Math and Science in daily life – Part 2


Last time, we discussed how Pythagorean theorem helps us in every day life. Today, I would like to invite your attention to Pi , a beautiful constant (yet one we cannot fully measure) that is in practically everything we see and use.

pie

Image courtesy https://firstnewnan.com/casual-pi-day/

So, what is Pi ?

The common definition is that it is the ratio of the circumference of a circle to its diameter. How cool is that ? ALL circles have the same ratio of circumference to diameter – and thus all circles are similar figures. In Math we represent it as the greek letter  π .  There is a lot of interesting trivia to know about pi – so lets start there

Pi goes on and on

π is an irrational number ( a geeky way of saying it cannot be represented as a regular fraction like 1/2 ). It goes 3.1415926535897932384626433… with no repeating pattern emerging in its decimals.  It can be approximated to 22/7 for use in daily math. Its a standard test for supercomputers to see how many digits after decimal point of pi can be computed. Practically only a few hundred are needed even for the most complex applications though.

Pi is the reason you can’t square a circle !

π is not just irrational – its also transcendental ( yet another geeky way of saying it cannot be expressed as the root of a polynomial ). Not all irrational numbers are transcendental – like for example the square root of 2 is irrational, but not transcendental. Its a very special class of numbers – mostly because it is very hard to prove a given number is transcendental. You may have heard of “Squaring the circle” as a way to say “trying the impossible”. It is because of pi being transcendental that you can’t create a square with the exact same area as a circle with a compass and straight edges. 

Pi has Indian connections 

While the origin of pi is attributed to the Greek mathematician Euclid, several Indian mathematicians like Aryabhatta, Madhava led the effort in providing proof and to calculate its value – usually based on infinite series of numbers. Its also interesting how the algorithms used to calculate pi has changed over time. It started with infinite series, then moved to iterative algorithms – and then switched back to infinite series again thanks to the use of an equation developed by the genius mathematician Srinivasa Ramanujan

OK, so now about the use of Pi in daily life

Finding the area, volume, circumference of anything that has a curve

Anything that is curved has some association to a circle in math – and circles all have pi as the ratio for their circumference to diameter. Consequently calculations for all those curved things like circles, spheres, cones, bell curves and so on ALL have pi in it.

Pi is also useful with straight lines and angles

While we commonly talk about measuring angles in degrees, math geeks measure it in radians. Quite simply a full circle is 360 degrees and in radians it is 2π radians. In other words 1 degree is π/180 radians. Thanks to pi finding its way into measurement of angles – it features in most of trigonometry as well – even though all you see is straight lines and angles between them. This extends to Calculus and other branches of math which we will explore later. Pi is the one constant that is everywhere !

Pi in the sky…and elsewhere

Every aspect of transportation uses Pi in its calculations. This one is personal for me – being a mechanical engineer by trade. Think of a plane or a car – everything from calculating surface area to wind resistance includes pi (remember all curves lead to pi !). Not only that – pi is integral to the calculation of all navigation be it in air, water or land – including finding the distance between stars and planets etc ! This is especially critical for planes that always have to fly in arcs, and where fuel consumption calculation can be a life or death issue.

Pi and the length of rivers

Prof Stolum of Cambridge University showed using Fractals that the ratio between the actual length of rivers from source to mouth and their direct length as the crow flies can be approximated to pi !

Pi is a friend of random numbers

We saw how Pi helps with curves and angles between lines. But it also shows up in seemingly unrelated domains. Here is an example – among a collection of random  numbers, the probability of two numbers with no common factor is 6/π2 . If you are a math geek – you can find many other examples like this – like Cesaro’s theorem and Buffon’s needle problem where pi comes up unexpectedly.

Pi as a safety net 

Whether we are designing a beam which should not break apart from vibrations ( My mind is racing back to my machine design classes) , or a video game (much more pleasant memories from creating games on BASIC) that should not crash when users do random things – the engineers factor in some “randomness” into their model to account for real life. No surprise – all those models typically include pi ! This is just a result of something we spoke above – such probability distributions typically use “area under a curve”, and that by definition needs pi

Math and Science in daily life – Part 1


In my job, I spend a lot of time explaining technology topics to my clients in simple english. Off late, a lot of such conversations are about AI, Data science, Quantum Computing etc . Those topics are rooted in maths and physics (amongst other things), and I often find there is some fear about math and science that exists in the minds of some people listening to me, and it gets in the way of their appreciation of technology.

I am also the dad of a teenage daughter who loves math and science. From my daughter and her friends and teachers – I got the feeling that many a time students get education in math and science in quite an abstract way, and it leads them to think “Why should I really learn any of this? I am not going to use this in real life”. In a few weeks, I am going to give a talk to high schoolers on how the math they learn in school manifests in real life in ways they may not have realized .

