0.1: Basics

Mathematical objects come into existence by definitions. 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 which help us to calculate 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

\(6 \div 2 \times 5-\dfrac{7}{5}-3= (( 6 \div ( 2 \times 5))- (7 \div 5))-3=-\dfrac{16}{5}.\)

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.

In short form:

Order of Operations

B Brackets

E Exponents

D Division

M Multiplication

An Addition

S Subtraction

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 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 \(a\) and is denoted \(b|a\). Further, \(b\) is called a divisor (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 \( 0 < xy.\)

 

Definition:

A theorem is a declarative statement about mathematics for which there is proof.