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Vitalik

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

(Edited)

Hackernoon

Hackernoon

4 years ago


👏 Awesome post! When is part 3 coming?

Trent Lapinski

Trent Lapinski

4 years ago

Very complex topic, great explanation

More on Web3 & Crypto

Scott Hickmann

Scott Hickmann

4 years ago

YouTube

This is a YouTube video:

Juxtathinka

Juxtathinka

3 years ago

Why Is Blockchain So Popular?

What is Bitcoin?

The blockchain is a shared, immutable ledger that helps businesses record transactions and track assets. The blockchain can track tangible assets like cars, houses, and land. Tangible assets like intellectual property can also be tracked on the blockchain.

Imagine a blockchain as a distributed database split among computer nodes. A blockchain stores data in blocks. When a block is full, it is closed and linked to the next. As a result, all subsequent information is compiled into a new block that will be added to the chain once it is filled.

The blockchain is designed so that adding a transaction requires consensus. That means a majority of network nodes must approve a transaction. No single authority can control transactions on the blockchain. The network nodes use cryptographic keys and passwords to validate each other's transactions.

Blockchain History

The blockchain was not as popular in 1991 when Stuart Haber and W. Scott Stornetta worked on it. The blocks were designed to prevent tampering with document timestamps. Stuart Haber and W. Scott Stornetta improved their work in 1992 by using Merkle trees to increase efficiency and collect more documents on a single block.

In 2004, he developed Reusable Proof of Work. This system allows users to verify token transfers in real time. Satoshi Nakamoto invented distributed blockchains in 2008. He improved the blockchain design so that new blocks could be added to the chain without being signed by trusted parties.

Satoshi Nakomoto mined the first Bitcoin block in 2009, earning 50 Bitcoins. Then, in 2013, Vitalik Buterin stated that Bitcoin needed a scripting language for building decentralized applications. He then created Ethereum, a new blockchain-based platform for decentralized apps. Since the Ethereum launch in 2015, different blockchain platforms have been launched: from Hyperledger by Linux Foundation, EOS.IO by block.one, IOTA, NEO and Monero dash blockchain. The block chain industry is still growing, and so are the businesses built on them.

Blockchain Components

The Blockchain is made up of many parts:

1. Node: The node is split into two parts: full and partial. The full node has the authority to validate, accept, or reject any transaction. Partial nodes or lightweight nodes only keep the transaction's hash value. It doesn't keep a full copy of the blockchain, so it has limited storage and processing power.

2. Ledger: A public database of information. A ledger can be public, decentralized, or distributed. Anyone on the blockchain can access the public ledger and add data to it. It allows each node to participate in every transaction. The distributed ledger copies the database to all nodes. A group of nodes can verify transactions or add data blocks to the blockchain.

3. Wallet: A blockchain wallet allows users to send, receive, store, and exchange digital assets, as well as monitor and manage their value. Wallets come in two flavors: hardware and software. Online or offline wallets exist. Online or hot wallets are used when online. Without an internet connection, offline wallets like paper and hardware wallets can store private keys and sign transactions. Wallets generally secure transactions with a private key and wallet address.

4. Nonce: A nonce is a short term for a "number used once''. It describes a unique random number. Nonces are frequently generated to modify cryptographic results. A nonce is a number that changes over time and is used to prevent value reuse. To prevent document reproduction, it can be a timestamp. A cryptographic hash function can also use it to vary input. Nonces can be used for authentication, hashing, or even electronic signatures.

5. Hash: A hash is a mathematical function that converts inputs of arbitrary length to outputs of fixed length. That is, regardless of file size, the hash will remain unique. A hash cannot generate input from hashed output, but it can identify a file. Hashes can be used to verify message integrity and authenticate data. Cryptographic hash functions add security to standard hash functions, making it difficult to decipher message contents or track senders.

Blockchain: Pros and Cons

The blockchain provides a trustworthy, secure, and trackable platform for business transactions quickly and affordably. The blockchain reduces paperwork, documentation errors, and the need for third parties to verify transactions.

Blockchain security relies on a system of unaltered transaction records with end-to-end encryption, reducing fraud and unauthorized activity. The blockchain also helps verify the authenticity of items like farm food, medicines, and even employee certification. The ability to control data gives users a level of privacy that no other platform can match.

In the case of Bitcoin, the blockchain can only handle seven transactions per second. Unlike Hyperledger and Visa, which can handle ten thousand transactions per second. Also, each participant node must verify and approve transactions, slowing down exchanges and limiting scalability.

The blockchain requires a lot of energy to run. In addition, the blockchain is not a hugely distributable system and it is destructible. The security of the block chain can be compromised by hackers; it is not completely foolproof. Also, since blockchain entries are immutable, data cannot be removed. The blockchain's high energy consumption and limited scalability reduce its efficiency.

Why Is Blockchain So Popular?
The blockchain is a technology giant. In 2018, 90% of US and European banks began exploring blockchain's potential. In 2021, 24% of companies are expected to invest $5 million to $10 million in blockchain. By the end of 2024, it is expected that corporations will spend $20 billion annually on blockchain technical services.

Blockchain is used in cryptocurrency, medical records storage, identity verification, election voting, security, agriculture, business, and many other fields. The blockchain offers a more secure, decentralized, and less corrupt system of making global payments, which cryptocurrency enthusiasts love. Users who want to save time and energy prefer it because it is faster and less bureaucratic than banking and healthcare systems.

Most organizations have jumped on the blockchain bandwagon, and for good reason: the blockchain industry has never had more potential. The launch of IBM's Blockchain Wire, Paystack, Aza Finance and Bloom are visible proof of the wonders that the blockchain has done. The blockchain's cryptocurrency segment may not be as popular in the future as the blockchain's other segments, as evidenced by the various industries where it is used. The blockchain is here to stay, and it will be discussed for a long time, not just in tech, but in many industries.

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Marco Manoppo

Marco Manoppo

3 years ago

Failures of DCG and Genesis

Don't sleep with your own sister.

70% of lottery winners go broke within five years. You've heard the last one. People who got rich quickly without setbacks and hard work often lose it all. My father said, "Easy money is easily lost," and a wealthy friend who owns a family office said, "The first generation makes it, the second generation spends it, and the third generation blows it."

This is evident. Corrupt politicians in developing countries live lavishly, buying their third wives' fifth Hermès bag and celebrating New Year's at The Brando Resort. A successful businessperson from humble beginnings is more conservative with money. More so if they're atom-based, not bit-based. They value money.

Crypto can "feel" easy. I have nothing against capital market investing. The global financial system is shady, but that's another topic. The problem started when those who took advantage of easy money started affecting other businesses. VCs did minimal due diligence on FTX because they needed deal flow and returns for their LPs. Lenders did minimum diligence and underwrote ludicrous loans to 3AC because they needed revenue.

Alameda (hence FTX) and 3AC made "easy money" Genesis and DCG aren't. Their businesses are more conventional, but they underestimated how "easy money" can hurt them.

Genesis has been the victim of easy money hubris and insolvency, losing $1 billion+ to 3AC and $200M to FTX. We discuss the implications for the broader crypto market.

Here are the quick takeaways:

  • Genesis is one of the largest and most notable crypto lenders and prime brokerage firms.

  • DCG and Genesis have done related party transactions, which can be done right but is a bad practice.

  • Genesis owes DCG $1.5 billion+.

  • If DCG unwinds Grayscale's GBTC, $9-10 billion in BTC will hit the market.

  • DCG will survive Genesis.

What happened?

Let's recap the FTX shenanigan from two weeks ago. Shenanigans! Delphi's tweet sums up the craziness. Genesis has $175M in FTX.

Cred's timeline: I hate bad crisis management. Yes, admitting their balance sheet hole right away might've sparked more panic, and there's no easy way to convey your trouble, but no one ever learns.

By November 23, rumors circulated online that the problem could affect Genesis' parent company, DCG. To address this, Barry Silbert, Founder, and CEO of DCG released a statement to shareholders.

  • A few things are confirmed thanks to this statement.

  • DCG owes $1.5 billion+ to Genesis.

  • $500M is due in 6 months, and the rest is due in 2032 (yes, that’s not a typo).

  • Unless Barry raises new cash, his last-ditch efforts to repay the money will likely push the crypto market lower.

  • Half a year of GBTC fees is approximately $100M.

  • They can pay $500M with GBTC.

  • With profits, sell another port.

Genesis has hired a restructuring adviser, indicating it is in trouble.

Rehypothecation

Every crypto problem in the past year seems to be rehypothecation between related parties, excessive leverage, hubris, and the removal of the money printer. The Bankless guys provided a chart showing 2021 crypto yield.

In June 2022, @DataFinnovation published a great investigation about 3AC and DCG. Here's a summary.

  • 3AC borrowed BTC from Genesis and pledged it to create Grayscale's GBTC shares.

  • 3AC uses GBTC to borrow more money from Genesis.

  • This lets 3AC leverage their capital.

  • 3AC's strategy made sense because GBTC had a premium, creating "free money."

  • GBTC's discount and LUNA's implosion caused problems.

  • 3AC lost its loan money in LUNA.

  • Margin called on 3ACs' GBTC collateral.

  • DCG bought GBTC to avoid a systemic collapse and a larger discount.

  • Genesis lost too much money because 3AC can't pay back its loan. DCG "saved" Genesis, but the FTX collapse hurt Genesis further, forcing DCG and Genesis to seek external funding.

bruh…

Learning Experience

Co-borrowing. Unnecessary rehypothecation. Extra space. Governance disaster. Greed, hubris. Crypto has repeatedly shown it can recreate traditional financial system disasters quickly. Working in crypto is one of the best ways to learn crazy financial tricks people will do for a quick buck much faster than if you dabble in traditional finance.

