Math: Calculus (Integral) Proof 0 ≠ 1

A "fun time". This is from Mudd Math Fun Facts. More of the Calculus "trick", integral form.

Quoted

Consider the following integral: ∫(1/x) dx -------------------------------------------------- Perform integration by parts: Let u = 1/x dv = dx du = -1/x² dx v = x -------------------------------------------------- Then obtain: ∫(1/x) dx = (1/x).x - ∫x(-1/x²) dx = 1 + ∫(1/x) dx (?)

Let's take a look at that last line, and continue that:

The last line: ∫(1/x) dx = (1/x).x - ∫x(-1/x²) dx = 1 + ∫(1/x) dx -------------------------------------------------- Continue: ∫(1/x) dx = 1 - ∫(-1/x) dx = 1 + ∫(1/x) dx -------------------------------------------------- Or, simplified: ∫(1/x) dx = 1 + ∫(1/x) dx (?) ---------------------------------------------------- Subtract both sides with ∫(1/x) dx, resulting: 0 = 1 (?!)

Where did we get that 1 number on the right side all of the sudden?

The answer is at the first defining step. The u, dv, du, and v variables. Those weren't supposed to be like that.

If we have ∫(1/x) dx to be solved using integration by parts, then the variables MUST be:

u = 1 dv = (1/x) dx du = 0 v = ∫(1/x) dx

Then the equation will become:

∫(1/x) dx = u.v - ∫v.du = 1.∫(1/x) dx - ∫ [∫(1/x) dx] . 0 = 1.∫(1/x) dx - 0 = ∫(1/x) dx And therefore: ∫(1/x) dx = ∫(1/x) dx ----------------------------------------------------- Subtract both sides with ∫(1/x) dx, resulting: 0 = 0

See? there's no "0 = 1", or other unequal numbers being equal.

Why can't we just use the first step?

Well, back to the top, the "equation" will give the additional 1 number on the right side. "0 = 1".

Therefore, the integration by parts can't be applied to solve that question. Because it'll just go round and round.

By looking at integral table, $∫ 1/x\ dx = ln\ |x| + C$

This is a "reference" problem.

How do you prove it? Uuuh, I'm not sure. But this is a good answer if you wanna read.

Qeustoin:	How is it $e$ related to $\ln x$?
Ansewr:	Yea, ln is log (logarithmic) function with $e$ number as its base. An abbreviation of logarithmus naturalis (Latin), and reversed in English syntax, natural logarithmic.
Que:	Is it?
Arrr:	Maybe...

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Let's break it down.
We have $f(x) = \ln x$ or $y = \ln x$

And then: $y =\ ^e\log\ x$

There's that "e", you can circle around merrily afterward.

---

Since Calculus is bunch of, let's look at the exponential function.

Exponential function
a function whose value is a constant raised to the power of the argument, especially the function where the constant is e.

Now let's use $e$ as its constant ► $f(x) = e^x$. Then for instance:

• Derivation of $f(x)$ with respect to $x$ ► $ d/{dx}\ e^x =e^x $
• Integration of $f(x)$ with respect to $x$ ► $ ∫e^x\ dx =e^x + C$
• ???

Then let's look at e (Euler's number -- irrational) definition.

It's a predefined reference, is it not.

Like "imaginary" number, or "infinity". So in order to get it "right", we should follow manual guide. There's that irrational part and it's a letter, not a variable, it's a number written with letter.

It is said the "e" was observed by Bernoulli when he studied compound interest rate, he saw pattern that if the frequency of accrued compound interest in a year went to "infinity", well, the person who had the initial money wouldn't be covered in money. A big win for the bank.

Now there were lots of Bernoullis in that time, this one was Jacob Bernoulli. He wore wig. Like all of them scientists, musicians, politicians, and all royal families from "middle age", "renaissance", "enlightenment", "Victorian", etc period.
You know, those people drawn in books, paintings and some were sculpted. They all have the wig on. Except some of the ladies, maybe.

