Maths - Blog Posts

Since 2009, Ada Lovelace Day has aimed “to raise the profile of women in science, technology, engineering and maths by encouraging people around the world to talk about the women whose work they admire.”  The day’s namesake, Ada Lovelace (1815-1852), was the daughter of Lord Byron and Anne Isabella Milbanke.  Ada, in possession of a keen intellect and deep passion for machinery, was educated in mathematics at the insistence of her mother. Later in life, Ada studied the workings of the Analytical Engine developed by mathematician and inventor Charles Babbage. In her notes on the engine, Ada described an algorithm for computing numbers – an algorithm which would distinguish Ada as one of the world’s “first computer programmers.”  

In honor of Ada Lovelace Day, we present some images from the CHF Archives of women working in various chemistry labs. Click on each photo for additional information.

And for more women in science content, consider taking a look at the films in The Catalyst Series: Women in Chemistry by the Chemical Heritage Foundation.

1729

The number 1729 has an interesting story in mathematics involving the extraordinary Indian mathematician Srinivasa Ramanujan. G. H. Hardy accounts:

“I remember once going to see him (Ramanujan) when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. ‘No’, he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.’”

Ramanujan had a knack for numbers. Growing up in India at the turn of the 20th century, Ramanujan was largely self-taught. Over his short life time (aged 32), he independently developed nearly 4,000 results in mathematics. He kept his results (without proofs) in notebooks that modern mathematicians are still looking into this day. Nearly all of his results have been proven to be true and have driven research in number theory for the past century. Recently, one of Rumanujan’s results, previously unknown to mathematicians, was an important piece to a 2006 publication.

More at https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

Fibonacci sequence in the hiding…

The Fibonacci sequence can help you quickly convert between miles and kilometers

The Fibonacci sequence is a series of numbers where every new number is the sum of the two previous ones in the series.

1, 1, 2, 3, 5, 8, 13, 21, etc. The next number would be 13 + 21 = 34.

Here’s the thing: 5 mi = 8 km. 8 mi = 13 km. 13 mi = 21 km, and so on.

Edit: You can also do this with multiples of these numbers (e.g. 5*10 = 8*10, 50 mi = 80 km). If you’ve got an odd number that doesn’t fit in the sequence, you can also just round to the nearest Fibonacci number and compensate for this in the answer. E.g. 70 mi ≈ 80 mi. 80 mi = 130 km. Subtract a small value like 15 km to compensate for the rounding, and the end result is 115 km.

This works because the Fibonacci sequence increases following the golden ratio (1:1.618). The ratio between miles and km is 1:1.609, or very, very close to the golden ratio. Hence, the Fibonacci sequence provides very good approximations when converting between km and miles.

Survivorship Bias

I have posted about survivorship bias and how it affects your career choices: how a Hollywood actor giving the classic “follow your dreams and never give up” line is bad advice and is pure survivorship bias at work.

When I read up on the wikipedia page, I encountered an interesting story:

During WWII the US  Air Force wanted to minimize bomber losses to enemy fire. The Center for Naval Analyses ran a research on where bombers tend to get hit with the explicit aim of enforcing the parts of the airframe that is most likely to receive incoming fire. This is what they came up with:

So, they said: the red dots are where bombers are most likely to be hit, so put some more armor on those parts to make the bombers more resilient. That looked like a logical conclusion, until Abraham Wald - a mathematician - started asking questions: 

- how did you obtain that data? - well, we looked at every bomber returning from a raid, marked the damages on the airframe on a sheet and collected the sheets from all allied air bases over months. What you see is the result of hundreds of those sheets. - and your conclusion? - well, the red dots are where the bombers were hit. So let’s enforce those parts because they are most exposed to enemy fire.  - no. the red dots are where a bomber can take a hit and return. The bombers that took a hit to the ailerons, the engines or the cockpit never made it home. That’s why they are absent in your data. The blank spots are exactly where you have to enforce the airframe, so those bombers can return.

This is survivorship bias. You only see a subset of the outcomes. The ones that made it far enough to be visible. Look out for absence of data. Sometimes they tell a story of their own.

BTW: You can see the result of this research today. This is the exact reason the A-10 has the pilot sitting in a titanium armor bathtub and has it’s engines placed high and shielded.

This is great :) My favorite is proof by ghost reference, what is yours?

