More on Personal Growth

Entreprogrammer
3 years ago
The Steve Jobs Formula: A Guide to Everything
A must-read for everyone
Jobs is well-known. You probably know the tall, thin guy who wore the same clothing every day. His influence is unavoidable. In fewer than 40 years, Jobs' innovations have impacted computers, movies, cellphones, music, and communication.
Steve Jobs may be more imaginative than the typical person, but if we can use some of his ingenuity, ambition, and good traits, we'll be successful. This essay explains how to follow his guidance and success secrets.
1. Repetition is necessary for success.
Be patient and diligent to master something. Practice makes perfect. This is why older workers are often more skilled.
When should you repeat a task? When you're confident and excited to share your product. It's when to stop tweaking and repeating.
Jobs stated he'd make the crowd sh** their pants with an iChat demo.
Use this in your daily life.
Start with the end in mind. You can put it in writing and be as detailed as you like with your plan's schedule and metrics. For instance, you have a goal of selling three coffee makers in a week.
Break it down, break the goal down into particular tasks you must complete, and then repeat those tasks. To sell your coffee maker, you might need to make 50 phone calls.
Be mindful of the amount of work necessary to produce the desired results. Continue doing this until you are happy with your product.
2. Acquire the ability to add and subtract.
How did Picasso invent cubism? Pablo Picasso was influenced by stylised, non-naturalistic African masks that depict a human figure.
Artists create. Constantly seeking inspiration. They think creatively about random objects. Jobs said creativity is linking things. Creative people feel terrible when asked how they achieved something unique because they didn't do it all. They saw innovation. They had mastered connecting and synthesizing experiences.
Use this in your daily life.
On your phone, there is a note-taking app. Ideas for what you desire to learn should be written down. It may be learning a new language, calligraphy, or anything else that inspires or intrigues you.
Note any ideas you have, quotations, or any information that strikes you as important.
Spend time with smart individuals, that is the most important thing. Jim Rohn, a well-known motivational speaker, has observed that we are the average of the five people with whom we spend the most time.
Learning alone won't get you very far. You need to put what you've learnt into practice. If you don't use your knowledge and skills, they are useless.
3. Develop the ability to refuse.
Steve Jobs deleted thousands of items when he created Apple's design ethic. Saying no to distractions meant upsetting customers and partners.
John Sculley, the former CEO of Apple, said something like this. According to Sculley, Steve’s methodology differs from others as he always believed that the most critical decisions are things you choose not to do.
Use this in your daily life.
Never be afraid to say "no," "I won't," or "I don't want to." Keep it simple. This method works well in some situations.
Give a different option. For instance, X might be interested even if I won't be able to achieve it.
Control your top priority. Before saying yes to anything, make sure your work schedule and priority list are up to date.
4. Follow your passion
“Follow your passion” is the worst advice people can give you. Steve Jobs didn't start Apple because he suddenly loved computers. He wanted to help others attain their maximum potential.
Great things take a lot of work, so quitting makes sense if you're not passionate. Jobs learned from history that successful people were passionate about their work and persisted through challenges.
Use this in your daily life.
Stay away from your passion. Allow it to develop daily. Keep working at your 9-5-hour job while carefully gauging your level of desire and endurance. Less risk exists.
The truth is that if you decide to work on a project by yourself rather than in a group, it will take you years to complete it instead of a week. Instead, network with others who have interests in common.
Prepare a fallback strategy in case things go wrong.
Success, this small two-syllable word eventually gives your life meaning, a perspective. What is success? For most, it's achieving their ambitions. However, there's a catch. Successful people aren't always happy.
Furthermore, where do people’s goals and achievements end? It’s a never-ending process. Success is a journey, not a destination. We wish you not to lose your way on this journey.

Tim Denning
3 years ago
Read These Books on Personal Finance to Boost Your Net Worth
And retire sooner.
Books can make you filthy rich.
If you apply what you learn. In 2011, I was broke and had broken dreams.
