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Properties of Real Numbers


We discuss the properties of real numbers including integer, rational and irrational numbers. This class of numbers plays a key role in algebra as well as in real analysis. For example, the density of rational numbers in the set of real numbers is the key to proving many known mathematical results.

A concise summary of properties of real numbers

We note that the modern real analysis is mainly based on the set of real numbers. In fact, this set enjoys very important properties that can be translated to the other general sets. Let us discover together this magic set.

The set of natural numbers, positive integers

We use natural numbers every day in real life. These numbers are 1,2,3, … For example to count money, to count the population of a city or a county. These numbers are called natural numbers and we have collected them in a set of the form \begin{align*} \{1,2,3,4,\cdots,1000,\cdots\}. \end{align*} The complete natural numbers set is the following set \begin{align*} \{0,1,2,3,4,\cdots,1000,\cdots\}. \end{align*} We have only added 0 to the first natural numbers set. For simplicity, this set will be denoted by $\mathbb{N}$. Natural numbers are called positive numbers ”think of a positive balance in a bank account”. One may wonder why we added zero to the set of natural numbers. Really, zero is very important in all of the algebra ”and of course in real life”.

The zero is the transition point from positive to negative ”if your bank account is exhausted, your bank tells you that you have 0 dollars, and sometimes the bank allows you to withdraw another sum of money so you will be debited it i.e. a negative balance, so to go from positive to negative, you have to go through zero”.

We recall that if $n$ is a natural number, then \begin{align*} \underset{n\;\text{times}}{\underbrace{1+1+\cdots+1}}=n. \end{align*}


Whole numbers also called whole numbers are only the union of positive natural numbers and negative numbers, i.e. the set \begin{align*} \mathbb{Z}=\{\cdots,-3,-2,-1,0,1,2,3,\cdots\}. \end{align*} For example, if your Bank account is debited by 100 dollars, then we say that your account balance is -100$.

If $n$ is natural positive number then $$n+(-n)=0.$$ We also write $n+(-n)=n-n=0$. For example $2+(-2)=0$. On the other hand if $n$ and $k$ integers then the following multiplication operations hold: \begin{align*} &(-n)\times k=-(n\times k)=-(nk),\cr & (-n)\times (-k)=n\times k=nk,\cr & n(k+p)=nk+np. \end{align*}

For example, \begin{align*} &(-2)\times 3=-(2\times 3)=-6,\cr & (-2)\times (-3)=2\times 3=6. \end{align*} The rule is $(-)\times (-)=(+)$ and $(-)\times (+)=(-)$.

Let $p$ be an integer and $n$ be a positive natural number. If we multiply $p$ by its self $n$ times then we get another integer denoted by $p^n$. We write \begin{align*} p^n=\underset{n\;text{times}}{\underbrace{p\times p\times \cdots\times p}} \end{align*} For example $2^3=2\times 2\times 2=8$. We mention that $p^0=1$. We have also $(-1)^2=(-1)\times (-1)=1$ and $(-1)^3=(-1)\times (-1)\times (-1)= 1\times (-1)=-1$. From this we deduce that $(-1)^n=1$ if $n$ is an even number and $(-1)^n=-1$ if $n$ is odd number.

if $a$ and $b$ are integers and $n$ is a natural number then \begin{align*} (ab)^n=a^nb^n. \end{align*}

Rational numbers

If $p$ and $q$ are two integers, then we say that p divides q if there exists an integer $k$ such that $q=kp$. The result of the division is $k$ and we write \begin{align*} k=\frac{q}{p}. \end{align*} Sometimes dividing $ p $ by $ q $ is not possible in the sense that the result $ \frac {q} {p} $ is not an integer.

A number $ r $ is called rational if it takes the following form
where $ p $ and $ q $ are integers with $ p $ different from zero. Consequently, a rational number can be an integer if $ p $ divides $ q $. So the set of rational numbers noted $ \mathbb{Q} $ contains the set of whole numbers and we write $ \mathbb{Z} \subset \mathbb{Q} $.

If $a,b,c$ and $d$ are integers such that $b$ and $d$ are different from zero, then \begin{align*} \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},\quad \frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}. \end{align*} This shows that some and the difference between two rational numbers are also rational numbers.

Real numbers

A number $x$ is called irrational if it can not be written as $\frac{p}{q}$ with $p$and $q$ are integers and $q$ no null. The union of rational and irrational number sets is a set called real numbers set and will be denoted by $\mathbb{R}$.

Some notation: if $x$ is a real number we write $x\in \mathbb{R}$ ”we say that $x$ belongs to $\mathbb{R}$”. If a real number is positive we then write $x\ge 0$ and if it is negative we write $x\le 0$ ”we say also strictly positive if we can replace $le$ by the symbol $<$ and strictly negative if we can replace $ge$ by $>$”. We note that if $x\le 0$ then $-x\ge 0$.

We have seen in algebra for beginners that any real number is a limit of a sequence of rational numbers “this property is called the density of the rational set $\mathbb{Q}$ in the real set $\mathbb{R}$”. It is one of the most important properties of real numbers.

We also have the following properties

  • If $x$ is a real number and $n$ a natural number then $x+x+\cdots+x=nx$,
  • If $n$ is integer then $1+2+3+\cdots+n=\frac{n(n+1)}{2}$.
  • If $a$ is a real number such that $a$ not equal to 1, and $n$ is a natural number, then $1+a+a^2+\cdots+a^n=\frac{1-a^{n+1}}{1-a}$. For proof of this property see the remarkable identities page.
  • For any real number $x$, we associate a positive number denoted by $|x|$ and defined by $|x|=x$ if $x\ge 0,$ and $|x|=-x$ if $x\le 0$, and also $|0|=0$. We note that for any real $x$, $|-x|=|x|$.
  • For any reals $a$ and $b$, we have $|a b|=|a||b|,$ $|a+b|le |a|+|b|$ and $|a-b|le |a|+|b|$.
  • For any non-null numbers $a$ and $b,$ we have \begin{align*} \frac{1}{\frac{a}{b}}=\frac{b}{a}. \end{align*}

Numbers worksheet

In this section, we shall use the properties of real numbers to solve some exercises.

Exercise: Simplify the number \begin{align*} A=1+\frac{2}{1+\frac{3}{2}}. \end{align*}

Solution: We have first $1+\frac{3}{2}=\frac{2}{2+3}=\frac{5}{2}$. Then $A$ becomes \begin{align*} A=1+\frac{2}{\frac{5}{2}}= 1+ 2 \frac{2}{5}=1+ \frac{4}{5}=\frac{5+4}{5}=\frac{9}{5}. \end{align*}

Exercise: Calculate \begin{align*} B=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}. \end{align*}

Solution: Remark that $4=2^2$ and $8=2^3$. Then \begin{align*} B&=1+\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3\cr &=\frac{1-\left(\frac{1}{2}\right)^{3+1}}{1-\frac{1}{2}}. \end{align*} Remark that $\left(\frac{1}{2}\right)^{3+1}=\frac{1}{2^4}=\frac{1}{16}$ and $1-\frac{1}{2}=\frac{1}{2}$. then \begin{align*} B&=\frac{1-\frac{1}{16}}{\frac{1}{2}}= 2 \left(1-\frac{1}{16}\right)\cr &= 2\;\frac{16-1}{16}=2\;\frac{15}{16}=\frac{15}{8}. \end{align*}

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