A limerick:

Idempotence.

A term I’d always found intriguing, mostly because it’s such an unusual word. It’s a concept from mathematics and computer science but can be applied more generally—not that it often is. Basically, it’s an operation that, no matter how many times you do it, you’ll still get the same result, at least without doing other operations in between. A classic example would be view_your_bank_balance being idempotent, and withdraw_1000 not being idempotent.

HTs: @aidmcg and Ewan Silver who kept saying it

Could you explain this tfw no ZF joke? I really dont get it... :D

Get ready for a long explanation! For everyone’s reference, the joke (supplied by @awesomepus​) was:

Q: What did the mathematician say when he encountered the paradoxes of naive set theory?A: tfw no ZF

You probably already know the ‘tfw no gf’ (that feel when no girlfriend) meme, which dates to 2010. I’m assuming you’re asking about the ZF part.

Mathematically, ZF is a reference to Zermelo-Fraenkel set theory, which is a set of axioms commonly accepted by mathematicians as the foundation of modern mathematics. As you probably know if you’ve taken geometry, axioms are super important: they are basic assumptions we make about the world we’re working in, and they have serious implications for what we can and can’t do in that world. 

For example, if you don’t assume the Parallel Postulate (that consecutive interior angle measures between two parallel lines and a transversal sum to 180°, or twice the size of a right angle), you can’t prove the Triangle Angle Sum Theorem (that the sum of the angle measures in any triangle is also 180°). It’s not that the Triangle Angle Sum Theorem theorem is not true without the Parallel Postulate — simply that it is unprovable, or put differently, neither true nor false, without that Postulate. Asking whether the Triangle Angle Sum Theorem is true without the Parallel Postulate is really a meaningless question, mathematically. But we understand that, in Euclidean geometry (not in curved geometries), both the postulate and the theorem are “true” in the sense that we have good reason to believe them (e.g., measuring lots of angles in physical parallel lines and triangles). Clearly, the axioms we choose are important.

Now, in the late 19th and early 20th century, mathematicians and logicians were interested in understanding the underpinnings of the basic structures we use in math — sets, or “collections,” being one of them, and arithmetic being another. In short, they were trying to come up with an axiomatic set theory. Cantor and Frege were doing a lot of this work, and made good progress using everyday language. They said that a set is any definable collection of elements, where “definable” means to provide a comprehension (a term you’re familiar with if you program in Python), or rule by which the set is constructed.

But along came Bertrand Russell. He pointed out a big problem in Cantor and Frege’s work, which is now called Russell’s paradox. Essentially, he made the following argument:

Y’all are saying any definable collection is a set. Well, how about this set: R, the set of all sets not contained within themselves. This is, according to you, a valid set, because I gave that comprehension. Now, R is not contained within itself, naturally: if it is contained within itself, then it being an element is a violation of my construction of R in the first place. But R must be contained within itself: if it’s not an element of itself, then it is a set that does not contain itself, and therefore it is an element of itself. So we have that R ∈ R and also R ∉ R. This is a contradiction! Obviously, your theory is seriously messed up.

This paradox is inherently a part of Cantor and Frege’s set theory — it shows that their system was inconsistent (with itself). As Qiaochu Yuan explains over at Quora, the problem is exactly what Russell pointed out: unrestricted comprehension — the idea that you can get away with defining any set you like simply by giving a comprehension. Zermelo and Fraenkel then came along and offered up a system of axioms that formalizes Cantor and Frege’s work logically, and restricts comprehension. This is called Zermelo-Fraenkel set theory (or ZF), and it is consistent (with itself). Cantor and Frege’s work was then retroactively called naive set theory, because it was, of course, pretty childish:

There are two more things worth knowing about axiomatic systems in mathematics. First, some people combine Zermelo-Fraenkel set theory with the Axiom of Choice¹, resulting in a set theory called ZFC. This is widely used as a standard by mathematicians today. Second, Gödel proved in 1931 that no system of axioms for arithmetic can be both consistent and complete — in every consistent axiomatization, there are “true” statements that are unprovable. Or put another way: in every consistent axiomatic system, there are statements which you can neither prove nor disprove.For example, in ZF, the Axiom of Choice is unprovable — you can’t prove it from the axioms in ZF. And in both ZF and ZFC, the continuum hypothesis² is unprovable.³ Gödel’s result is called the incompleteness theorem, and it’s a little depressing, because it means you can’t have any good logical basis for all of mathematics (but don’t tell anyone that, or we might all be out of a job). Luckily, ZF or ZFC has been good enough for virtually all of the mathematics we as a species have done so far!