Someone suggested I read finance books. One Up On Wall Street was his first recommendation.
Finance books were my crack.
I've read every money book since then. Some are good, but most stink.
These books will make you rich.
The Almanack of Naval Ravikant by Eric Jorgenson
This isn't a cliche book.
This book was inspired by a How to Get Rich tweet thread.
It’s one of the best tweets I’ve ever read.
Naval thinks differently. He nukes ordinary ideas. I've never heard better money advice.
Eric Jorgenson wrote a book about this tweet thread with Navals permission. A must-read, easy-to-digest book.
Best quote
Seek wealth, not money or status. Wealth is having assets that earn while you sleep. Money is how we transfer time and wealth. Status is your place in the social hierarchy — Naval
Morgan Housel's The Psychology of Money
Many finance books advise investing like a dunce.
They almost all peddle the buy an index fund BS. Different book.
It's about money-making psychology. Because any fool can get rich and drunk on their ego. Few can consistently make money.
Each chapter is short. A single-page chapter breaks all book publishing rules.
Best quote
Spending money to show people how much money you have is the fastest way to have less money — Morgan Housel
J.L. Collins' The Simple Path to Wealth
Most of the best money books were written by bloggers.
JL Collins blogs. This easy-to-read book was written for his daughter.
This book popularized the phrase F You Money. With enough money in your bank account and investment portfolio, you can say F You more.
A bad boss is an example. You can leave instead of enduring his wrath.
You can then sit at home and look for another job while financially secure. JL says its mind-freedom is powerful.
Best phrasing
You own the things you own and they in turn own you — J.L. Collins
Tony Robbins' Unshakeable
I like Tony. This book makes me sweaty.
Tony interviews the world's top financiers. He interviews people who rarely do so.
This book taught me all-weather portfolio. It's a way to invest in different asset classes in good, bad, recession, or depression times.
Look at it:
Investing isn’t about buying one big winner — that’s gambling. It’s about investing in a diversified portfolio of assets.
Best phrasing
The best opportunities come in times of maximum pessimism — Tony Robbins
Ben Graham's The Intelligent Investor
This book helped me distinguish between a spectator and an investor.
Spectators are those who shout that crypto, NFTs, or XYZ platform will die.
Tourists. They want attention and to say "I told you so." They make short-term and long-term predictions like fortunetellers. LOL. Idiots.
Benjamin Graham teaches smart investing. You'll buy a long-term asset. To be confident in recessions, use dollar-cost averaging.
Best phrasing
Those who do not remember the past are condemned to repeat it. — Benjamin Graham
The Napoleon Hill book Think and Grow Rich
This classic book introduced positive thinking to modern self-help.
Lazy pessimists can't become rich. No way.
Napoleon said, "Thoughts create reality."
No surprise that he discusses obsession and focus in this book. They are the fastest ways to make more money to invest in time and wealth-protecting assets.
Best phrasing
The starting point of all achievement is DESIRE. Keep this constantly in mind. Weak desire brings weak results, just as a small fire makes a small amount of heat — Napoleon Hill
Ramit Sethi's book I Will Teach You To Be Rich
This book is mostly good. The part about credit cards is trash.
Avoid credit card temptations. I don't care about their airline points.
This book teaches you to master money basics (that many people mess up) then automate it so your monkey brain doesn't ruin your financial future.
The book includes great negotiation tactics to help you make more money in less time.
Best quote
The 85 Percent Solution: Getting started is more important than becoming an expert — Ramit Sethi
David Bach's The Automatic Millionaire
You've probably met a six- or seven-figure earner who's broke. All their money goes to useless things like cars.
Money isn't as essential as what you do with it. David teaches how to automate your earnings for more money.
Compounding works once investing is automated. So you get rich.
His strategy eliminates luck and (almost) guarantees millionaire status.
Best phrasing
Every time you earn one dollar, make sure to pay yourself first — David Bach
Thomas J. Stanley's The Millionaire Next Door
Thomas defies the definition of rich.
He spends much of the book highlighting millionaire traits he's studied.
