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Leah

Leah

3 years ago

The Burnout Recovery Secrets Nobody Is Talking About

More on Personal Growth

Jari Roomer

Jari Roomer

3 years ago

Successful people have this one skill.

Without self-control, you'll waste time chasing dopamine fixes.

I found a powerful quote in Tony Robbins' Awaken The Giant Within:

“Most of the challenges that we have in our personal lives come from a short-term focus” — Tony Robbins

Most people are short-term oriented, but highly successful people are long-term oriented.

Successful people act in line with their long-term goals and values, while the rest are distracted by short-term pleasures and dopamine fixes.

Instant gratification wrecks lives

Instant pleasure is fleeting. Quickly fading effects leave you craving more stimulation.

Before you know it, you're in a cycle of quick fixes. This explains binging on food, social media, and Netflix.

These things cause a dopamine spike, which is entertaining. This dopamine spike crashes quickly, leaving you craving more stimulation.

It's fine to watch TV or play video games occasionally. Problems arise when brain impulses aren't controlled. You waste hours chasing dopamine fixes.

Instant gratification becomes problematic when it interferes with long-term goals, happiness, and life fulfillment.

Most rewarding things require delay

Life's greatest rewards require patience and delayed gratification. They must be earned through patience, consistency, and effort.

Ex:

  • A fit, healthy body

  • A deep connection with your spouse

  • A thriving career/business

  • A healthy financial situation

These are some of life's most rewarding things, but they take work and patience. They all require the ability to delay gratification.

To have a healthy bank account, you must save (and invest) a large portion of your monthly income. This means no new tech or clothes.

If you want a fit, healthy body, you must eat better and exercise three times a week. So no fast food and Netflix.

It's a battle between what you want now and what you want most.

Successful people choose what they want most over what they want now. It's a major difference.

Instant vs. delayed gratification

Most people subconsciously prefer instant rewards over future rewards, even if the future rewards are more significant.

We humans aren't logical. Emotions and instincts drive us. So we act against our goals and values.

Fortunately, instant gratification bias can be overridden. This is a modern superpower. Effective methods include:

#1: Train your brain to handle overstimulation

Training your brain to function without constant stimulation is a powerful change. Boredom can lead to long-term rewards.

Unlike impulsive shopping, saving money is boring. Having lots of cash is amazing.

Compared to video games, deep work is boring. A successful online business is rewarding.

Reading books is boring compared to scrolling through funny videos on social media. Knowledge is invaluable.

You can't do these things if your brain is overstimulated. Your impulses will control you. To reduce overstimulation addiction, try:

  • Daily meditation (10 minutes is enough)

  • Daily study/work for 90 minutes (no distractions allowed)

  • First hour of the day without phone, social media, and Netflix

  • Nature walks, journaling, reading, sports, etc.

#2: Make Important Activities Less Intimidating

Instant gratification helps us cope with stress. Starting a book or business can be intimidating. Video games and social media offer a quick escape in such situations.

Make intimidating tasks less so. Break them down into small tasks. Start a new business/side-hustle by:

  • Get domain name

  • Design website

  • Write out a business plan

  • Research competition/peers

  • Approach first potential client

Instead of one big mountain, divide it into smaller sub-tasks. This makes a task easier and less intimidating.

#3: Plan ahead for important activities

Distractions will invade unplanned time. Your time is dictated by your impulses, which are usually Netflix, social media, fast food, and video games. It wants quick rewards and dopamine fixes.

Plan your days and be proactive with your time. Studies show that scheduling activities makes you 3x more likely to do them.

To achieve big goals, you must plan. Don't gamble.

Want to get fit? Schedule next week's workouts. Want a side-job? Schedule your work time.

Nitin Sharma

Nitin Sharma

2 years ago

Quietly Create a side business that will revolutionize everything in a year.

Quitting your job for a side gig isn't smart.

Photo by Artur Voznenko on Unsplash

A few years ago, I would have laughed at the idea of starting a side business.

I never thought a side gig could earn more than my 9-to-5. My side gig pays more than my main job now.

You may then tell me to leave your job.  But I don't want to gamble, and my side gig is important. Programming and web development help me write better because of my job.

Yes, I share work-related knowledge. Web development, web3, programming, money, investment, and side hustles are key.

