Different Types of Real Nos.

• A number that can be represented as a point on the number line is a Real No.
• Such number line is called Real No. Line.
• There are several types of numbers within the Real No. Line.
• In Elementary/ Primary schools, understanding of Natural nos. and some knowledge on fractions is good enough for Math study.
• In middle/ high schools, Directed nos., Rational and Irrational nos. are introduced.

Definitions

1. Natural nos. : \( \{1, 2, 3, \ldots \} \ \) denoted by the set \( \mathbb{N} \ \) [some authors include integer 0]
2. Integers : \( \{\ldots , -2, -1, 0, 1, 2, \ldots \} \) denoted by the set \(\mathbb{Z}\)
3. Rational nos. : $\big\{ \frac{p}{q}\big |\; p, q \in \mathbb{Z},\; q \neq 0 \big\} \ $ denoted by the set $\mathbb{Q}$
4. Irrational nos. : $ \mathbb{R} {\setminus} \mathbb{Q}$
5. Surds $\subseteq \mathbb{R} {\setminus} \mathbb{Q}$. Examples are: square root $ (\sqrt{\phantom{x}}) $, cube root $ (\sqrt[3]{\phantom{x}}) $ form, and so on.

Misconceptions

1. Fractions and Rational nos. seems to be the same but they are different. All fractions are Rational nos., not all rational nos. are fractions. By definition, all integers are Rational nos., but integers are commonly considered different from fractions.
2. All decimal nos. can be expressed in the form of Rational nos. no matter recurring or finite decimal. Thus, the set of decimal nos. is not a separate type of Real nos., it is a subset of Rational nos.

Problem Solving Techniques

Definitions

1. $ i = \sqrt{-1},  \mbox{ i.e. }  i^2 = -1 $
2. For all $C = a + bi \ $ where $ a, b $ are Real nos., we call $C$ a Complex No., and denote the set of Complex No. as $ \mathbb{C} $.
3. For all $C = a+ bi, $
   • $a$ is the Real part
   • $b$ is the Imaginary part
   • $bi$ is the Imaginary unit
4. in particular if $b = 0, $ then $C$ is also a Real no., otherwise $b\neq 0 , $ it is either a non-Real Complex no. or purely Imaginary no.

Misconceptions

1. Both the Real part, $ a $ and Imaginary part, \(b\) are Real nos.
2. Don’t confuse the Imaginary part, $b$ with the Imaginary unit \(bi\)!
3. Don’t assume any Complex No. be non-Real! Acutally, $\, \mathbb{R} \subseteq \mathbb{C} $. That is, all Real nos. belong to the set of Complex Nos. — extension of Real no. set.

Problem Solving Techniques

Why need Functions?

Finding out the relationship between variables is useful in all kinds of knowledge. In real world, it is common that one quantity depends on the another quantity.

For example, the overtime salary of a worker depends on the quantity of extra working hours and the hourly rate of overtime.

Definitions

A relation between the elements of two sets \(A\) and \(B\) is a function, \(f\) from \(A\) to \(B\) if:

1. for each element $x \in A, $ there is an assigned element $y \in B $ usually written as $ y = f(x) ; $
2. for any $ x_1, x_2 \in A $ assigned to $ y_1, y_2 \in B $ respectively,  if $x_1 = x_2 $ then $y_1 = y_2. $

1. set \(A\) is called the Domain of $f$
2. set \(B\) is called the Codomain of $f$
3. Range (Image) of $ \, f: $ $ \quad f(A) = \{f(x) \in B \; | \;  x \in A\} $
4. $ f \ $ is also called a mapping from $ A \mbox{ to } B, \ $ denoted by $ \ f:A \to B $

A function can be represented in several forms, including:

II.  Functions in Graphical Form

Functions are commonly represented by graphs, especially continuous function.
The graph of a function \(f\) is the collection of all ordered pairs \((x, f(x)) \\[.2em]\).

Example – Quadratic Functions

In this example,

• Domain: \( \mathbb{R}\)
• Range: \(\{y \in \mathbb{R}\; | \; y \geq 0 \}\)

IV.  Functions in Venn Diagram

Because sets can usually be represented by Venn diagrams, sometimes a function is represented by Venn diagrams with arrows connecting the elements from the Domain to Codomain of the function.

