## What does exponential mean?

Exponential is a mathematical term,which indicates the variation of anything in terms of the exponential constant -**e.**Whenever something gets increased or gets reduced in terms of

**2.718 times**,we call it

**exponential**.

####
What is **e**?

**e**is a

**universal constant**, having a value approximately 2.718

#### What does e represent?

The e represent the scale of growth of such an exponential action.In the physical world,we can see things

**grows and dies**. Mostly living things,or running phenomenon's.

**Nature have a policy of growth**. We don't know why; may be because that's easier,faster and comfortable. The term exponential refers to such a growth.

Generally,

**everything grows by multiplying**. The micro organisms like

**bacteria**,

**Radio Active decaying**etc

#### So How do we interpret e?

Before interpreting e and exponential growth,lets examine how things multiply.#### How do nature multiplies things?

We know, the easier and simplest multiplication is doubling. Nature too ,depends on it.1 become 2,2 become 4,4 become 8 and so on.

Mathematically we can represent this in terms of 2's power or 2's multiples.

**2^x**is the general equation of multiplication. It continues as 2*2*2*2*2*2...

#### Then what Is the need of exponent?

We already said,**exponential means growth/multiplying**. So,lets consider this 2*2*2*2*2...as the growing rate.

#### Growth = 2^x

#### But from Where did we got this 2?

Actually it is 1 in the beginning. 1 becomes 2,2 becomes 4...So,in the beginning there was 1,it multiplied into 2. Which means,the

**multiplication was 100%**.

A 100% multiplication means,doubling. Getting the equal amount again,means getting the whole again. The whole is 100%. Thus we can think, the

**2 is actually the initial 1 and the multiplied 100%**

**2 = ( initial condition + finalising condition)**

**Initial condition is the whatever amount we had. If we wanna double; the finalizing condition is a 1. If we only wanted to increase the growth by a small percentage, the value of 1 reduces up to 0, or becomes larger when we want a big growth.**

Generally big and low growths are artificial. Every growth

*(positive or negative)*depends upon the current condition

*(current condition will be the initial condition of that particular instant)*

A

**negative growth**is nothing but decay.

So,try drawing a flow chart. 1 becomes 2,2 becomes 4 and so on.

The interesting twisting fact is that, at the beginning the responsibility of growth was with the 1.

In the next step the growth,which is at a constant rate,became the responsibility of 2.

This two divides the responsibility and creates another 2-the new 4.

So totally the responsibility of growth resides now with 4.

To easily visualize this, consider a family. Parents give birth two 2 children. The responsibility of family now resides within the two children. Who will marry, give birth to 2 each. The total 4 children again marries and give birth to 8.

*The only difference here is no marriage is taking pace.*

Flow Chart |

Now remember, initially the first grandfather had the responsibility to start his family. Only him can - 100%.

Next the two children had the equal responsibility to grow the family 50% and 50%.

Generation by generation, it becomes - 25%,25%,25%,25%.

Then next generation will be 12.5%,12.5%,12.5%,12.5%,12.5%,12.5%,12.5% ,12.5% and so on.

Multiplication / Growth (The Green and Blue markings represents the indivitual growth, which we finally adds) |

Individual Contribution |

nth Generation |

Generation Growth |

Finding the Limiting Value of e |

The responsibility of each generation reduces,but the

**growth rate**of family is equal ,if we consider the

**whole members of each generation.**

Now we can consider our doubling equation.

2 = (1 + 1) becomes, 2 = (1 + 0.5 + 0.5),It again becomes 2 = (1+ 0.25 + 0.25 + 0.25 + 0.25)

When it undergoes

**x times growth**, the final part,

**splits**into

**x equal parts.**Where the

**individual contribution of a single one (the single family of nth generation) will be 1/n**

We take the

**2^x**equation,and substitute

**(1 + 1/n)**to get the individual growth sequence.

To get the total value of entire families of

**nth generation**, we add them,n times. We don't know n, so we

**consider n is approaching infinity**,and applies the techniques from

**calculus**to solve. This will give us a value Approximately

**2.71828**... That's how

**e**got born.

*This is not the exact mathematical approach,yet it is simpler than any textbook language.*

**(I hope)**

Every Process in nature,multiplies in this mode, at the rate of 2.718. It is a factor like

**Pi**,which is

**3.1428**...

**Pi is actually the ratio between circumference and diameter of a circle,while e is the constant growth factor of nature**.

Now we know,

**what is e**.

When we make a function,on its terms,which is causing the growth ,we call the growth is exponential,and function is -

**Exponential Function**

This is how,if we analyse the limit graphically(image credit : math is fun) |

**I will be telling more about e,in my next article :**

**what is the application of e?**

**what is the physical meaning of e intuitively?**

**what is the significance of e?**

**why always e?**

**where all we can find e?**

**How is e related to logarithm?**