We are so happy, so happy . . . 

- Why you are so happy? 

We are so happy because the mother-nature gifted us with so beautiful environment . . . 

- What do you mean? 

I mean the numbers that we know and can use to count things . . . 

- What the numbers? 

First of all the whole numbers like  1,  2,  3,  4  and so on.  Indeed,  using these numbers you can count a lot of things. 

You can count your toys,  your pens,  apples in a dish on a table,  pages in your book,  your books itself,  cars in a street or bricks in pyramids,  and even the stars on the sky and grains of sand in a beach.  The numbers that you use for it are the numbers  1,  2,  3,  4  and so on,  and they are called the  whole numbers

People invented these numbers thousand years ago and use them till our days.  With these numbers you can count elements in any finite set of objects. 

And even if you have nothing,  you still have the number to express it.  It is the number  "zero",  or  0 The set of  whole  numbers plus the number of  0  is called the  natural numbers

You can imagine that these numbers are comfortably placed in a straight line as the points equidistantly distributed along this line.  This line is called the  number line


This is not all the story. 

You can count not only the whole things,  but their parts also.  For example,  you can count every half an hour.  If the apples are cut in fourths,  you can count fourths parts.  If your mother cut a pie or a pizza into  12  parts,  you can count that pieces.  Our numerical system has special numbers for it.  These numbers are  rational numbers  like  ,  ,  ,  ,    and so on. 

Similar to the whole numbers,  you can place the rational numbers in the number line too.  Moreover,  you can place them in the number line in a way that their ordering is in place:  each bigger number is located to the right of a smaller number. 

The rational numbers fill the number line by a dense way.  It means that for any small interval in the number line you can find a rational number inside this interval. 

But this is not the full story yet.  By the miraculous way,  there are numbers that are not rational.  For example,  the number    is not rational.  In opposite,  it is  irrational.  The ancient Greece mathematicians knew it.  They found the proof of the fact that the number    is not rational.  The elementary proof works and shows that there are infinitely many  irrational numbers.  For instance,  all the numbers  ,  ,  ,  , . . . ,  , ... are irrational,  if the whole number    under the square root sign is not exactly the full square.  (For the proof see the lesson  Proving irrationality of some real numbers  under the current topic in this site). 

By the similar way,  all the numbers  ,  ,  ,  ,  ,  ,  , . . . ,  , ... are irrational,  if the whole number    under the cubic root sign is not exactly the full cube.  (For the proof see the same lesson  Proving irrationality of some real numbers  under the current topic in this site). 

There are many other irrational numbers.  If you are acquainted with the famous numbers    and    from  Geometry  and  Calculus,  you probably know that they are irrational numbers too. 

Actually,  the amount of irrational numbers is even greater than the amount of rational numbers. 

Still there is a miraculous way to place all irrational numbers in the number line in a tiny gaps between the rational numbers in a way that every larger number lies to the right of the smaller one. 

Now,  when we put all the whole numbers,  all the integer numbers,  all the rational numbers and all the irrational numbers in the number line,  this line is full at last.  It is fully filled with the numbers.  There is no more place in the number line.  All  real numbers  are here.  All of them are in your possession and are ready to work for you. 

Thanks to having so many numbers people can measure and calculate such different things as the speed of light,  the mass of the proton,  the charge of the electron,  the radius of the Earth,  the distance to planets and stars and even more. 

That is why we are so happy.

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