0.1: Basics

Mathematical objects come into existence by definition. These definitions must give an absolutely clear picture of the object or concept. We don't need to prove them; we simply need to define them clearly. We are going to state some basic facts that are needed in this course:

Basic Facts on Sets:

• The collection of counting numbers, otherwise known as the collection of natural numbers, is usually denoted by \(\mathbb{N}.\) We write \(\mathbb{N} = \{ 1,2,3,4, \dots\}.\)
• The collection of the integers is usually denoted by \(\mathbb{Z}\) and we write \({\mathbb{Z}} = \{ \dots,-3,-2,-1,0,1,2,3,4, \dots\}.\)
• The collection of the positive integers is usually denoted by \(\mathbb{Z_+}\) and we write \({\mathbb{Z_+}} = \{ 1,2,3,4, \dots\}.\)
• The collection of the negative integers is usually denoted by \(\mathbb{Z_-}\) and we write \({\mathbb{Z_-}} = \{ -1,-2,-3,-4, \dots\}.\)
• The collection of all rational numbers (fractions) is usually denoted by \(\mathbb{Q}\), and we write \({\mathbb{Q}} = \left\{ \frac{a}{b}: a \mbox{ and }b \mbox{ are integers}, \, b \ne 0 \right\}.\)
• The collection of all irrational numbers is denoted by \({\mathbb{Q^c}}\).
• The collection of all real numbers is denoted by \(\mathbb{R}\). This set contains all of the rational numbers and all of the irrational numbers.

Basic Facts:

We shall assume the use of the usual addition, subtraction, multiplication, and division as operations and inequalities (\(<, >, \leq, \geq)\) and equality (\(=\)) are relations on \(\mathbb{R}\).

1. The distributive law: If \(a,b\) and \(c\) are real numbers, then \(a(b+c)=ab+ac\) and \((b+c)a=ba+ca.\)
2. The commutative law: If \(a\) and \(b\) are real numbers, then \(ab=ba\) and \(a+b=b+a.\)
3. The associative law: If \(a,b\) and \(c\) are real numbers, then \(a+(b+c)=(a+b)+c\) and \(a(bc)=(ab)c.\)
4. The existence of \(0\): The real number \(0\) exists so that, for any real number \(a, a+0=0+a=a.\)
5. The existence of \(1\): The real number \(1\) exists so that, for any real number \(a, a \cdot 1=1 \cdot a=a.\)
6. Subtraction: For each real number \(a,\) there exists a real number \(-a,\) so that \(a+(-a)=0=(-a)+a.\)
7. Division: For each nonzero real number \(a,\) there exists a real number \(\displaystyle\frac{1}{a},\) so that \(a\left(\frac{1}{a}\right)=\left(\frac{1}{a}\right)a=1.\)

The laws above form the foundation of arithmetic and algebra of real numbers. They are the laws that we have accepted and used with no reserve. They are mentioned here to encourage the reader to develop an appreciation for them and an awareness that they must be respected in all calculations involving real numbers. Further, there are rules of precedence that help us evaluate any valid arithmetic expression. For example, if given the following \(6 \div 2 \times 5-\dfrac{7}{5}-3\), we will apply the rule to calculate.

Example \(\PageIndex{1}\):

Evaluate \(6 \div 2 \times 5-\dfrac{7}{5}-3\).

Solution

Multiplication and division have the same priority and should be performed
from left to right. Therefore,
\(6\div 2\times 5=3\times 5=15.\)

Hence,
\(6\div 2\times 5-\frac{7}{5}-3=15-\frac{7}{5}-3.\)

Thus,
\(12-\frac{7}{5}=\frac{60}{5}-\frac{7}{5}=\frac{53}{5}.\)

 

Rules of Precedence

1. Functions are evaluated first.

2. Expressions inside parentheses or brackets are evaluated next.

3. Multiplication and division are next and evaluated left to right.

4. Addition and subtraction are last and are evaluated left to right.

Caution

BEDMAS stands for: \( \boxed{ \begin{array}{ll} \text{B} & = \text{Brackets}\\ \text{E} & = \text{Exponents}\\ \text{D} & = \text{Division}\\ \text{M} & = \text{Multiplication}\\ \text{A} & = \text{Addition}\\ \text{S} & = \text{Subtraction} \end{array} } \)

