In this post I’m going to talk about the natural logarithm, continuous growth and their relationship. In the process, I’m also going to derive * e*.

The story begins with the exponential function: . We want to know its derivative.

To be continued…

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# Category: Mathematics

# What’s so natural about ‘e’?

# Euler’s number

# Bayes’ Rule Explained

# Mandelbrot Set

exploring chaos

In this post I’m going to talk about the natural logarithm, continuous growth and their relationship. In the process, I’m also going to derive * e*.

The story begins with the exponential function: . We want to know its derivative.

To be continued…

Euler’s number (*e*) and an understanding of its not-so-obvious relationship with continuous growth has eluded so many of us, it’s only apt that we take some time out of our daily lives to sit down and have a little chat.

**An Example**

Interest earned on an investment of $100 @ 10% annual interest-rate is $100 a year. Usually, this $10 will be paid out at the end of the year. This can be modeled as:

If you don’t want to wait for one whole year to earn interest, you may ask for the accrued interest to be paid every six months. In that case, you earn $5 twice a year.

Following this logic, interest could be earned as frequently as desired while the total interest earned in a year remains the same. For example, you could demand 100 interest payments to be made at equal intervals of time within a year, net interest earned still being $10 per annum.

Okay, so now you’re getting 100 interest payments a year. By the end of the year, you’re able to add 10% to your investment.

You’re happy with the way things are until a friend shows up who tells you there’s a way to make more money, and it’s by re-investing the interest earned! Between each interest payment, you add the previously earned interest to the principal amount, thus changing the interest that will be earned next. Unlike previous scheme, the amount of interest paid keeps increasing. As the number of payments (n) approach infinity, then:

To generalize, let’s say x = 0.1. Then:

And so, can be rewritten as:

You wake up one morning and feel something isn’t right. Your head hurts. You’re aware that recently there’s this virus on the loose that’s causing people to lose their minds and you worry that you might be infected. This disease is aptly named Mindlostisis and no cure exists. So you rush to the nearest hospital and get yourself tested. Test comes positive. Doctor tells you 90% of people who have Mindlostisis test positive. Should you be worried?

No, not yet. ** Because what if 90% of people who don’t have Mindlostisis also test positive? ** The doctor has given you a number that you don’t care about, namely the probability of testing positive if you have the disease. Instead, what you

**P(T)** = Probability of the test coming positive.

**P(M)** = Probability of having Mindlostisis.

**P(T|M)** = Probability of a positive test given that you have Mindlostisis.

**P(M|T)** = Probability of having Mindlostisis given that the test comes positive.

Rephrasing what’s been said earlier, the doctor gave you **P****(T|M)** when it’s **P(M|T)** that you’re really interested in. And this is where Bayes’ Rule comes in:

This equation is far from obvious so let’s try and understand it using a Venn diagram.

** U** = Set of all people in the city.

Additionally, total number of people in any set is denoted in lowercase. For example, total number of people in ** X** is

What you want to know is **P(M|T)**, which is the ratio of number of people in * A* and number of people in

In order to calculate **P(M|T)**, we do need **P(T|M)** but that’s not enough. Something more is required, and let that something be * s*.

Putting it all together we get:

Clearly, to be able to tell your chances of having the disease given a positive test, you not only need to know what the doctor said, but also: **(P(M) / P(T))**.

For my viewing pleasure.

R code:

library(ggplot2) mandelbrot_set <- function(precision) { is_mandelbrot_point <- function(a, b) { orig_a <- a orig_b <- b n_iters <- 30 # defines escape propensity for (i in seq(n_iters)) { x <- a y <- b a <- (x^2 - y^2) + orig_a b <- (2*x*y) + orig_b } # If the number escapes to infinity, it's not a mandelbrot point. if (is.nan(a) || is.nan(b)) F else T } n <- 3 # axes restricted to -3 to +3 x <- NULL y <- NULL for (i in seq(-n, n, precision)) for (j in seq(-n, n, precision)) { flag <- is_mandelbrot_point(i, j) if (flag) { x <- c(x, i) y <- c(y, j) } } data.frame(x, y) } mdat <- mandelbrot_set(.01) ggplot(mdat, aes(x, y)) + geom_point(color="brown", size=.1) + theme(axis.title=element_blank(), axis.text=element_blank(), axis.ticks=element_blank())