Rich people are quiet, so you wouldn't know they're wealthy. They don't earn much money or drive a BMW.
Thomas will give you the math to get started.
Best phrasing
I am not impressed with what people own. But I’m impressed with what they achieve. I’m proud to be a physician. Always strive to be the best in your field…. Don’t chase money. If you are the best in your field, money will find you. — Thomas J. Stanley
by Bill Perkins "Die With Zero"
Let’s end with one last book.
Bill's book angered many people. He says we spend too much time saving for retirement and die rich. That bank money is lost time.
Your grandkids could use the money. When children inherit money, they become lazy, entitled a-holes.
Bill wants us to spend our money on life-enhancing experiences. Stop saving money like monopoly monkeys.
Best phrasing
You should be focusing on maximizing your life enjoyment rather than on maximizing your wealth. Those are two very different goals. Money is just a means to an end: Having money helps you to achieve the more important goal of enjoying your life. But trying to maximize money actually gets in the way of achieving the more important goal — Bill Perkins

Tim Denning
3 years ago
In this recession, according to Mark Cuban, you need to outwork everyone
Here’s why that’s baloney
Mark Cuban popularized entrepreneurship.
Shark Tank (which made Mark famous) made starting a business glamorous to attract more entrepreneurs. First off
This isn't an anti-billionaire rant.
Mark Cuban has done excellent. He's a smart, principled businessman. I enjoy his Web3 work. But Mark's work and productivity theories are absurd.
You don't need to outwork everyone in this recession to live well.
You won't be able to outwork me.
Yuck! Mark's words made me gag.
Why do boys think working is a football game where the winner wins a Super Bowl trophy? To outwork you.
Hard work doesn't equal intelligence.
Highly clever professionals spend 4 hours a day in a flow state, then go home to relax with family.
If you don't put forth the effort, someone else will.
- Mark.
He'll burn out. He's delusional and doesn't understand productivity. Boredom or disconnection spark our best thoughts.
TikTok outlaws boredom.
In a spare minute, we check our phones because we can't stand stillness.
All this work p*rn makes things worse. When is it okay to feel again? Because I can’t feel anything when I’m drowning in work and haven’t had a holiday in 2 years.
Your rivals are actively attempting to undermine you.
Ohhh please Mark…seriously.
This isn't a Tom Hanks war film. Relax. Not everyone is a rival. Only yourself is your competitor. To survive the recession, be better than a year ago.
If you get rich, great. If not, there's more to life than Lambos and angel investments.
Some want to relax and enjoy life. No competition. We witness people with lives trying to endure the recession and record-high prices.
This fictitious rival worsens life and work.
If you are truly talented, you will motivate others to work more diligently and effectively.
No Mark. Soz.
If you're a good leader, you won't brag about working hard and treating others like cogs. Treat them like humans. You'll have EQ.
Silly statements like this are caused by an out-of-control ego. No longer watch Shark Tank.
Ego over humanity.
Good leaders will urge people to keep together during the recession. Good leaders support those who are laid off and need a reference.
Not harder, quicker, better. That created my mental health problems 10 years ago.
Truth: we want to work less.
The promotion of entrepreneurship is ludicrous.
Marvel superheroes. Seriously, relax Max.
I used to write about entrepreneurship, then I quit. Many WeWork Adam Neumanns. Carelessness.
I now utilize the side hustle title when writing about online company or entrepreneurship. Humanizes.
Stop glorifying. Thinking we'll all be Elon Musks who send rockets to Mars is delusional. Most of us won't create companies employing hundreds.
OK.
The true epidemic is glorification. fewer selfies Little birdy needs less bank account screenshots. Less Uber talk.
We're exhausted.
Fun, ego-free business can transform the world. Take a relax pill.
Work as if someone were attempting to take everything from you.
I've seen people lose everything.
Myself included. My 20s startup failed. I was almost bankrupt. I thought I'd never recover. Nope.
Best thing ever.
