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# 0: Introduction

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## Real Number Arithmetic

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 laws: 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.

## The partitioning of the real number system

The collection of all real numbers contains a number of important sets. These are introduced next together with the appropriate standard notation.

1. The collection of  counting numbers otherwise known as the collection  of  natural numbers is usually denoted by $$\mathbb{N}.$$ We write $$\mathbb{N} = \{ 1,2,3,4, \dots\}.$$
2. The collection of the integers is usually denoted by $$\mathbb{Z}$$ and we write $$\bf{\mathbb{Z}} = \{ \dots,-3,-2,-1,0,1,2,3,4, \dots\}.$$ Notice that  $$\bf{\mathbb{N}}$$ is properly contained in $$\mathbb{Z},$$ denoted by $${\mathbb{N}} \subsetneqq \mathbb{Z}.$$
3.  The collection of all rational numbers (fractions) is usually denoted by $$\mathbb{Q}$$ and We write $$\bf{\mathbb{Q}} = \left\{ \frac{a}{b}: a \mbox{ and }b \mbox{ are integers}, \, b \ne 0 \right\}.$$ Notice that  $$\bf{\mathbb{Z}}$$ is properly contained in $$\bf{\mathbb{Q}}$$ That is, $${\mathbb{Z}} \subsetneqq \mathbb{Q}.$$
4. The collection of all irrational numbers is denoted by $$\bf{\mathbb{I}}$$. This set contains all the real numbers which are not rational numbers. The rational numbers and irrational numbers have no elements in common. That means $$\bf{\mathbb{Q}} \cap \mathbb{I} = \emptyset.$$
5.  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. We write $$\bf{\mathbb{Q}} \cup \mathbb{I} = \mathbb{R}.$$

Definitions

1.  A real number is called 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 $$n=2m+1$$.

## Special properties of the real numbers

It is worthwhile mentioning the following three properties:

1.  The collection of the real numbers has an order:

Given any two distinct real numbers $$a$$ and $$b$$, one of the following statements
will be true:

1. $$a<b,$$
2. $$b<a.$$

Example $$\PageIndex{1}$$:

Consider the real numbers $$3.111$$ and $$3.11101,$$ it is not difficult to see that
$$3.111<3.11101.$$

2.  The collection of the real numbers is complete:

Given any two distinct real numbers, there will always be a third real number which will lie in between
the two given.

Example $$\PageIndex{2}$$:

Given the real numbers $$1.99999$$ and $$1.999991,$$ we can find the real number $$1.9999905$$ which certainly lies in
between two.

3. The collection of the real numbers has neither a greatest element nor does it have a least element.

## Useful hints  for calculations without your calculator

Often in calculations, multiplication and division will be necessary, and although calculators are now
available to help us through these chores, it is useful to develop a facility with these two operations.
To this end, we conclude here some very useful facts.

1. An integer is an even number if and only if it is divisible by $$2..$$
2. An integer  is divisible by $$3$$ if and only if the sum of its digits is divisible by $$3.$$

Example $$\PageIndex{3}$$:

$3579$  is divisible by $3$ since the sum of its digits is $24$ which is divisible by $3.$

$476911$  is {\bf not} divisible by $3$ since the sum of its digits is $28$ which is not divisible by $3.$

​​​​​​​​​​​​​​​​​​​3. An integer  is divisible by $5$ if and only if  its last digit (the units digit) is either $0$ or $5.$

Example $$\PageIndex{4}$$:

$27795$  and $46790$ are both divisible by $5.$

$3714$ is not divisible by $5$ since its last digit (the units digit) is $4.$

​​​​​​​4. Given any integer, if the difference between the sum of every other digit and the sum of the digits which remain is divisible by $11$, then the integer is itself divisible by $11$.

Example $$\PageIndex{5}$$:

$(4+7+1+7)=19$ and $(6+9+2+2)=19,$   $(4+7+1+7)-(6+9+2+2)=0$ and $46791272$ is divisible by $11.$
\item $9123917$ is divisible by $11$, since
$9+2+9+7=27$ and $1+3+1=5,$  $(9+2+9+7)-(1+3+1)=22,$ which  is divisible by $11.$

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