Introduction to e: The Magical Number of Growth
Hello! Welcome to your study notes for one of the most fascinating numbers in all of mathematics: the number e. If you've ever heard of pi (π), then think of e as its equally cool cousin. It might seem a bit mysterious at first, but don't worry! We're going to break it down step-by-step.
In this chapter, we'll discover what e is, where it comes from, and meet its best friend, the natural logarithm (ln). Understanding e is super important because it's the key to describing all sorts of real-world phenomena, from population growth to radioactive decay, and it's a superstar in calculus!
What is 'e'? The Story of Continuous Growth
The best way to understand e is to think about money and interest. It's an analogy that makes everything much clearer.
An Analogy: The Magical Bank Account
Imagine you put $1 in a special bank that offers an amazing 100% interest rate per year. Let's see how much money you'd have after one year depending on how often they calculate your interest.
The formula we'll use is $$A = P(1 + \frac{r}{n})^{nt}$$, where:
P = initial amount ($1)
r = annual interest rate (100% or 1)
t = number of years (1)
n = number of times interest is calculated per year
For our example, the formula simplifies to: $$A = (1 + \frac{1}{n})^{n}$$
Case 1: Interest calculated ONCE a year (n=1)
You get 100% of your $1 at the end of the year.
Amount = $$(1 + \frac{1}{1})^1 = 2^1 = $2.00$$
Case 2: Interest calculated TWICE a year (n=2)
You get 50% interest after 6 months, and then 50% on the new total for the next 6 months.
Amount = $$(1 + \frac{1}{2})^2 = (1.5)^2 = $2.25$$
Hey, that's more money!
Case 3: Interest calculated FOUR times a year (n=4)
Amount = $$(1 + \frac{1}{4})^4 \approx (1.25)^4 \approx $2.44$$
What happens if we keep increasing n? Let's make it calculate interest every day, every second, every instant!
- If n = 12 (monthly): $$(1 + \frac{1}{12})^{12} \approx $2.61$$
- If n = 365 (daily): $$(1 + \frac{1}{365})^{365} \approx $2.714$$
- If n = 1,000,000 (a million times): $$(1 + \frac{1}{1000000})^{1000000} \approx $2.71828...$$
Notice something amazing? As n gets bigger and bigger, the final amount gets closer and closer to a specific number. It doesn't grow to infinity! This special number, the limit of this process, is what we call e.
The Formal Definition of 'e'
In mathematics, we express this idea using a limit. This is the first official definition you need to know.
Definition 1: The Limit Definition
The number e is the value that the expression $$(1 + \frac{1}{n})^n$$ approaches as n becomes infinitely large. We write this as:
$$e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$$
e is an irrational number, just like π. This means its decimal places go on forever without repeating a pattern.
e ≈ 2.718281828...
Did you know?
The number e is often called Euler's Number, named after the brilliant Swiss mathematician Leonhard Euler, who did extensive work with it.
Key Takeaway
e represents the result of 100% continuous growth over one period. It's the natural limit to growth.
Another Way to Look at 'e': An Infinite Series
There's a second way to define e, which is also very powerful. It involves adding up an infinite number of terms. Don't worry, you just need to recognise this formula.
Quick Review: Factorials!
The exclamation mark in maths is called a factorial. It means you multiply a whole number by every whole number below it down to 1.
Example: $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$
Also, a special case: $$0! = 1$$
Definition 2: The Infinite Series Definition
We can define the function $$e^x$$ as an infinite sum:
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...$$
To find the value of e itself, we just let x = 1 in the formula above: $$e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ...$$ $$e = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + ...$$ If you add these terms up, you'll see they also get closer and closer to 2.71828...
Key Takeaway
There are two main ways to define e: as a limit representing continuous growth, and as an infinite series. Both lead to the same magical number!
Meet the Natural Logarithm (ln)
Every superhero has a sidekick, and for the function $$e^x$$, its sidekick is the natural logarithm, written as ln(x).
Quick Review: What is a Logarithm?
A logarithm is just a question. The expression $$log_b(a)$$ asks: "What power do I need to raise the base b to, in order to get the number a?"
Example: $$log_{10}(100) = 2$$, because you need to raise the base 10 to the power of 2 to get 100 ($$10^2 = 100$$).
Defining the Natural Logarithm
The natural logarithm is simply a logarithm with the base e.
ln(x) is the same as loge(x)
So, when you see ln(x), it's asking the question: "e to what power gives me x?"
Step-by-Step Examples:
- What is ln(e)?
This asks: "e to what power equals e?" The answer is clearly 1. So, ln(e) = 1. - What is ln(1)?
This asks: "e to what power equals 1?" Any number to the power of 0 is 1. So, ln(1) = 0. - What is ln(e5)?
This asks: "e to what power equals e5?" The answer is right there! It's 5. So, ln(e5) = 5.
The Most Important Relationship: They are Inverses!
The functions $$e^x$$ and $$ln(x)$$ are inverse functions. This means they "undo" each other, just like multiplication undoes division.
This gives us two incredibly useful rules: $$e^{\ln(x)} = x$$ $$\ln(e^x) = x$$
This property is essential for solving equations involving e or ln.
Key Takeaway
The natural logarithm, or ln(x), is just a logarithm with base e. It is the inverse function of $$e^x$$.
Why Do We Care? The Magic of 'e' in Calculus
So why is this number so special? The true beauty of e shines in calculus.
The function $$f(x) = e^x$$ has a mind-blowing property:
The derivative (or gradient) of ex is simply ex.
This means at any point on the graph of $$y = e^x$$, the value of the function is exactly equal to the slope of the tangent at that point. It is the only function (besides y=0) where its value and its rate of change are the same. This makes countless calculations in physics, engineering, and finance much, much simpler. It's why e is considered the "natural" choice for an exponential base.
Chapter Summary and Key Points
Great job making it through! It's a lot to take in, but here are the absolute must-know points.
- What is e? It's a special irrational number, approximately 2.718. It represents the limit of continuous growth.
- Definition 1 (Limit): $$e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$$
- Definition 2 (Series): $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$
- What is ln(x)? It's the natural logarithm, which means a logarithm with base e. ($$\ln(x) = \log_e(x)$$)
- Inverse Relationship: $$e^x$$ and $$\ln(x)$$ undo each other. This means $$\ln(e^x) = x$$ and $$e^{\ln(x)} = x$$.
- Why is it special for Calculus? The derivative of $$e^x$$ is itself, $$e^x$$. This makes it the "natural" base for calculus.
Keep reviewing these core ideas, and you'll be a master of e in no time. You've got this!