Let me now show you how to make one.

Create a side business based on your profession or your interests.

I'd be direct.

Most people don't know where to start or which side business to pursue.

You can make money by taking online surveys, starting a YouTube channel, or playing web3 games, according to several blogs.

You won't make enough money and will waste time.

Nitin directs our efforts. My friend, you've worked and have talent. Profit from your talent.

Example:

College taught me web development. I soon created websites, freelanced, and made money. First year was hardest for me financially and personally.

As I worked, I became more skilled. Soon after, I got more work, wrote about web development on Medium, and started selling products.

I've built multiple income streams from web development. It wasn't easy. Web development skills got me a 9-to-5 job.

Focus on a specific skill and earn money in many ways. Most people start with something they hate or are bad at; the rest is predictable.

Result? They give up, frustrated.

Quietly focus for a year.

I started my side business in college and never told anyone. My parents didn't know what I did for fun.

The only motivation is time constraints. So I focused.

As I've said, I focused on my strengths (learned skills) and made money. Yes, I was among Medium's top 500 authors in a year and got a bonus.

How did I succeed? Since I know success takes time, I never imagined making enough money in a month. I spent a year concentrating.

I became wealthy. Now that I have multiple income sources, some businesses pay me based on my skill.

I recommend learning skills and working quietly for a year. You can do anything with this.

The hardest part will always be the beginning.

When someone says you can make more money working four hours a week. Leave that, it's bad advice.

If someone recommends a paid course to help you succeed, think twice.

The beginning is always the hardest.

I made many mistakes learning web development. When I started my technical content side gig, it was tough. I made mistakes and changed how I create content, which helped.

And it’s applicable everywhere.

Don't worry if you face problems at first. Time and effort heal all wounds.

Quitting your job to work a side job is not a good idea.

Some honest opinions.

Most online gurus encourage side businesses. It takes time to start and grow a side business.

Suppose you quit and started a side business.

After six months, what happens? Your side business won't provide enough money to survive.

Indeed. Later, you'll become demotivated and tense and look for work.

Instead, work 9-5, and start a side business. You decide. Stop watching Netflix and focus on your side business.

I know you're busy, but do it.

Next? It'll succeed or fail in six months. You can continue your side gig for another six months because you have a job and have tried it.

You'll probably make money, but you may need to change your side gig.

That’s it.

You've created a new revenue stream.

Remember.

Starting a side business, a company, or finding work is difficult. There's no free money in a competitive world. You'll only succeed with skill.

Read it again.

Focusing silently for a year can help you succeed.

I studied web development and wrote about it. First year was tough. I went viral, hit the top 500, and other firms asked me to write for them. So, my life changed.

Yours can too. One year of silence is required.

Enjoy!

Theo Seeds

Theo Seeds

3 years ago

The nine novels that have fundamentally altered the way I view the world

I read 53 novels last year and hope to do so again.

Books are best if you love learning. You get a range of perspectives, unlike podcasts and YouTube channels where you get the same ones.

Book quality varies. I've read useless books. Most books teach me something.

These 9 novels have changed my outlook in recent years. They've made me rethink what I believed or introduced me to a fresh perspective that changed my worldview.

You can order these books yourself. Or, read my summaries to learn what I've synthesized.

Enjoy!

Fooled By Randomness

Nassim Taleb worked as a Wall Street analyst. He used options trading to bet on unlikely events like stock market crashes.

Using financial models, investors predict stock prices. The models assume constant, predictable company growth.

These models base their assumptions on historical data, so they assume the future will be like the past.

Fooled By Randomness argues that the future won't be like the past. We often see impossible market crashes like 2008's housing market collapse. The world changes too quickly to use historical data: by the time we understand how it works, it's changed.

Most people don't live to see history unfold. We think our childhood world will last forever. That goes double for stable societies like the U.S., which hasn't seen major turbulence in anyone's lifetime.

Fooled By Randomness taught me to expect the unexpected. The world is deceptive and rarely works as we expect. You can't always trust your past successes or what you've learned.

Antifragile

More Taleb. Some things, like the restaurant industry and the human body, improve under conditions of volatility and turbulence.

