More on Personal Growth

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.

Mia Gradelski
3 years ago
Six Things Best-With-Money People Do Follow
I shouldn't generalize, yet this is true.
Spending is simpler than earning.
Prove me wrong, but with home debt at $145k in 2020 and individual debt at $67k, people don't have their priorities straight.
Where does this loan originate?
Under-50 Americans owed $7.86 trillion in Q4 20T. That's more than the US's 3-trillion-dollar deficit.
Here’s a breakdown:
🏡 Mortgages/Home Equity Loans = $5.28 trillion (67%)
🎓 Student Loans = $1.20 trillion (15%)
🚗 Auto Loans = $0.80 trillion (10%)
💳 Credit Cards = $0.37 trillion (5%)
🏥 Other/Medical = $0.20 trillion (3%)
Images.google.com
At least the Fed and government can explain themselves with their debt balance which includes:
-Providing stimulus packages 2x for Covid relief
-Stabilizing the economy
-Reducing inflation and unemployment
-Providing for the military, education and farmers
No American should have this much debt.
Don’t get me wrong. Debt isn’t all the same. Yes, it’s a negative number but it carries different purposes which may not be all bad.
Good debt: Use those funds in hopes of them appreciating as an investment in the future
-Student loans
-Business loan
-Mortgage, home equity loan
-Experiences
Paying cash for a home is wasteful. Just if the home is exceptionally uncommon, only 1 in a million on the market, and has an incredible bargain with numerous bidders seeking higher prices should you do so.
To impress the vendor, pay cash so they can sell it quickly. Most people can't afford most properties outright. Only 15% of U.S. homebuyers can afford their home. Zillow reports that only 37% of homes are mortgage-free.
People have clearly overreached.
Ignore appearances.
5% down can buy a 10-bedroom mansion.
Not paying in cash isn't necessarily a negative thing given property prices have increased by 30% since 2008, and throughout the epidemic, we've seen work-from-homers resort to the midwest, avoiding pricey coastal cities like NYC and San Francisco.
By no means do I think NYC is dead, nothing will replace this beautiful city that never sleeps, and now is the perfect time to rent or buy when everything is below average value for people who always wanted to come but never could. Once social distance ends, cities will recover. 24/7 sardine-packed subways prove New York isn't designed for isolation.
When buying a home, pay 20% cash and the balance with a mortgage. A mortgage must be incorporated into other costs such as maintenance, brokerage fees, property taxes, etc. If you're stuck on why a home isn't right for you, read here. A mortgage must be paid until the term date. Whether its a 10 year or 30 year fixed mortgage, depending on interest rates, especially now as the 10-year yield is inching towards 1.25%, it's better to refinance in a lower interest rate environment and pay off your debt as well since the Fed will be inching interest rates up following the 10-year eventually to stabilize the economy, but I believe that won't be until after Covid and when businesses like luxury, air travel, and tourism will get bashed.
Bad debt: I guess the contrary must be true. There is no way to profit from the loan in the future, therefore it is just money down the drain.
-Luxury goods
-Credit card debt
-Fancy junk
-Vacations, weddings, parties, etc.
Credit cards and school loans are the two largest risks to the financial security of those under 50 since banks love to compound interest to affect your credit score and make it tougher to take out more loans, not that you should with that much debt anyhow. With a low credit score and heavy debt, banks take advantage of you because you need aid to pay more for their services. Paying back debt is the challenge for most.
Choose Not Chosen
As a financial literacy advocate and blogger, I prefer not to brag, but I will now. I know what to buy and what to avoid. My parents educated me to live a frugal, minimalist stealth wealth lifestyle by choice, not because we had to.
That's the lesson.
The poorest person who shows off with bling is trying to seem rich.
Rich people know garbage is a bad investment. Investing in education is one of the best long-term investments. With information, you can do anything.
Good with money shun some items out of respect and appreciation for what they have.
Less is more.
Instead of copying the Joneses, use what you have. They may look cheerful and stylish in their 20k ft home, yet they may be as broke as OJ Simpson in his 20-bedroom mansion.
Let's look at what appears good to follow and maintain your wealth.
#1: Quality comes before quantity
Being frugal doesn't entail being cheap and cruel. Rich individuals care about relationships and treating others correctly, not impressing them. You don't have to be rich to be good with money, although most are since they don't live the fantasy lifestyle.
Underspending is appreciating what you have.
Many people believe organic food is the same as washing chemical-laden produce. Hopefully. Organic, vegan, fresh vegetables from upstate may be more expensive in the short term, but they will help you live longer and save you money in the long run.
Consider. You'll save thousands a month eating McDonalds 3x a day instead of fresh seafood, veggies, and organic fruit, but your life will be shortened. If you want to save money and die early, go ahead, but I assume we all want to break the world record for longest person living and would rather spend less. Plus, elderly people get tax breaks, medicare, pensions, 401ks, etc. You're living for free, therefore eating fast food forever is a terrible decision.
With a few longer years, you may make hundreds or millions more in the stock market, spend more time with family, and just live.
Folks, health is wealth.
Consider the future benefit, not simply the cash sign. Cheapness is useless.
Same with stuff. Don't stock your closet with fast-fashion you can't wear for years. Buying inexpensive goods that will fail tomorrow is stupid.
Investing isn't only in stocks. You're living. Consume less.
#2: If you cannot afford it twice, you cannot afford it once
I learned this from my dad in 6th grade. I've been lucky to travel, experience things, go to a great university, and conduct many experiments that others without a stable, decent lifestyle can afford.
I didn't live this way because of my parents' paycheck or financial knowledge.
Saving and choosing caused it.
I always bring cash when I shop. I ditch Apple Pay and credit cards since I can spend all I want on even if my account bounces.
Banks are nasty. When you lose it, they profit.
Cash hinders banks' profits. Carrying a big, hefty wallet with cash is lame and annoying, but it's the best method to only spend what you need. Not for vacation, but for tiny daily expenses.
Physical currency lets you know how much you have for lunch or a taxi.
It's physical, thus losing it prevents debt.
If you can't afford it, it will harm more than help.
#3: You really can purchase happiness with money.
If used correctly, yes.
Happiness and satisfaction differ.
It won't bring you fulfillment because you must work hard on your own to help others, but you can travel and meet individuals you wouldn't otherwise meet.
You can meet your future co-worker or strike a deal while waiting an hour in first class for takeoff, or you can meet renowned people at a networking brunch.
Seen a pattern here?
Your time and money are best spent on connections. Not automobiles or firearms. That’s just stuff. It doesn’t make you a better person.
Be different if you've earned less. Instead of trying to win the lotto or become an NFL star for your first big salary, network online for free.
Be resourceful. Sign up for LinkedIn, post regularly, and leave unengaged posts up because that shows power.
Consistency is beneficial.
I did that for a few months and met amazing people who helped me get jobs. Money doesn't create jobs, it creates opportunities.
Resist social media and scammers that peddle false hopes.
Choose wisely.
#4: Avoid gushing over titles and purchasing trash.
As Insider’s Hillary Hoffower reports, “Showing off wealth is no longer the way to signify having wealth. In the US particularly, the top 1% have been spending less on material goods since 2007.”
I checked my closet. No brand comes to mind. I've never worn a brand's logo and rotate 6 white shirts daily. I have my priorities and don't waste money or effort on clothing that won't fit me in a year.
Unless it's your full-time work, clothing shouldn't be part of our mornings.
Lifestyle of stealth wealth. You're so fulfilled that seeming homeless won't hurt your self-esteem.
That's self-assurance.
Extroverts aren't required.
That's irrelevant.
Showing off won't win you friends.
They'll like your personality.
#5: Time is the most valuable commodity.
Being rich doesn't entail working 24/7 M-F.
They work when they are ready to work.
Waking up at 5 a.m. won't make you a millionaire, but it will inculcate diligence and tenacity in you.
You have a busy day yet want to exercise. You can skip the workout or wake up at 4am instead of 6am to do it.
Emotion-driven lazy bums stay in bed.
Those that are accountable keep their promises because they know breaking one will destroy their week.
Since 7th grade, I've worked out at 5am for myself, not to impress others. It gives me greater energy to contribute to others, especially on weekends and holidays.
It's a habit that I have in my life.
Find something that you take seriously and makes you a better person.
As someone who is close to becoming a millionaire and has encountered them throughout my life, I can share with you a few important differences that have shaped who we are as a society based on the weekends:
-Read
-Sleep
-Best time to work with no distractions
-Eat together
-Take walks and be in nature
-Gratitude
-Major family time
-Plan out weeks
-Go grocery shopping because health = wealth
#6. Perspective is Important
Timing the markets will slow down your career. Professors preach scarcity, not abundance. Why should school teach success? They give us bad advice.
If you trust in abundance and luck by attempting and experimenting, growth will come effortlessly. Passion isn't a term that just appears. Mistakes and fresh people help. You can get money. If you don't think it's worth it, you won't.
You don’t have to be wealthy to be good at money, but most are for these reasons. Rich is a mindset, wealth is power. Prioritize your resources. Invest in yourself, knowing the toughest part is starting.
Thanks for reading!

Glorin Santhosh
3 years ago
Start organizing your ideas by using The Second Brain.
Building A Second Brain helps us remember connections, ideas, inspirations, and insights. Using contemporary technologies and networks increases our intelligence.
This approach makes and preserves concepts. It's a straightforward, practical way to construct a second brain—a remote, centralized digital store for your knowledge and its sources.
How to build ‘The Second Brain’
Have you forgotten any brilliant ideas? What insights have you ignored?
We're pressured to read, listen, and watch informative content. Where did the data go? What happened?
Our brains can store few thoughts at once. Our brains aren't idea banks.
Building a Second Brain helps us remember thoughts, connections, and insights. Using digital technologies and networks expands our minds.
Ten Rules for Creating a Second Brain
1. Creative Stealing
Instead of starting from scratch, integrate other people's ideas with your own.
This way, you won't waste hours starting from scratch and can focus on achieving your goals.
Users of Notion can utilize and customize each other's templates.
2. The Habit of Capture
We must record every idea, concept, or piece of information that catches our attention since our minds are fragile.
When reading a book, listening to a podcast, or engaging in any other topic-related activity, save and use anything that resonates with you.
3. Recycle Your Ideas
Reusing our own ideas across projects might be advantageous since it helps us tie new information to what we already know and avoids us from starting a project with no ideas.
4. Projects Outside of Category
Instead of saving an idea in a folder, group it with documents for a project or activity.
If you want to be more productive, gather suggestions.
5. Burns Slowly
Even if you could finish a job, work, or activity if you focused on it, you shouldn't.
You'll get tired and can't advance many projects. It's easier to divide your routine into daily tasks.
Few hours of daily study is more productive and healthier than entire nights.
6. Begin with a surplus
Instead of starting with a blank sheet when tackling a new subject, utilise previous articles and research.
You may have read or saved related material.
7. Intermediate Packets
A bunch of essay facts.
You can utilize it as a document's section or paragraph for different tasks.
Memorize useful information so you can use it later.
8. You only know what you make
We can see, hear, and read about anything.
What matters is what we do with the information, whether that's summarizing it or writing about it.
9. Make it simpler for yourself in the future.
Create documents or files that your future self can easily understand. Use your own words, mind maps, or explanations.
10. Keep your thoughts flowing.
If you don't employ the knowledge in your second brain, it's useless.
Few people exercise despite knowing its benefits.
Conclusion:
You may continually move your activities and goals closer to completion by organizing and applying your information in a way that is results-focused.
Profit from the information economy's explosive growth by turning your specialized knowledge into cash.
Make up original patterns and linkages between topics.
You may reduce stress and information overload by appropriately curating and managing your personal information stream.
Learn how to apply your significant experience and specific knowledge to a new job, business, or profession.
Without having to adhere to tight, time-consuming constraints, accumulate a body of relevant knowledge and concepts over time.
Take advantage of all the learning materials that are at your disposal, including podcasts, online courses, webinars, books, and articles.
You might also like

Victoria Kurichenko
3 years ago
My Blog Is in Google's Top 10—Here's How to Compete
"Competition" is beautiful and hateful.
Some people bury their dreams because they are afraid of competition. Others challenge themselves, shaping our world.
Competition is normal.
It spurs innovation and progress.
I wish more people agreed.
As a marketer, content writer, and solopreneur, my readers often ask:
"I want to create a niche website, but I have no ideas. Everything's done"
"Is a website worthwhile?"
I can't count how many times I said, "Yes, it makes sense, and you can succeed in a competitive market."
I encourage and share examples, but it's not enough to overcome competition anxiety.
I launched an SEO writing website for content creators a year ago, knowing it wouldn't beat Ahrefs, Semrush, Backlinko, etc.
Not needed.
Many of my website's pages rank highly on Google.
Everyone can eat the pie.
In a competitive niche, I took a different approach.
Look farther
When chatting with bloggers that want a website, I discovered something fascinating.
They want to launch a website but have no ideas. As a next step, they start listing the interests they believe they should work on, like wellness, lifestyle, investments, etc. I could keep going.
Too many generalists who claim to know everything confuse many.
Generalists aren't trusted.
We want someone to fix our problems immediately.
I don't think broad-spectrum experts are undervalued. People have many demands that go beyond generalists' work. Narrow-niche experts can help.
I've done SEO for three years. I learned from experts and courses. I couldn't find a comprehensive SEO writing resource.
I read tons of articles before realizing that wasn't it. I took courses that covered SEO basics eventually.
I had a demand for learning SEO writing, but there was no solution on the market. My website fills this micro-niche.
Have you ever had trouble online?
Professional courses too general, boring, etc.?
You've bought off-topic books, right?
You're not alone.
Niche ideas!
Big players often disregard new opportunities. Too small. Individual content creators can succeed here.
In a competitive market:
Never choose wide subjects
Think about issues you can relate to and have direct experience with.
Be a consumer to discover both the positive and negative aspects of a good or service.
Merchandise your annoyances.
Consider ways to transform your frustrations into opportunities.
The right niche is half-success. Here is what else I did to hit the Google front page with my website.
An innovative method for choosing subjects
Why publish on social media and websites?
Want likes, shares, followers, or fame?
Some people do it for fun. No judgment.
I bet you want more.
You want to make decent money from blogging.
Writing about random topics, even if they are related to your niche, won’t help you attract an audience from organic search. I'm a marketer and writer.
I worked at companies with dead blogs because they posted for themselves, not readers. They did not follow SEO writing rules; that’s why most of their content flopped.
I learned these hard lessons and grew my website from 0 to 3,000+ visitors per month while working on it a few hours a week only. Evidence:
I choose website topics using these criteria:
- Business potential. The information should benefit my audience and generate revenue. There would be no use in having it otherwise.
My topics should help me:
Attract organic search traffic with my "fluff-free" content -> Subscribers > SEO ebook sales.
Simple and effective.
- traffic on search engines. The number of monthly searches reveals how popular my topic is all across the world. If I find that no one is interested in my suggested topic, I don't write a blog article.
- Competition. Every search term is up against rivals. Some are more popular (thus competitive) since more websites target them in organic search. A new website won't score highly for keywords that are too competitive. On the other side, keywords with moderate to light competition can help you rank higher on Google more quickly.
- Search purpose. The "why" underlying users' search requests is revealed. I analyze search intent to understand what users need when they plug various queries in the search bar and what content can perfectly meet their needs.
My specialty website produces money, ranks well, and attracts the target audience because I handpick high-traffic themes.
Following these guidelines, even a new website can stand out.
I wrote a 50-page SEO writing guide where I detailed topic selection and share my front-page Google strategy.
My guide can help you run a successful niche website.
In summary
You're not late to the niche-website party.
The Internet offers many untapped opportunities.
We need new solutions and are willing to listen.
There are unexplored niches in any topic.
Don't fight giants. They have their piece of the pie. They might overlook new opportunities while trying to keep that piece of the pie. You should act now.

Will Leitch
3 years ago
Don't treat Elon Musk like Trump.
He’s not the President. Stop treating him like one.
Elon Musk tweeted from Qatar, where he was watching the World Cup Final with Jared Kushner.
Musk's subsequent Tweets were as normal, basic, and bland as anyone's from a World Cup Final: It's depressing to see the world's richest man looking at his phone during a grand ceremony. Rich guy goes to rich guy event didn't seem important.
Before Musk posted his should-I-step-down-at-Twitter poll, CNN ran a long segment asking if it was hypocritical for him to reveal his real-time location after defending his (very dumb) suspension of several journalists for (supposedly) revealing his assassination coordinates by linking to a site that tracks Musks private jet. It was hard to ignore CNN's hypocrisy: It covered Musk as Twitter CEO like President Trump. EVERY TRUMP STORY WAS BASED ON HIM SAYING X, THEN DOING Y. Trump would do something horrific, lie about it, then pretend it was fine, then condemn a political rival who did the same thing, be called hypocritical, and so on. It lasted four years. Exhausting.
It made sense because Trump was the President of the United States. The press's main purpose is to relentlessly cover and question the president.
It's strange to say this out. Twitter isn't America. Elon Musk isn't a president. He maintains a money-losing social media service to harass and mock people he doesn't like. Treating Musk like Trump, as if he should be held accountable like Trump, shows a startling lack of perspective. Some journalists treat Twitter like a country.
The compulsive, desperate way many journalists utilize the site suggests as much. Twitter isn't the town square, despite popular belief. It's a place for obsessives to meet and converse. Journalists say they're breaking news. Their careers depend on it. They can argue it's a public service. Nope. It's a place lonely people go to speak all day. Twitter. So do journalists, Trump, and Musk. Acting as if it has a greater purpose, as if it's impossible to break news without it, or as if the republic is in peril is ludicrous. Only 23% of Americans are on Twitter, while 25% account for 97% of Tweets. I'd think a large portion of that 25% are journalists (or attention addicts) chatting to other journalists. Their loudness makes Twitter seem more important than it is. Nope. It's another stupid website. They were there before Twitter; they will be there after Twitter. It’s just a website. We can all get off it if we want. Most of us aren’t even on it in the first place.
Musk is a website-owner. No world leader. He's not as accountable as Trump was. Musk is cable news's primary character now that Trump isn't (at least for now). Becoming a TV news anchor isn't as significant as being president. Elon Musk isn't as important as we all pretend, and Twitter isn't even close. Twitter is a dumb website, Elon Musk is a rich guy going through a midlife crisis, and cable news is lazy because its leaders thought the entire world was on Twitter and are now freaking out that their playground is being disturbed.
I’ve said before that you need to leave Twitter, now. But even if you’re still on it, we need to stop pretending it matters more than it does. It’s a site for lonely attention addicts, from the man who runs it to the journalists who can’t let go of it. It’s not a town square. It’s not a country. It’s not even a successful website. Let’s stop pretending any of it’s real. It’s not.

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
