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ALL MATHEMATICS IN JUST ONE EQUATION

    Say, you are dropping a ball.  The height it had descended at the time 't' is given by the formula 1/2gt^2.  Here g=9.8 acceleration due to gravity.

S= 1/2gt^2
1/2g =9.8/2 =4.9
hence S= 4.9t^2

    The simple formula gives the distance fallen at any instant 't'.  The distance increases as the square of the time since the ball accelerates continuously due to gravity.  This is the meaning of the formula which we are going to explore.  You understand that y=x^2 is similar to that formula.

1. Let us plot a curve for this equation.-figure 1
  We get a parabola, an important shape in math and physics.  Here, for  every y, there are two values for x. 
x= -2 and +2, both satisfies the equation.
y= (-2)^2=(+2)^2 =4
This kind of equation is called quadratic equation in Algebra

2.  If x^2=y or x^2 =-1, what is the solution.  X can be root of minus one. But "root of -1" does not exist in real numbers.  So we call it imaginary number and denote it by letter 'i'. Now , we can have new number system as 1i,2i,3i... or 2+3i, 1+i, etc.  These are called complex numbers (real + imaginary).  Complex Algebra is born.

3. Y=x^2 .  What is 2 in terms of y and x.  2 is the logarithm of y for the base x.
consider, 25 =5^2
Hence, Log 25 =2
            
This is the principle of Logarithm.

3. Once again draw the parabola-figure 2
What is the slope of the curve at any point x.  The calculus (differentiation) says, it is 2x.  At x=2, it is 2*2=4.  At x = 5, it is 2*5=10.  It is obvious from the curve.
    What is the area under the curve between two points.  The 'integration' says it is x^3/3.  Between the points x =-1 and x=+1,
the area is 1^3/3 - (-1)^3/3
                  =1/3 +1/3 = 2/3 units.
    The answer can be verified using above figure-2.
    Differentiation is the rate of change of y with respect to x.  Integration of summation of y from one point to another(x).

4. Throw a ball horizontally from a top of the building.  It will follow a parabolic path as given in figure 3
    The parabolic path is again due to the formula component 1/2gt^2.  So, this is the physical and visual simulation of the equation y =x^2.

5.y=x^2 is also the area of the square of side x.  As x increases, x^2 =y increases rapidly.  The above geometrical figures-4  illustrate it clearly.

6. Y=x^2.  It is x to the power of 2.  That means multiplying x by itself.  It also means adding x, x times.  Every mathematical operations originates from addition.  Computer works on this principle.

7.  Y=x^2 is a function, a model, a machine.  X takes in certain values as input and gives out y as output.  It is math engine of input -process-output.

   Y=x^2 is a simple equation using which we can understand many fields in mathematics.  

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