We didn't have a word for this counterintuitive concept until Taleb wrote Antifragile. The human body (which responds to some stressors, like exercise, by getting stronger) and the restaurant industry both benefit long-term from disorder (when economic turbulence happens, bad restaurants go out of business, improving the industry as a whole).

Many human systems are designed to minimize short-term variance because humans don't understand it. By eliminating short-term variation, we increase the likelihood of a major disaster.

Once, we put out every forest fire we found. Then, dead wood piled up in forests, causing catastrophic fires.

We don't like price changes, so politicians prop up markets with stimulus packages and printing money. This leads to a bigger crash later. Two years ago, we printed a ton of money for stimulus checks, and now we have double-digit inflation.

Antifragile taught me how important Plan B is. A system with one or two major weaknesses will fail. Make large systems redundant, foolproof, and change-responsive.

Reality is broken

We dread work. Work is tedious. Right?

Wrong. Work gives many people purpose. People are happiest when working. (That's why some are workaholics.)

Factory work saps your soul, office work is boring, and working for a large company you don't believe in and that operates unethically isn't satisfying.

Jane McGonigal says in Reality Is Broken that meaningful work makes us happy. People love games because they simulate good work. McGonigal says work should be more fun.

Some think they'd be happy on a private island sipping cocktails all day. That's not true. Without anything to do, most people would be bored. Unemployed people are miserable. Many retirees die within 2 years, much more than expected.

Instead of complaining, find meaningful work. If you don't like your job, it's because you're in the wrong environment. Find the right setting.

The Lean Startup

Before the airplane was invented, Harvard scientists researched flying machines. Who knew two North Carolina weirdos would beat them?

The Wright Brothers' plane design was key. Harvard researchers were mostly theoretical, designing an airplane on paper and trying to make it fly in theory. They'd build it, test it, and it wouldn't fly.

The Wright Brothers were different. They'd build a cheap plane, test it, and it'd crash. Then they'd learn from their mistakes, build another plane, and it'd crash.

They repeated this until they fixed all the problems and one of their planes stayed aloft.

Mistakes are considered bad. On the African savannah, one mistake meant death. Even today, if you make a costly mistake at work, you'll be fired as a scapegoat. Most people avoid failing.

In reality, making mistakes is the best way to learn.

Eric Reis offers an unintuitive recipe in The Lean Startup: come up with a hypothesis, test it, and fail. Then, try again with a new hypothesis. Keep trying, learning from each failure.

This is a great startup strategy. Startups are new businesses. Startups face uncertainty. Run lots of low-cost experiments to fail, learn, and succeed.

Don't fear failing. Low-cost failure is good because you learn more from it than you lose. As long as your worst-case scenario is acceptable, risk-taking is good.

The Sovereign Individual

Today, nation-states rule the world. The UN recognizes 195 countries, and they claim almost all land outside of Antarctica.

We agree. For the past 2,000 years, much of the world's territory was ungoverned.

Why today? Because technology has created incentives for nation-states for most of the past 500 years. The logic of violence favors nation-states, according to James Dale Davidson, author of the Sovereign Individual. Governments have a lot to gain by conquering as much territory as possible, so they do.

Not always. During the Dark Ages, Europe was fragmented and had few central governments. Partly because of armor. With armor, a sword, and a horse, you couldn't be stopped. Large states were hard to form because they rely on the threat of violence.

When gunpowder became popular in Europe, violence changed. In a world with guns, assembling large armies and conquest are cheaper.

James Dale Davidson says the internet will make nation-states obsolete. Most of the world's wealth will be online and in people's heads, making capital mobile.

Nation-states rely on predatory taxation of the rich to fund large militaries and welfare programs.

When capital is mobile, people can live anywhere in the world, Davidson says, making predatory taxation impossible. They're not bound by their job, land, or factory location. Wherever they're treated best.

Davidson says that over the next century, nation-states will collapse because they won't have enough money to operate as they do now. He imagines a world of small city-states, like Italy before 1900. (or Singapore today).

We've already seen some movement toward a more Sovereign Individual-like world. The pandemic proved large-scale remote work is possible, freeing workers from their location. Many cities and countries offer remote workers incentives to relocate.

Many Western businesspeople live in tax havens, and more people are renouncing their US citizenship due to high taxes. Increasing globalization has led to poor economic conditions and resentment among average people in the West, which is why politicians like Trump and Sanders rose to popularity with angry rhetoric, even though Obama rose to popularity with a more hopeful message.

The Sovereign Individual convinced me that the future will be different than Nassim Taleb's. Large countries like the U.S. will likely lose influence in the coming decades, while Portugal, Singapore, and Turkey will rise. If the trend toward less freedom continues, people may flee the West en masse.

So a traditional life of college, a big firm job, hard work, and corporate advancement may not be wise. Young people should learn as much as possible and develop flexible skills to adapt to the future.

Sapiens

Sapiens is a history of humanity, from proto-humans in Ethiopia to our internet society today, with some future speculation.

Sapiens views humans (and Homo sapiens) as a unique species on Earth. We were animals 100,000 years ago. We're slowly becoming gods, able to affect the climate, travel to every corner of the Earth (and the Moon), build weapons that can kill us all, and wipe out thousands of species.

Sapiens examines what makes Homo sapiens unique. Humans can believe in myths like religion, money, and human-made entities like countries and LLCs.

These myths facilitate large-scale cooperation. Ants from the same colony can cooperate. Any two humans can trade, though. Even if they're not genetically related, large groups can bond over religion and nationality.

Combine that with intelligence, and you have a species capable of amazing feats.

Sapiens may make your head explode because it looks at the world without presupposing values, unlike most books. It questions things that aren't usually questioned and says provocative things.

It also shows how human history works. It may help you understand and predict the world. Maybe.

The 4-hour Workweek

Things can be done better.

Tradition, laziness, bad bosses, or incentive structures cause complacency. If you're willing to make changes and not settle for the status quo, you can do whatever you do better and achieve more in less time.

The Four-Hour Work Week advocates this. Tim Ferriss explains how he made more sales in 2 hours than his 8-hour-a-day colleagues.

By firing 2 of his most annoying customers and empowering his customer service reps to make more decisions, he was able to leave his business and travel to Europe.

Ferriss shows how to escape your 9-to-5, outsource your life, develop a business that feeds you with little time, and go on mini-retirement adventures abroad.

Don't accept the status quo. Instead, level up. Find a way to improve your results. And try new things.

Why Nations Fail

Nogales, Arizona and Mexico were once one town. The US/Mexico border was arbitrarily drawn.

Both towns have similar cultures and populations. Nogales, Arizona is well-developed and has a high standard of living. Nogales, Mexico is underdeveloped and has a low standard of living. Whoa!

Why Nations Fail explains how government-created institutions affect country development. Strong property rights, capitalism, and non-corrupt governments promote development. Countries without capitalism, strong property rights, or corrupt governments don't develop.

Successful countries must also embrace creative destruction. They must offer ordinary citizens a way to improve their lot by creating value for others, not reducing them to slaves, serfs, or peasants. Authors say that ordinary people could get rich on trading expeditions in 11th-century Venice.

East and West Germany and North and South Korea have different economies because their citizens are motivated differently. It explains why Chile, China, and Singapore grow so quickly after becoming market economies.

People have spent a lot of money on third-world poverty. According to Why Nations Fail, education and infrastructure aren't the answer. Developing nations must adopt free-market economic policies.

Elon Musk

Elon Musk is the world's richest man, but that’s not a good way to describe him. Elon Musk is the world's richest man, which is like calling Steve Jobs a turtleneck-wearer or Benjamin Franklin a printer.

Elon Musk does cool sci-fi stuff to help humanity avoid existential threats.

Oil will run out. We've delayed this by developing better extraction methods. We only have so much nonrenewable oil.

Our society is doomed if it depends on oil. Elon Musk invested heavily in Tesla and SolarCity to speed the shift to renewable energy.

Musk worries about AI: we'll build machines smarter than us. We won't be able to stop these machines if something goes wrong, just like cows can't fight humans. Neuralink: we need to be smarter to compete with AI when the time comes.

If Earth becomes uninhabitable, we need a backup plan. Asteroid or nuclear war could strike Earth at any moment. We may not have much time to react if it happens in a few days. We must build a new civilization while times are good and resources are plentiful.

Short-term problems dominate our politics, but long-term issues are more important. Long-term problems can cause mass casualties and homelessness. Musk demonstrates how to think long-term.

The main reason people are impressed by Elon Musk, and why Ashlee Vances' biography influenced me so much, is that he does impossible things.

Electric cars were once considered unprofitable, but Tesla has made them mainstream. SpaceX is the world's largest private space company.

People lack imagination and dismiss ununderstood ideas as impossible. Humanity is about pushing limits. Don't worry if your dreams seem impossible. Try it.

Thanks for reading.

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Monroe Mayfield

Monroe Mayfield

2 years ago

CES 2023: A Third Look At Upcoming Trends

Las Vegas hosted CES 2023. This third and last look at CES 2023 previews upcoming consumer electronics trends that will be crucial for market share.

Photo by Willow Findlay on Unsplash

Definitely start with ICT. Qualcomm CEO Cristiano Amon spoke to CNBC from Las Vegas on China's crackdown and the company's automated driving systems for electric vehicles (EV). The business showed a concept car and its latest Snapdragon processor designs, which offer expanded digital interactions through SalesForce-partnered CRM platforms.

Qualcomm CEO Meets SK Hynix Vice Chairman at CES 2023 On Jan. 6, SK hynix Inc.'s vice chairman and co-CEO Park Jung-ho discussed strengthening www.businesskorea.co.kr.

Electrification is reviving Michigan's automobile industry. Michigan Local News reports that $14 billion in EV and battery manufacturing investments will benefit the state. The report also revealed that the Strategic Outreach and Attraction Reserve (SOAR) fund had generated roughly $1 billion for the state's automotive sector.

Michigan to "dominate" EV battery manufacturing after $2B investment. Michigan spent $2 billion to safeguard www.mlive.com.

Ars Technica is great for technology, society, and the future. After CES 2023, Jonathan M. Gitlin published How many electric car chargers are enough? Read about EV charging network issues and infrastructure spending. Politics aside, rapid technological advances enable EV charging network expansion in American cities and abroad.

New research says US needs 8x more EV chargers by 2030. Electric vehicle skepticism—which is widespread—is fundamentally about infrastructure. arstechnica.com

Finally, the UNEP's The Future of Electric Vehicles and Material Resources: A Foresight Brief. Understanding how lithium-ion batteries will affect EV sales is crucial. Climate change affects EVs in various ways, but electrification and mining trends stand out because more EVs demand more energy-intensive metals and rare earths. Areas & Producers has been publishing my electrification and mining trends articles. Follow me if you wish to write for the publication.

Producers This magazine analyzes medium.com-related corporate, legal, and international news to examine a paradigm shift.

The Weekend Brief (TWB) will routinely cover tech, industrials, and global commodities in global markets, including stock markets. Read more about the future of key areas and critical producers of the global economy in Areas & Producers.

TotalEnergies, Stellantis Form Automotive Cells Company (ACC) A joint-venture to design and build electric vehicles (EVs) was formed in 2020.

Matt Nutsch

Matt Nutsch

3 years ago

Most people are unaware of how artificial intelligence (A.I.) is changing the world.

Image created by MidjourneyAI user Dreamland3K

Recently, I saw an interesting social media post. In an entrepreneurship forum. A blogger asked for help because he/she couldn't find customers. I now suspect that the writer’s occupation is being disrupted by A.I.

Introduction

Artificial Intelligence (A.I.) has been a hot topic since the 1950s. With recent advances in machine learning, A.I. will touch almost every aspect of our lives. This article will discuss A.I. technology and its social and economic implications.

What's AI?

A computer program or machine with A.I. can think and learn. In general, it's a way to make a computer smart. Able to understand and execute complex tasks. Machine learning, NLP, and robotics are common types of A.I.

AI's global impact

MidjourneyAI image generated by user Desmesne

AI will change the world, but probably faster than you think. A.I. already affects our daily lives. It improves our decision-making, efficiency, and productivity.

A.I. is transforming our lives and the global economy. It will create new business and job opportunities but eliminate others. Affected workers may face financial hardship.

AI examples:

OpenAI's GPT-3 text-generation

MidjourneyAI generated image of robot typing

Developers can train, deploy, and manage models on GPT-3. It handles data preparation, model training, deployment, and inference for machine learning workloads. GPT-3 is easy to use for both experienced and new data scientists.

My team conducted an experiment. We needed to generate some blog posts for a website. We hired a blogger on Upwork. OpenAI created a blog post. The A.I.-generated blog post was of higher quality and lower cost.

MidjourneyAI's Art Contests

Théâtre D’opéra Spatial by Jason M. Allen via MidjourneyAI

AI already affects artists. Artists use A.I. to create realistic 3D images and videos for digital art. A.I. is also used to generate new art ideas and methods.

MidjourneyAI and GigapixelAI won a contest last month. It's AI. created a beautiful piece of art that captured the contest's spirit. AI triumphs. It could open future doors.

After the art contest win, I registered to try out these new image generating A.I.s. In the MidjourneyAI chat forum, I noticed an artist's plea. The artist begged others to stop flooding RedBubble with AI-generated art.

Shutterstock and Getty Images have halted user uploads. AI-generated images flooded online marketplaces.

Imagining Videos with Meta

AI generated video example from Meta AI

Meta released Make-a-Video this week. It's an A.I. app that creates videos from text. What you type creates a video.

This technology will impact TV, movies, and video games greatly. Imagine a movie or game that's personalized to your tastes. It's closer than you think.

Uses and Abuses of Deepfakes

Carrie Fischer’s likeness in the movie The Rise of Skywalker

Deepfake videos are computer-generated images of people. AI creates realistic images and videos of people.

Deepfakes are entertaining but have social implications. Porn introduced deepfakes in 2017. People put famous faces on porn actors and actresses without permission.

Soon, deepfakes were used to show dead actors/actresses or make them look younger. Carrie Fischer was included in films after her death using deepfake technology.

Deepfakes can be used to create fake news or manipulate public opinion, according to an AI.

Voices for Darth Vader and Iceman

James Earl Jones, who voiced Darth Vader, sold his voice rights this week. Aged actor won't be in those movies. Respeecher will use AI to mimic Jones's voice. This technology could change the entertainment industry. One actor can now voice many characters.

Val Kilmer in Top Gun as imagined by MidjourneyAI

AI can generate realistic voice audio from text. Top Gun 2 actor Val Kilmer can't speak for medical reasons. Sonantic created Kilmer's voice from the movie script. This entertaining technology has social implications. It blurs authentic recordings and fake media.

Medical A.I. fights viruses

MidjourneyAI generated image of virus

A team of Chinese scientists used machine learning to predict effective antiviral drugs last year. They started with a large dataset of virus-drug interactions. Researchers combined that with medication and virus information. Finally, they used machine learning to predict effective anti-virus medicines. This technology could solve medical problems.

AI ideas AI-generated Itself

MidjourneyAI image generated by user SubjectChunchunmaru

OpenAI's GPT-3 predicted future A.I. uses. Here's what it told me:

AI will affect the economy. Businesses can operate more efficiently and reinvest resources with A.I.-enabled automation. AI can automate customer service tasks, reducing costs and improving satisfaction.

A.I. makes better pricing, inventory, and marketing decisions. AI automates tasks and makes decisions. A.I.-powered robots could help the elderly or disabled. Self-driving cars could reduce accidents.

A.I. predictive analytics can predict stock market or consumer behavior trends and patterns. A.I. also personalizes recommendations. sways. A.I. recommends products and movies. AI can generate new ideas based on data analysis.

Conclusion

Image generated from MidjourneyAI by user PuddingPants.”

A.I. will change business as it becomes more common. It will change how we live and work by creating growth and prosperity.

Exciting times,  but also one which should give us all pause. Technology can be good or evil. We must use new technologies ethically, fairly, and honestly.

“The author generated some sentences in this text in part with GPT-3, OpenAI’s large-scale language-generation model. Upon generating draft language, the author reviewed, edited, and revised the language to their own liking and takes ultimate responsibility for the content of this publication. The text of this post was further edited using HemingWayApp. Many of the images used were generated using A.I. as described in the captions.”

Vitalik

Vitalik

3 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

  • x+5
  • x^4
  • x^3+3x^2+3x+1
  • 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

  • A(0)+B(0)=C(0)
  • A(1)+B(1)=C(1)
  • A(2)+B(2)=C(2)
  • A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

  • 12+1=13
  • 10+8=18
  • 15+8=23
  • 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

  • A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
  • B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
  • C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3