Misconceptions

1. Though a function is a graph, by definition, not all graphs are functions, e.g. $ \ y^2=x \enspace $
   [why? To be discussed in online class]
2. Only test condition ii. above in definitions whether a relation is a function, but missed to testing condition i. (for each element in \(A\) there is assigned element in \(B\))
3. The notation \(f(x)\) appears in Algebraic functions, but not necessary for all types functions, e.g. in Table-form.

Why need Exponents

When a number, say \(a\) multiplies itself several times, the expression can be written as either the LHS or RHS of the equation below:\[ \underbrace{a \times a \times \ldots \times a}_{\color{RoyalBlue}{k\text{ times}}} = a^k \]

• Using exponent in the RHS of the above identity makes LHS simpler.
• The exponent \(k\) can be extended from a positive integer to any Rational no. and even beyond [to be covered by college-level Math]
• If \(k\) is positive non-integer, then \[k = p/q \quad \text{where } p,q \in \mathbb{Z}\] and \(a^k\) means \(a\) multiplies itself \(p\) times and takes \(q^{\rm th}\) root, i.e. $$ \begin{align} a^\frac{p}{q} &= (a^p)^\frac{1}{q} \\ &= \raise{.8ex} \sqrt[q \uproot1]{ } \underbrace{ \overline{a \times a \times \cdots \times a}}_{ \color{RoyalBlue}{p\ \text{times}}} \end{align} $$
• In measuring extremely tiny or extremely big quantities, e.g. the term “nanometer” means \(10^{-9}\) meter; speed of light is approximately \(3 \times 10^8\) m/s.

Definitions

Exponents

1. In an expression \(a^k\),  \(a\) is called the base and \(k\) is called the exponent or index.
2. A function involving exponent is called exponential function.
3. \(a^0 = 1,\) for all \( a \neq 0 \)
4. \(0^k = 0,\) for all \( k \neq 0 \)
5. \(0^0\) is undefined

Logarithms

1. $ \log_a x = y \text{ means } a^y = x , $ $ a$ is a non-zero Real no.
2. \(a\) is called the base and \(y\) is called the exponent.
3. In log expressions, when the base \(a\) is equal to 10, i.e. \( \log_{10}x \), it is usually denoted by \(\log x\) (no need to write 10 as the base).
4. When studying Further Math, need to know that when the base \(a\) is equal to constant e, i.e. \(\log_e x\), we call it the natural log of \(x\), written as \(\ln x\).

Properties

For the properties below, \(\, a, b, m, n \in \mathbb{R} {\setminus}  \{0\} \\[1em]\)

Exponents

1. \(a^m \cdot a^n = a^{m+n}\)
2. \( (a^m)^n = a^{m \cdot n} \)

Logorithms

1. $ \log_a mn = \log_a m + \log_a n $
2. $ \log_a \dfrac{m}{n} = \log_a m-\log_a n $
3. $ \log_a m^n = n\log_am $
4. $ \log_a m = \dfrac{\log_b m}{\log_b a} $

Problem Solving Techniques

Why are Inequalities Important?

Sometimes we do not know exactly the value of an unknown, but we know the boundaries of possible values.

Long before having the knowledge of Irrational nos., people knew that the circumference to diameter ratio, \(\pi\) is greater than 3. And 3 is then the lower bound of \(\pi\). People then worked hard to narrow down the boundaries (both lower and upper bounds) of \(\pi\) and getting more and more accurate value of it although its exact value can neither be written as a fraction nor a decimal figure because it is an irrational no. 

Certain inequalities are particularly important in algebra. Besides Linear Programming, concepts of inequalities are frequently used in other areas, like Graph of Functions, Geometric Sequences and Arithmetic Sequences, etc.

Elementary Properties

Please note that in (1.3), \(ab >0\) means they are of the same sign.

Example

Quadratic Inequalities

Considering a quadratic equation, if the equal sign is replaced by an inequality sign, then it becomes a Quadratic Inequality.

Steps of Solving Quadratic Inequalities:

Recall the Quadratic Formula: