, d\}\)? Any finite set \(X\) with \(N\) elements (i.e., \(|X| = N\)) may be w

In order to study the elements of a set, we compare or relate each element with others. We already know some relations such as "\(=\)" (equal), "\(<\)" (less than), "\(\subset\)" (subset), etc.  We generalize this idea. A relation is defined on a set. For example, \(<\) is (usually) used for comparing two numbers that are well-ordered. So \(<\) is defined on \(\mathbb{R}\), for example. Similarly, \(\subset\) is the subset relation, so it is defined on a set of sets. To denote a relation in general, we often use the symbol \(\sim\). But there is nothing special about this particular symbol. We could instead use \(\bowtie\), \(\heartsuit\), \(\equiv\), etc. Or we can use any alphabets like \(A, b, \rho,\cdots\). Instead of writing \(a < b\), we could write \(a~ \mathcal{L}~ b\) (as long as everybody agrees on this convention). It's just a matter of definition. But what is a relation, anyway? Definition (Relation) A relation \(R\) on a set

We can define a new map by composing two or more maps. Map composition can be regarded as an operation between maps. As such, we can make an algebra of maps. Definition (map composition) Let \(f: A \to B\) and \(g: B \to C\) be maps. In particular, note that \(\text{cod}f = \text{dom}g\). Then, we define a new map, denoted \(g\circ f: A \to C\), by \[(g\circ f)(x) = g(f(x))\] for each \(x \in A\). This new map \(g\circ f\) is called the composition of \(f\) with \(g\). It is convenient to read \(g\circ f\) as "\(g\) after \(f\)." The composition can be graphically represented as in the following commutative diagram : Remark . As noted in the definition, we can compose \(g\circ f\) if and only if \(\text{cod}f = \text{dom}g\). Remark . Some authors (e.g., in computer science) prefer to write \(f; g\) ("\(f\), then \(g\)") rather than \(g\circ f\). Example . Suppose that \[f: \mathbb{Z} \to \mathbb{Z}; f(x) = x - 1,\] and \[g: \mathbb{Z} \to \mathbb{N}_{0}; g(x) = x^

 We often study the properties of a set by comparing it with another set. The "comparison"' is done by a  map  between the sets. Roughly speaking, a map (also called a  function ) is a rule to assign an element of a set to an element of another set .  Let \(A\) and \(B\) be sets. Then we write \(f: A \to B\) to mean that \(f\) is a map from \(A\) to \(B\). That is, for each \(a\in A\), we assign an element \(f(a) \in B\). We often use a "diagram" such as the following: The mapping of each element \(x \in A\) to \(f(x) \in B\) is written as \(x \mapsto f(x).\) Example . Let us define a map \(f: \mathbb{Z} \to \mathbb{N}\) by \[f(x) = 1 + x^2, ~~ (x \in \mathbb{Z}).\] Note that \(f(x)\in \mathbb{N}\) for any \(x \in \mathbb{Z}\), indeed. □ Definition (domain, codomain) Let \(A\) and \(B\) be sets, and \(f: A \to B\) be a map. Then, \(A\) and \(B\) are called the domain and codomain of \(f\), respectively. We write \(\text{dom} f = A\) and \(\text{cod} f = B\). De

We may construct a new set by combining existing sets through set operations . Thus, sets equipped with set operations form an algebra of sets, so to speak. We will see set operations are closely related to logical operations. Definition (union of sets) Let \(A\) and \(B\) be sets. Then, their union, denoted \(A \cup B\), is defined as \[A\cup B = \{x | x \in A \lor x \in B\}.\] That is, \(A \cup B\) is a set consisting of elements that are in \(A\) or in \(B\). Remark: "\(x \in A\)" and "\(x \in B\)" are logical propositions (predicates). Example: If \(A = \{1, 2, 3\}\) and \(B = \{1, 3, 5\}\), then \(A \cup B = \{1, 2, 3, 5\}\). Definition (intersection of sets) Let \(A\) and \(B\) be sets. Then, their intersection, denoted \(A \cap B\), is defined as \[A\cap B = \{x | x \in A \land x \in B\}.\] That is, \(A \cap B\) is a set consisting of elements that are in \(A\) and in \(B\). Remark: The logical "and" (i.e., conjunction) is often written as a comma (