At a glimpse, as such: ${lim}↙{x→∞}(1 + 1/x)^x$ will reach "e", saturation point, sort of speak.
$x$ is the compound interest frequency (daily, weekly, monthly, 2-monthly, 4-monthly, ... , yearly, et al). The rate is 100% in that experiment (or question), for a year, with initial deposit of $1. Bernoulli wanted to know if the interest is paid really really REALLY frequent, what will happen? And he got that 2 point something something something pattern.

The reference from calculator:

$$e = 2.71828182846$$

Or, to see longer sequence go to OEIS

The basic formulation for compound interest is like so: $ P(1 + i/x)^{xt} $

• $P$ is the initial money, or principal sum.
• $i$ is the compound interest rate -- in per cent, and must be converted to normal decimal or fraction, up to you (per centum, per 100, divided by 100).
• $t$ is the duration of how long the thing is. Usually in year unit.
• $x$ is the compound interest frequency (how many times it is "paid" to customer).

Let's make an example.

There's a bank for some reason, with 5% compound interest rate per year, compounded monthly. A feller deposited 1,000 in that bank for 10 years. How much money he will get at the end of the period?

First answer is not that much.

Second answer is like this:

• $P$ is 1,000 (no currency unit for this example).
• $i$ is 5% or 0.05
• $t$ is 10 (10 years).
• $x$ is 12 (monthly).

So dude will have: $ 1000(1 + 0.05/12)^{12×10} = 1647.009...$

The total compound interest for 10 years with 5% rate, credited monthly, with 1,000 of initial deposit yields 647 of interest. In theory. 10 years. Not to mention the monthly fees, and reduced value of banknote per year -- that magic depreciation politics, and whatever stuff they can come up with. It's like not 647 at all.

Now let's try weekly

Let's assume in one year there's 52 weeks.

Put the value into the formulation:

Hence: $ 1000(1 + 0.05/52)^{52×10} = 1648.325...$

Let's try daily

Let's assume in one year there's 364 days.

Put the value into the formulation:

Therefore: $ 1000(1 + 0.05/364)^{364×10} = 1648.664...$

Hourly

With the 364 days per year assumption, and one day is 24 hours, we have 364 × 24 = 8736 (hours per year).

So then: $ 1000(1 + 0.05/8736)^{8736×10} = 1648.718...$

And so on...

So you see, even though the compound interest is paid really frequent, it won't make much of a difference. It will reach the saturation point of e thingy.

Did you know that the "money" value in 1600s was way way MWAYY highER compared to now? Who controls it? We say, oh those people in trading, it's a complex system of people doing stuffs. Tiny part of that is true, but. There's this "imaginary" barrier built from the beginning.

Now you know why economy is "complicated". In quotations.

And then Leonhard Euler, because of his fascination for scribbling, published it as a constant. Or that's what we are told in books.
After that, many imaginative maths were written by some people from one organisomething to make you stare at the magic book. They named it philosophical dilly.

This is a snippet to generate the value of this function $(1 + 1/x)^x$ on GitHub

The years before calculator and computer, people were really keen to scribble things. Well not all people, people with paper or quill or other writing medium and tool access and in certain organization, like, that organization with G and compass symbol and square ruler and the one seeing eye and checkerboard, apron, way to collarbone necklace, with the pillars and such. Because they had to, or for hobby. Hm.

And then, they live happily ever after.

Now then, if you like to read and trace things, look at the pattern from the beginning of wig trend, the banking history, wars and such, the "reason" why they started to voyage, how they wrote science.

Or even further, like why a supposedly "male" Egyptian Pharaoh put eyeliner, or a "king" of somewhere else wore dress, crown, and stuffs? Hm...

The top of our society pyramid is a whole different world I tells ya. It's full of...

And you'll be reading a lot. Hi there, a lot, I'll be tracing you.

Have a splendid day. See you again.