PEMDAS stands for: \( \boxed{ \begin{array}{ll} \text{P} & = \text{Parentheses}\\ \text{E} & = \text{Exponents}\\ \text{M} & = \text{Multiplication}\\ \text{D} & = \text{Division}\\ \text{A} & = \text{Addition}\\ \text{S} & = \text{Subtraction} \end{array} } \)

why  ''PEMDAS'' or ''BEDMAS'' can cause misunderstandings. see below

Order of Operations

The mnemonics PEMDAS and BEDMAS can cause misunderstandings because they may suggest that operations are always performed in a strict left-to-right order. In fact, multiplication and division have the same precedence, and addition and subtraction have the same precedence. Operations with the same precedence are performed from left to right.

For example, consider \( 8\div 2\times 4. \) A student who interprets PEMDAS as ``multiply before divide'' might calculate \(8\div(2\times4)=1. \) However, multiplication and division have the same precedence, so we work from left to right: \( 8\div2\times4 =4\times4 =16. \)

Similarly, addition and subtraction have the same precedence. For example, \( 10-3+2=7+2=9, \) not \( 10-(3+2)=5. \)

A better way to think about the order of operations:

\( \boxed{ \begin{array}{c} \text{Grouping Symbols}\\\downarrow\\\text{Exponents}\\ \downarrow\\ \text{Multiplication and Division, from left to right}\\ \downarrow\\\text{Addition and Subtraction, from left to right} \end{array}} \)

Thus, multiplication does not always come before division, and addition does not always come before subtraction. Operations at the same level of precedence are performed from left to right.

 

 

Recall that, if \(a\) and \(b\) are real numbers or \(a, \, b \in \mathbb{R}\) as written in mathematical language, then

1. \(a < b\) means that \(a\) is less than \(b.\)
2. \(a > b\) means that \(a\) is greater than \(b.\)

Definitions:

1. A real number is positive if it is greater than \(0\).
2. A real number is called non-negative if it is greater than or equal to \(0\).
3. An integer \(n\) is an even number if there is an integer \(m\) such that \(n=2m\).
4. An integer \(n\) is an odd number if there is an integer \(m\) such that \(n=2m+1\).
5. An integer \(a\) is said to be divisible by an integer \(b\) if there is an integer \(m\) such that \(a=bm\). In this case, we can say that \(b\) divides \(a\), denoted by \(b|a\). Furthermore, \(b\) is called a divisor (or factor) of \(a\).
6. A positive integer \(p\) is called prime if \(p>1\) and the only positive divisors of \(p\) are \(1\) and \(p\).
7. A positive integer \(n\) is called composite if there is a positive integer \(m\) such that \(1<m< n\) and \(m|n\).

Note

Note that \(1\) is neither prime nor composite.

Axioms for Inequalities

The following are axioms for inequalities:

• Trichotomy Law: if \( x \) and \( y \) are real numbers, then one and only one of the three statements \(x < y, x = y\) and \(y < x\) is true.
• Transitivity: if \(x, y \) and \(z\) are real numbers and if \( x < y\) and \(y < z\) then \(x < z.\)
• if \(x, y \) and \(z\) are real numbers and if \( x < y\) then \(x + z < y + z.\)
• if \( x \) and \( y \) are real numbers which satisfy \(0 < x \) and \( 0 < y\) then \( 0 < xy.\)

 

Definitions:

• A theorem is a mathematical statement that has been proved to be true using definitions, axioms, previously established results, and logical reasoning.
• A conjecture is a mathematical statement that is thought to be true but has not yet been proven formally in the field.​
•  An axiom is a statement accepted without proof.
• A definition is a precise description of a mathematical concept.
• A lemma is a proved result used mainly to prove another result.
• A corollary is a result that follows directly from a theorem.
• A proposition is a proved mathematical statement, often of moderate importance.​