So I am going to try a couple of posts here to see if I can explain how simple math leads to powerful and beautiful things we see and use in our daily lives. I would really love your feedback on each topic including alternate/better explanations, and also your suggestions on what else would make good examples. If anyone wants to post a guest blog – we can consider that too.

Let me ease into it with simple math and see how it goes . We can build up from there . Here we go !

Pythagorean theorem

Lets start with the very simple and quite powerful Pythagorean theorem . It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. We remember it via this elegant equation

 A2  + B2 = C2 Screen Shot 2018-03-02 at 4.33.04 PM

This theorem has been proved in several ways , so we will skip that part and get to the fun aspects . If there are three positive numbers A, B and C where A2  + B2 = C2    is true, then it means that there exists a right angled triangle with A and B as the short sides and C as hypotenuse. Some of the fun is based on this converse property of this simple theorem.

How to make square edges on your garden bed

Any three numbers which make true is called a pythagorean triple. The numbers 3,4,5 form one such triple ( 3 squared is 9, 4 squared is 16. the sum of 9 and 16 is 25 which also happens to be the square of 5 ).

Here is a simple problem – lets say you are making a garden bed to plant vegetables.  How do we make the edges square if all you have is a tape measure ? Easy – just make a a triangle with sides 3, 4 and 5 units using the tape and mark the end points on the ground. Voila – You have edges at perfect square !

How big of a ladder do you need ?

An extension of this is also true for finding the length of a ladder required to paint a wall – probably a question kids get in school exams. If the point where the ladder needs to touch the wall is 4 meters, and the bottom of the ladder is 3 meters away from the wall on the floor – the length of the ladder needed is 5 meters !

Or let’s say you want to convert stairs into a ramp on your back porch – the same concept applies .

Real engineers use this information frequently – the one I remember the most from my college days is in surveying land, finding length of trusses etc .

Planning your next painting project

Lets say we draw three similar figures using the sides of the right angled triangle forming one side of those figures. Similar figures just mean the lengths of sides have same ratios, the included angle between sides are the same.  Easy way to think about it is zooming a picture on your phone . All parts increase or decrease in size – but their ratio to each other stays the same . It does not matter what the figure is – it could be a square, a pentagon or another triangle. The area of the largest figure (the one using hypotenuse as one of its sides) is ALWAYS the same as the sum of areas of the two smaller figures. How cool is that ! See this picture from wikipedia to see what I mean

520px-Pythagorean.svg

Lets say these three squares are walls you need to paint . For the same quantity of paint you will need to paint the largest square – you can paint both the smaller squares !

How much marinara do you need ?

It’s not just about squares either – this area computation via Pythagorean triples also applies to circles . So for the the quantity of marinara you need for a 5 inch pizza can be used to coat both a 3 inch and 4 inch pizza ! If your mom is like mine – I wouldn’t advice trying this math on her while she is making you pizza . Theory of “mom is always right” over rules every other rule 🙂

If you are a social media marketer

You probably are a fan of Metcalfe’s law which says the value of a network is proportional to the square of the number of connected users in it . Let’s say you are given access to three potential networks . A has 3 connections , B has 4 connections and C has 5 connections . Since The squares of A and B add to the square of C – you get as much out of a network with 5 connections as you would get from A and B which together add to 7 connections . Spend your time and money wisely !

PS : I have my own misgivings on this – but would like to hear your feedback anyway . I didn’t come up with this myself and can’t remember anymore who told me this originally . But I have teased many a marketer friend with this at social events 🙂

What’s next

I think I would like to example the wonderful pi next . There is pi in pie !

Looking ahead – what jobs will technology take away ?


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As with a lot of things like politics, religion and so on – the world is sharply divided between people who believe AI and Robots (or automation in general) will take more jobs away than it creates.  I was drawn into this debate yet again by a few friends couple of weeks ago – so let me jot down while I still have it fresh in my mind. My crystal ball is not any more effective than yours when it comes to looking into the future – but there are a few scenarios where I do think jobs will be taken away. If your job is in one of these categories, the smart thing to do is to gain additional skills. Just to be clear – I also think there won’t be any net job losses. As always – this is all strictly my personal views on the topic, and not that of my employer.

Another way to look at this is – many companies will automate tasks and eliminate labor where they can to save costs. If you have skills that can make them earn more revenue directly or indirectly, you get to stay employed. Otherwise instead of reinvesting the savings, the company will probably treat it as profit, or keep the cash for future. The job itself might stay in many cases – but there just won’t be a need for as many people to do it. Granted – there will always be exceptions. Technology will also create a bunch of new jobs – which I will write about in another post.

I think there are at least four categories of jobs that will get disrupted soon.

1.If most of what you know is public knowledge

This is especially true for my own profession, which is consulting. In the 90s when I got out of college – there was no google. If I knew something special ( from books , professional magazines, training etc ) – a client would pay some money for me to tell them what I knew that they did not. That does not happen much any more – there is no premium for internet access . Clients and consultants both have access to similar information – so you need to know more than what is available on internet to fetch a premium. It might sound ridiculously obvious – but this is a bigger threat to (especially junior) consultants than almost anything else .

You absolutely need to stay couple of steps ahead of market to add value to a client today. Having a logically defensible point of view, knowing what others in the industry are up to, what disruptions are on the horizon, what untapped opportunities exist are all still things a client will pay a premium for.

Consultants are not the only ones at risk either. As an example – A hotel concierge function could also fall in this category. You don’t need a human to get you a restaurant reservation, check weather, know the local tourist spots and  so on. However, it will be hard to replace a human who can help you score a last minute Hamilton ticket in Broadway, or one who can answer questions from four different customers in parallel and make them all feel special.

2.If your work is all about short tail questions from a customer 

A lot of systems we use were not designed with end users in mind. Thanks to that, a lot of human intervention is still needed for people to use things they bought. A good part of customer service calls are about answering questions like “whats my account balance”, “can you reset my password”, “Can I set up a payment plan”, “Can I use a different credit card” etc. Automation is mature enough already to do those things without human intervention . If that is all your skill is – your job probably will be taken away soon.

But there are lot of things automation cannot do in this scenario  – at least not yet. For example , talking a customer out of canceling a service is not something AI can do very effectively like a trained retention specialist. From a customer’s point of view – an automated way of resetting a password, or making a routine payment would be easier/faster than needing to talk to someone. But when you are upset with poor service , or want to talk through multiple options – there is nothing worse than listening to a machine with a long menu. Also think of this – as tech (and laws) improves all around, in most categories customers will have zero or low switching costs.

So if you are skilled at higher value service – you should be in hot demand. The money an employer saves by automating the short tail responses is what lets them invest more in higher value services. Of course we can also take a cynical view that some companies will just add it to bottom line and not bother re-investing. While that is a short term possibility, I doubt they can do it in long term without risking their whole business.

3.If you are in a job where process trumps thinking 

There are several jobs where the job needs very little original thinking. The critical thinking is done by few people who designed the workflow, and not by people executing. This would include things like preparing fast food, paying invoices, checking totals, scanning documents etc. These jobs are generally at risk given they are easily automated – and probably the only reason they are still around is because of the one time cost of implementing new technology. Given all tech will eventually commoditize, this is only a temporary safety net.

The human intervention will be limited to exception processing in these workflows, especially those that involve safety, brand issues, downtime issues etc like – what if the lettuce delivered is rotten and you need to run to local grocery to buy some ? What happens if the scanner stops working the last day of the fiscal period ? Do you want to harass a customer on collecting $100, when you know in 3 months they are due for a $1000 renewal?

4.If your job is only about answering questions, and not about asking questions

Computers – and all the advances in AI and Quantum computing and whatever comes next – will keep getting better at answering more and more complex questions. There are questions a computer can answer faster and more frequently than humans today – like who was the 44th President of US? What planet is closest to earth in the solar system ? . There are questions that are really hard for computers too, where a human can often answer effortlessly – like who was the quarterback of the super bowl winning team the year the 44th President of US took office ?  But over time, we should expect computers to generally be able to answer most questions we ask.

But humans are way better than computers when it comes to asking questions. At some point, computers probably can interpret a medical image better , and compare it against a million other images faster than any trained human medical expert. However, that is only a starting point – human experts are way better at asking better/unique/complex questions and explore any body of knowledge and expand on it. This is why I think no expert system will eliminate doctors – they will just make the quality of medical service a doctor can provide a lot better, and reduce mistakes. In short – We get to ask the smart questions, and mostly leave finding the answers to machines.

In various forms – this phenomena will play out in every job . People who have access to smart machines that can find better answers get to make decisions faster and cheaper than others, and that is how competitive advantages will be created in the market.

So in a nutshell – differentiation in future will be based on humans who can ask better questions than they can ask today, and machines which can answer better, faster and cheaper than they can today.

Sounds pretty straightforward, but we will of course fight this every step of the way. When horse drawn fire engines were first introduced, humans used to race them on foot to prove their superiority. We know what happened after that. For many reasons – political, legal, social and economic – just because technology can be used to effectively solve a problem does not mean that it will happen fast. So in my view, there is practically very low risk of massive unemployment any time soon. But without a doubt , every job around us will evolve in a way that human value add will become all about asking better questions and technology’s value add will be about giving better answers.