The joke is that, when confronted with Russell’s paradox in naive set theory, the mathematician despairs, and wishes he could use Zermelo-Fraenkel set theory instead — ‘that feel when no ZF.’

I thought the joke was incredibly funny, specifically because of the reference to ‘tfw no gf’ and the implication that mathematicians romanticize ZF (which we totally do). I’ve definitely borrowed the joke to impress friends and faculty in the math department…a sort of fringe benefit of having a math blog.

– CJH

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Tutorials (programming) and other online resources:

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Memory Management in C

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Websites for learning and tools:

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Recommended books:

Computer organization and design: the hardware/software interface. David A.Patterson & John L. Hennessy.

Artificial intelligence: a modern approac. Stuart J. Russel & Peter Norvig.

Database systems: the complete book. Hector Garcia-Molina, Jeffrey D. Ullman, Jennifer Widom.

Algorithms: a functional programming approach. Fethi Rabbi & Guy Lapalme.

Data Structures & Algorithms in Java: Michael T. Goodrich & Roberto Tamassia.

The C programming language: Kernighan, D. & Ritchie.

Operating System Concepts: Avi Silberschatz, Peter Baer Galvin, Greg Gagne.

Study Tips:

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Writing in Sciences, Mathematics and Engineering:

Writing a Technical Report

Writing in the Sciences (Stanford online course)

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Useful :)

Laplace transform table. Source. (I’m obsessed. <3 And figured y’all would like this one, too!)

Curves of constant width

                                                    Source

The width of a circle is constant: its diameter.

But the circle is not the only shape that holds this pristine title. For instance let’s look at the Reuleaux triangle

Reuleaux triangle

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two.

The Reuleaux triangle is the first of a sequence of Reuleaux polygons, curves of constant width formed from regular polygons with an odd number of sides.

Some of these curves have been used as the shapes of coins

To drill square holes.

They are not entirely square, their edges are fillets i.e the edges are rounded and not sharp.

This animation offers a good insight as to why that is so.

And in china, apparently on bicycles.

The man Guan Baihua shows his self-made multi-angle-wheel bicycle on May 6, 2009 in Qingdao of Shandong Province, China. Guan Baihua spent 18 months to complete this strange bicycle.

Other shapes of constant width

There are other shapes of constant width beside the Reuleaux triangle ( that has been discussed in this post ), a whole bunch of them really. Do take a look at them. ( links below )

I will leave you guys with my favorite one.

More:

If this post fascinated you, i strongly suggest you check these out. They go in-depth with the mathematics that underlies these curves and talk about other cool stuff:

An animation of non-circular rollers

Shapes and Solids of Constant Width - Numberphile  

Shapes of constant width

Reuleaux Polygons,           

Edit:

For those who are wondering if these are something that one would stumble upon on a regular basis. You may not find perfect ones but similiar ones definitely.

I found mine on a really old BMI calculator thingy. ( not sure what you would call it )

Have fun exploring !

Aristotle’s Wheel Paradox. Can you figure out what the paradox is? (What doesn’t make sense?) More info at http://mathworld.wolfram.com/AristotlesWheelParadox.html

I just completed the finishing touches on my new poster, a detailed map of the Mandelbrot Set in a vintage style. I’m calling it the Mandelmap.

The Mandelbrot Set is a fractal shape with infinite detail that you can zoom in on. I often explore the Mandelbrot Set to find trippy patterns to create gifs with, but when I started I felt like I was just poking around at random. So I wanted to create a printed guide for myself to find my way around… I soon realized this was going to be a lot of work, so I decided I might as well take it to the next level and make an awesome poster that would be not just for myself but for everyone else to enjoy too.

What you see here is the result of more than a year’s research, planning, and execution. It’s a 36x24 inch poster rendered fully at 300 dpi, and everything you see was created from scratch. I will be posting more updates and information as I get the test prints in, and I hope to have this poster available to buy within the next couple months!

www.mandelmap.com

Heya!!

Today I revised some stuff (90% of the day lol) and did chapter 10.Chapter 10 is about the scenic viewpoint theorem,Bolzano-Weierstrass theorem for real and complex numbers.

