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THE NUMBER LAND



Natural numbers:
     To count apples and oranges, we need numbers like 1,2,3....These whole numbers are the natural numbers.  They are marked on the right side of the number line.

 Negative numbers:
     A car runs in the north direction with the speed of 50 km per hour.  Another car travels in south direction with the same speed.  The second car is said to be running with the speed of  -50 km per hour with respect to the starting point.  So we need negative numbers like -1,-2, -3.  They are marked on the left side of the number line.

Zero:
     How many apples are there in the empty basket? - nothing.
So, we need a symbol for nothing and that is 0.  0 is marked in the middle of the number line separating negative and positive numbers.

 Rational numbers:
     To share one apple among 4 people, we cut the apple into 4 pieces.  So each one get one fourth(1/4) of a apple.  So the rational numbers are those numbers which can be expressed as a ratio between two integers.
example: 5/2;   827/100
     There are many rational numbers between any two natural numbers.

 Irrational numbers:
     It is a number which cannot be written as a perfect fraction.
example; pi=22/7=3.141....
                     1/3=0.3333....  not ends.

 So for, the variety of numbers we have seen are known as real numbers since they exist in reality.

 Imaginary numbers:
     The only solution for the equation   x^2+1=0  is  x=⎷-1  . But square root of -1 does not have a real meaning.  So we call it imaginary number and give it a symbol  i

                                                                   x^2+4=0
                                                                   x^2= -4

                                                                       x= ⎷-4

                                                                        x=2*⎷-1
                                                                        x=2i
So 2i is the solution for the  x^2+4=0.   Like that , we may need imaginary numbers like 1i, 2i,3i...  Now, where to represent the imaginary numbers.
When 1 is given 180 degree rotation, it becomes -1 and settles on left side of number line.  Similarly, when all the positive numbers are given 2 pi radian rotation, they become negative numbers and fall on the left side.

                                    Now   i=⎷-1 =  -1^(1/2)  so i is the -1 to the power half.

     So logically, all positive numbers should be given half rotation(90 degrees).  Then they will fall as 1i,2i,3i,...  on the vertical line passing through the 0. Negative imaginary numbers are marked in the bottom side of the vertical line.

 Complex numbers:
     Take a point on the plane having number lines.  The point will be represented by both the real number and the imaginary number.
                  example;3+2i
    This number is called complex number.

      To conclude; real numbers are represented by horizontal line and the imaginary numbers are by the vertical line.  The plane contains complex numbers.  The mathematics is built on this number land.
   


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