Losing everything reveals your true self. Unintelligent entrepreneur egos perish instantly. Regaining humility revitalizes relationships.
Money's significance shifts. Stop chasing it like a puppy with a bone.
Fearing loss is unfounded.
Here is a more effective approach than outworking nobody.
(You'll thrive in the recession and become wealthy.)
Smarter work
Overworking is donkey work.
You don't want to be a career-long overworker. Instead than wasting time, write down what you do. List tasks and processes.
Keep doing/outsource the list. Step-by-step each task. Continuously systematize.
Then recruit a digital employee like Zapier or a virtual assistant in the same country.
Intelligent, not difficult.
If your big break could burn in hell, diversify like it will.
People err by focusing on one chance.
Chances can vanish. All-in risky. Instead of working like a Mark Cuban groupie, diversify your income.
If you're employed, your customer is your employer.
Sell the same abilities twice and add 2-3 contract clients. Reduce your hours at your main job and take on more clients.
Leave brand loyalty behind
Mark desires his employees' worship.
That's stupid. When times are bad, layoffs multiply. The problem is the false belief that companies care. No. A business maximizes profit and pays you the least.
To care or overpay is anti-capitalist (that run the world). Be honest.
I was a banker. Then the bat virus hit and jobs disappeared faster than I urinate after a night of drinking.
Start being disloyal now since your company will cheerfully replace you with a better applicant. Meet recruiters and hiring managers on LinkedIn. Whenever something goes wrong at work, act.
Loyalty to self and family. Nobody.
Outwork this instead
Mark doesn't suggest outworking inflation instead of people.
Inflation erodes your time on earth. If you ignore inflation, you'll work harder for less pay every minute.
Financial literacy beats inflation.
Get a side job and earn money online
So you can stop outworking everyone.
Internet leverages time. Same effort today yields exponential results later. There are still whole places not online.
Instead of working forever, generate money online.
Final Words
Overworking is stupid. Don't listen to wealthy football jocks.
Work isn't everything. Prioritize diversification, internet income streams, boredom, and financial knowledge throughout the recession.
That’s how to get wealthy rather than burnout-rich.
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DC Palter
2 years ago
Why Are There So Few Startups in Japan?
Japan's startup challenge: 7 reasons
Every day, another Silicon Valley business is bought for a billion dollars, making its founders rich while growing the economy and improving consumers' lives.
Google, Amazon, Twitter, and Medium dominate our daily lives. Tesla automobiles and Moderna Covid vaccinations.
The startup movement started in Silicon Valley, California, but the rest of the world is catching up. Global startup buzz is rising. Except Japan.
644 of CB Insights' 1170 unicorns—successful firms valued at over $1 billion—are US-based. China follows with 302 and India third with 108.
Japan? 6!
1% of US startups succeed. The third-largest economy is tied with small Switzerland for startup success.
Mexico (8), Indonesia (12), and Brazil (12) have more successful startups than Japan (16). South Korea has 16. Yikes! Problem?
Why Don't Startups Exist in Japan More?
Not about money. Japanese firms invest in startups. To invest in startups, big Japanese firms create Silicon Valley offices instead of Tokyo.
Startups aren't the issue either. Local governments are competing to be Japan's Shirikon Tani, providing entrepreneurs financing, office space, and founder visas.
Startup accelerators like Plug and Play in Tokyo, Osaka, and Kyoto, the Startup Hub in Kobe, and Google for Startups are many.
Most of the companies I've encountered in Japan are either local offices of foreign firms aiming to expand into the Japanese market or small businesses offering local services rather than disrupting a staid industry with new ideas.
There must be a reason Japan can develop world-beating giant corporations like Toyota, Nintendo, Shiseido, and Suntory but not inventive startups.
Culture, obviously. Japanese culture excels in teamwork, craftsmanship, and quality, but it hates moving fast, making mistakes, and breaking things.
If you have a brilliant idea in Silicon Valley, quit your job, get money from friends and family, and build a prototype. To fund the business, you approach angel investors and VCs.
Most non-startup folks don't aware that venture capitalists don't want good, profitable enterprises. That's wonderful if you're developing a solid small business to consult, open shops, or make a specialty product. However, you must pay for it or borrow money. Venture capitalists want moon rockets. Silicon Valley is big or bust. Almost 90% will explode and crash. The few successes are remarkable enough to make up for the failures.
Silicon Valley's high-risk, high-reward attitude contrasts with Japan's incrementalism. Japan makes the best automobiles and cleanrooms, but it fails to produce new items that grow the economy.
Changeable? Absolutely. But, what makes huge manufacturing enterprises successful and what makes Japan a safe and comfortable place to live are inextricably connected with the lack of startups.
Barriers to Startup Development in Japan
These are the 7 biggest obstacles to Japanese startup success.
Unresponsive Employment Market
While the lifelong employment system in Japan is evolving, the average employee stays at their firm for 12 years (15 years for men at large organizations) compared to 4.3 years in the US. Seniority, not experience or aptitude, determines career routes, making it tough to quit a job to join a startup and then return to corporate work if it fails.
Conservative Buyers
Even if your product is buggy and undocumented, US customers will migrate to a cheaper, superior one. Japanese corporations demand perfection from their trusted suppliers and keep with them forever. Startups need income fast, yet product evaluation takes forever.
Failure intolerance
Japanese business failures harm lives. Failed forever. It hinders risk-taking. Silicon Valley embraces failure. Build another startup if your first fails. Build a third if that fails. Every setback is viewed as a learning opportunity for success.
4. No Corporate Purchases
Silicon Valley industrial giants will buy fast-growing startups for a lot of money. Many huge firms have stopped developing new goods and instead buy startups after the product is validated.
Japanese companies prefer in-house product development over startup acquisitions. No acquisitions mean no startup investment and no investor reward.
Startup investments can also be monetized through stock market listings. Public stock listings in Japan are risky because the Nikkei was stagnant for 35 years while the S&P rose 14x.
5. Social Unity Above Wealth
In Silicon Valley, everyone wants to be rich. That creates a competitive environment where everyone wants to succeed, but it also promotes fraud and societal problems.
Japan values communal harmony above individual success. Wealthy folks and overachievers are avoided. In Japan, renegades are nearly impossible.
6. Rote Learning Education System
Japanese high school graduates outperform most Americans. Nonetheless, Japanese education is known for its rote memorization. The American system, which fails too many kids, emphasizes creativity to create new products.
Immigration.
Immigrants start 55% of successful Silicon Valley firms. Some come for university, some to escape poverty and war, and some are recruited by Silicon Valley startups and stay to start their own.
Japan is difficult for immigrants to start a business due to language barriers, visa restrictions, and social isolation.
How Japan Can Promote Innovation
Patchwork solutions to deep-rooted cultural issues will not work. If customers don't buy things, immigration visas won't aid startups. Startups must have a chance of being acquired for a huge sum to attract investors. If risky startups fail, employees won't join.
Will Japan never have a startup culture?
Once a consensus is reached, Japan changes rapidly. A dwindling population and standard of living may lead to such consensus.
Toyota and Sony were firms with renowned founders who used technology to transform the world. Repeatable.
Silicon Valley is flawed too. Many people struggle due to wealth disparities, job churn and layoffs, and the tremendous ups and downs of the economy caused by stock market fluctuations.
The founders of the 10% successful startups are heroes. The 90% that fail and return to good-paying jobs with benefits are never mentioned.
Silicon Valley startup culture and Japanese corporate culture are opposites. Each have pros and cons. Big Japanese corporations make the most reliable, dependable, high-quality products yet move too slowly. That's good for creating cars, not social networking apps.
Can innovation and success be encouraged without eroding social cohesion? That can motivate software firms to move fast and break things while recognizing the beauty and precision of expert craftsmen? A hybrid culture where Japan can make the world's best and most original items. Hopefully.