Fun fact: Bolzano- Weiestrass theorem for reals makes use of the scenic view point theorem and monotonic sequence theorem but both of them are not valid in the complex numbers because complex numbers don't have order.

This chapter also discussed about Cauchy sequences (it has to satisfy a condition to be called that),some facts (and proofs!) about when it is bounded and such and finally the Cauchy convergent criteria.

I also did chapter 11.It is about the convergence of series.It initially told me what series means and some popular series ( telescoping,harmonic,geometric etc) and what do we mean when we say a series is convergent.It spoke about some nice theorems (involving series of non negative terms,comparison test,cauchy criteria,absolute convergence) and some examples based on the theorems.I did get one basic doubt and I'm sure my friend will be able to clear that out as well.He really has my back when it comes to these silly doubts ^_^

Heya!!

Today I revised all the topics I covered in Analysis 1 so far and did chapter 8 and 9.

Chapter 8 is about the algebra of limits.In there I learnt how to write proofs using ε-N method.There were quite a number of cases (with examples!).Most of them were good and could be deduced easily from previous theorems and axioms but I did find 4 proofs a little challenging.I have noted them down and I’ll be discussing those with my prof soon.

Chapter 9 was about monotonic sequences.In there I learnt about the monotonic sequence theorem and the proof of this theorem is very cute.This theorem's proof is very intresting because it connects to the theorem where square root of 2 is proved as irrational but it is more general.So it patches all the gaps on the real number line quit nicely.There were some examples too which I liked and I also learnt the steps of proving a theorem when monotonicity is involved.

It was a pretty cool day.

I hope you had a good day too^_^

***

The Mayuriit Project

Will I make it? Stay tuned to find out!

206 days left…

[3.06 pm] I have not yet started my day.I watched a movie in the morning (Mona Lisa Smile) I am in love with Julia Roberts Help Me!!!.I am feeling so good about myself and my situation right now.I love these days.

***

[9.57 pm] I finished two chapters of Differential equations and revised previous week's stuff.I put a face mask on, lit a candle and just chilled in the evening after the work.I am thinking of switching my moisturising cream so hunting for a new one.I'm thinking of ordering some more candles as well because they are helping me in reducing my study anxiety lol.

I hope you guys had a nice day.

***

216 days left to go...

Will I make it?

The Mayuriit Project, Stay tuned on the Tumblr blog near you.

26/05/2021

Salut 👋

I just finished 30 minutes of French 🇫🇷 in Duolingo.I love the way it sounds 🤭So today I have decided to do the following stuff

Simple Harmonic Motion

Group theory Problems

Mathematics and statistics revision

I'm hungry,I'll go prepare my breakfast now.Ba bye have a wonderful day!

What does FTC say?

It says that if a person takes the derivative of a function and then integrates it over a region on the number line say [a, b] then this is the same as evaluating the function on its endpoints.

What does the Green's Theorem say?

Green's Theorem is the fundamental theorem of calculus in 2 dimensions.Instead of taking the derivative of a single variable function we take the curl of a 2 variable function.Instead of integrating this over a number line we integrate it on the xy plane.Instead of evaluating the function at the two endpoints a and b and taking the difference, we take the line integral of the function and integrate it around the curve in a counterclockwise direction.

What does the stokes' theorem say?

Stokes' theorem is the fundamental theorem of calculus in 3 dimensions. Instead of taking the derivative of a single variable function, we take the three-dimensional curl. Instead of integrating this over a number line, we integrate it on the surface (To evaluate the surface integral one has to dot the vector field with unit normal vectors). Instead of evaluating the function at the two endpoints a and b and taking the difference, we take the line integral of the function and integrate it around the curve on a surface in a counterclockwise direction just like in Green's Theorem.

16/05/2015

I just finished revising for group theory. It's about 5.30 in the evening and I thought I'd have it completed by 9 in the morning lol. So anyways now I plan on covering the following topics

Group acting on sets

Orbits and stabilizers

Revise maths and catch up with this week's work

Solve 10 problems on Group Theory

Wish me luck and I hope you are having a better day than me!❣

FIBONACCI IDENTITY

If you could instantly be granted fluency in 5 languages—not taking away your existing language proficiency in any way, solely a gain—what 5 would you choose?