Duane Michael
3 years ago
Don't Fall Behind: 7 Subjects You Must Understand to Keep Up with Technology
As technology develops, you should stay up to date
You don't want to fall behind, do you? This post covers 7 tech-related things you should know.
You'll learn how to operate your computer (and other electronic devices) like an expert and how to leverage the Internet and social media to create your brand and business. Read on to stay relevant in today's tech-driven environment.
You must learn how to code.
Future-language is coding. It's how we and computers talk. Learn coding to keep ahead.
Try Codecademy or Code School. There are also numerous free courses like Coursera or Udacity, but they take a long time and aren't necessarily self-paced, so it can be challenging to find the time.
Artificial intelligence (AI) will transform all jobs.
Our skillsets must adapt with technology. AI is a must-know topic. AI will revolutionize every employment due to advances in machine learning.
Here are seven AI subjects you must know.
What is artificial intelligence?
How does artificial intelligence work?
What are some examples of AI applications?
How can I use artificial intelligence in my day-to-day life?
What jobs have a high chance of being replaced by artificial intelligence and how can I prepare for this?
Can machines replace humans? What would happen if they did?
How can we manage the social impact of artificial intelligence and automation on human society and individual people?
Blockchain Is Changing the Future
Few of us know how Bitcoin and blockchain technology function or what impact they will have on our lives. Blockchain offers safe, transparent, tamper-proof transactions.
It may alter everything from business to voting. Seven must-know blockchain topics:
Describe blockchain.
How does the blockchain function?
What advantages does blockchain offer?
What possible uses for blockchain are there?
What are the dangers of blockchain technology?
What are my options for using blockchain technology?
What does blockchain technology's future hold?
Cryptocurrencies are here to stay
Cryptocurrencies employ cryptography to safeguard transactions and manage unit creation. Decentralized cryptocurrencies aren't controlled by governments or financial institutions.
Bitcoin, the first cryptocurrency, was launched in 2009. Cryptocurrencies can be bought and sold on decentralized exchanges.
Bitcoin is here to stay.
Bitcoin isn't a fad, despite what some say. Since 2009, Bitcoin's popularity has grown. Bitcoin is worth learning about now. Since 2009, Bitcoin has developed steadily.
With other cryptocurrencies emerging, many people are wondering if Bitcoin still has a bright future. Curiosity is natural. Millions of individuals hope their Bitcoin investments will pay off since they're popular now.
Thankfully, they will. Bitcoin is still running strong a decade after its birth. Here's why.
The Internet of Things (IoT) is no longer just a trendy term.
IoT consists of internet-connected physical items. These items can share data. IoT is young but developing fast.
20 billion IoT-connected devices are expected by 2023. So much data! All IT teams must keep up with quickly expanding technologies. Four must-know IoT topics:
Recognize the fundamentals: Priorities first! Before diving into more technical lingo, you should have a fundamental understanding of what an IoT system is. Before exploring how something works, it's crucial to understand what you're working with.
Recognize Security: Security does not stand still, even as technology advances at a dizzying pace. As IT professionals, it is our duty to be aware of the ways in which our systems are susceptible to intrusion and to ensure that the necessary precautions are taken to protect them.
Be able to discuss cloud computing: The cloud has seen various modifications over the past several years once again. The use of cloud computing is also continually changing. Knowing what kind of cloud computing your firm or clients utilize will enable you to make the appropriate recommendations.
Bring Your Own Device (BYOD)/Mobile Device Management (MDM) is a topic worth discussing (MDM). The ability of BYOD and MDM rules to lower expenses while boosting productivity among employees who use these services responsibly is a major factor in their continued growth in popularity.
IoT Security is key
As more gadgets connect, they must be secure. IoT security includes securing devices and encrypting data. Seven IoT security must-knows:
fundamental security ideas
Authorization and identification
Cryptography
electronic certificates
electronic signatures
Private key encryption
Public key encryption
Final Thoughts
With so much going on in the globe, it can be hard to stay up with technology. We've produced a list of seven tech must-knows.

Vitalik
4 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