I'm sick and tired of the lack of stem dark academia vibes in modern literature. Obviously we have Frankenstein, which I love, but I am foaming at the mouth for a book like The Secret History, or Bunny, or Dead Poet Society but stem.

Imagine a murder mystery at a prestigious university, but it's a sleep-deprived physics student who's obsessed with space, an equally as tired chemistry student that's only a ~little~ unhinged and overexcited for orgo lab, and a medicine student who spent an entire week studying anatomy getting together to solve it.

Or, they all suspect each other, though nobody knows that they do, and the actual murderer attempts to sabotage the group effort without being caught.

Do you think two weeks is enough for revisions for a math exam..?

Hey Anon! :D

I would say yeah, it's more than enough if you know how to manage your time and revise effectively. Personally, the time doesn't really matter much if you know how to revise well :)

Math Tips

(Pictures are not mine)

Well, let me tell you, we all have this love-hate relationship with this subject, right? The worst part is that when you don't know what the heck is going on, so, as a girl who studied maths (2 Volumes/textbooks) on her own during the year she was homeschooled, here are some tips and tricks that I did to get an A+ in my math finals!

Get your syllabus together

In the beginning I had no damn idea what was going on and it was just confusing. I had to do the first thing I did was taken my index/table of contents and mark the chapters which i knew very well and the ones I had no clue about. And then i arranged them with the marking scheme, like which one carries the most marks etc etc and study accordingly.

Complete lessons/chapters that you already know

When you finish off the things you already know then that's gonna give you the confidence you need even if you know only 1-2 chapters, learn it throughout and make sure that you'll get the answer no matter how twisted the sum is. If you're doubtful about the whole textbook like any normal person.... Start with the easy ones. (I know there are literally really no "easy" chapters, spare me)

Harder chapters need hard work

Most chapters like Trigonometry proofs, Geometry proofs, Algebra, Graphs, Mensuration and Calculus etc need more than minimum effort but here's a trick, what is the common thing in this? Yes, they're all formulae and theorem based which goes to my next point. These chapters are completely based on how much you've understood your basics.

Formulae and theorem cheatsheets

Make a list of all formulae and the theorem used in the book, write them chapter wise and no printouts or digital notes. Take a paper and write it down, no excuses. It helps you while you're practicing, revising and in the last minute review, it helped me damn much. Remember, maths is a sport. The basic formulae must come to you like reflexes.

YouTube is your best friend.

For every single chapter, go and watch the basics and how a sum is done step by step. A recommendation for this is Organic Chemistry Tutor who literally is one of the reasons i passed. He has videos from basic geometry, trigonometry, statistics to calculus. Search for your own YouTubers and be clear with concepts.

Math is fully memorization

Memorize formulae and theorems with the back of your hand, you should be able to recall them within seconds. Be thorough.

Memorize basic math values (if calculator isn't allowed)

Do this if you have a majority of chapters like Statistics, Mensuration, Profit/loss calculation etc, where large numbers are concerned. Memorize the first 10 square, cube, decimal and multiplication values. It may be dry but there are literally songs available for these things, I'm serious, i learnt the first 10 cube roots by listening to Senorita xD Search for rhymes and they'll definitely be many!!

Work it out!!!!!!

Can't stress this enough, atleast 30-40 mins is the minimum for maths. I'm serious, work out each sum, don't ever think it's a waste, you'll see the results. Practice makes perfect. Work out every single sum, from examples to exercise ones cause let's be honest, our examiners love to take problems from every nook and cranny of the book.

Whiteboard method

So, I made this up and it actually works, if you have a whiteboard or anything else, once you completed a chapter, take a random page and whatever sums you have on those two pages, you need to complete within a given time limit. It helps you to identify your weak points and where the hell you're losing both time and effort and not to mention that it gives you confidence boost up.

Hope this helps :))

I have a problem

I answered a question incorrectly today in Maths and I had to count back from ten and focus on my breathing because I could feel tears welling in my eyes. This isn't even the first time this has happened. What the hell is wrong with me?

"I can stay awake for just a bit longer!"

- Me at 3:00 knowing that my Maths Assessment is tomorrow

Math is really tiring, im so glad i finally get to relax and do some knitting and crochet and i oh god oh my what the fuck

i drew hatsune mik uin school today an dmy teacher didnt like it so maybe yous'll'll