Moving Forward

I think the crypto industry needs to consider its future. This is especially true for professionals. I'm not trying to scare you. In 2018 and 2020, I had doubts. No doubts now. Detailing the crypto industry's potential outcomes helped me gain certainty and confidence in its future. This includes VCs' benefits and talking points during the bull market, as well as what would happen if government regulations became hostile, etc. Even if that happens, I'm certain. This is permanent. I may write a post about that soon.

Sincerely,

M.

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Tim Denning

Tim Denning

3 years ago

I Posted Six Times a Day for 210 Days on Twitter. Here's What Happened.

I'd spend hours composing articles only to find out they were useless. Twitter solved the problem.

Photo by Humphrey Muleba on Unsplash

Twitter is wrinkled, say critics.

Nope. Writing is different. It won't make sense until you write there.

Twitter is resurgent. People are reading again. 15-second TikToks overloaded our senses.

After nuking my 20,000-follower Twitter account and starting again, I wrote every day for 210 days.

I'll explain.

I came across the strange world of microblogging.

Traditional web writing is filler-heavy.

On Twitter, you must be brief. I played Wordle.

Twitter Threads are the most popular writing format. Like a blog post. It reminds me of the famous broetry posts on LinkedIn a few years ago.

Image Credit: Josh Fetcher via LinkedIn

Threads combine tweets into an article.

  • Sharp, concise sentences

  • No regard for grammar

  • As important as the information is how the text looks.

Twitter Threads are like Michael Angelo's David monument. He chipped away at an enormous piece of marble until a man with a big willy appeared.

That's Twitter Threads.

I tried to remove unnecessary layers from several of my Wordpress blog posts. Then I realized something.

Tweeting from scratch is easier and more entertaining. It's quicker and makes you think more concisely.

Superpower: saying much with little words. My long-form writing has improved. My article sentences resemble tweets.

You never know what will happen.

Twitter's subcultures are odd. Best-performing tweets are strange.

Unusual trend: working alone and without telling anyone. It's a rebellion against Instagram influencers who share their every moment.

Early on, random thoughts worked:

My friend’s wife is Ukrainian. Her family are trapped in the warzone. He is devastated. And here I was complaining about my broken garage door. War puts everything in perspective. Today is a day to be grateful for peace.

Documenting what's happening triggers writing. It's not about viral tweets. Helping others matters.

There are numerous anonymous users.

Twitter uses pseudonyms.

You don't matter. On sites like LinkedIn, you must use your real name. Welcome to the Cyberpunk metaverse of Twitter :)

One daily piece of writing is a powerful habit.

Habits build creator careers. Read that again.

Twitter is an easy habit to pick up. If you can't tweet in one sentence, something's wrong. Easy-peasy-japanese.

Not what I tweeted, but my constancy, made the difference.

Daily writing is challenging, especially if your supervisor is on your back. Twitter encourages writing.

Tweets evolved as the foundation of all other material.

During my experiment, I enjoyed Twitter's speed.

Tweets get immediate responses, comments, and feedback. My popular tweets become newspaper headlines. I've also written essays from tweet discussions.

Sometimes the tweet and article were clear. Twitter sometimes helped me overcome writer's block.

I used to spend hours composing big things that had little real-world use.

Twitter helped me. No guessing. Data guides my coverage and validates concepts.

Test ideas on Twitter.

It took some time for my email list to grow.

Subscribers are a writer's lifeblood.

Without them, you're broke and homeless when Mark Zuckerberg tweaks the algorithms for ad dollars. Twitter has three ways to obtain email subscribers:

1. Add a link to your bio.

Twitter allows bio links (LinkedIn now does too). My eBook's landing page is linked. I collect emails there.

2. Start an online newsletter.

Twitter bought newsletter app Revue. They promote what they own.

I just established up a Revue email newsletter. I imported them weekly into my ConvertKit email list.

3. Create Twitter threads and include a link to your email list in the final tweet.

Write Twitter Threads and link the last tweet to your email list (example below).

Initial email subscribers were modest.

Numbers are growing. Twitter provides 25% of my new email subscribers. Some days, 50 people join.

Without them, my writing career is over. I'd be back at a 9-5 job begging for time off to spend with my newborn daughter. Nope.

Collect email addresses or die trying.

As insurance against unsubscribes and Zucks, use a second email list or Discord community.

What I still need to do

Twitter's fun. I'm wiser. I need to enable auto-replies and auto-DMs (direct messages).

This adds another way to attract subscribers. I schedule tweets with Tweet Hunter.

It’s best to go slow. People assume you're an internet marketer if you spam them with click requests.

A human internet marketer is preferable to a robot. My opinion.

210 days on Twitter taught me that. I plan to use the platform until I'm a grandfather unless Elon ruins it.

Athirah Syamimi

Athirah Syamimi

3 years ago

Here's How I Built A Business Offering Unlimited Design Services in Just One Weekend.

Weekend project: limitless design service. It was fun to see whether I could start a business quickly.

I use no-code apps to save time and resources.

TL;DR I started a business utilizing EditorX for my website, Notion for client project management, and a few favors to finish my portfolio.

First step: research (Day 1)

I got this concept from a Kimp Instagram ad. The Minimalist Hustler Daily newsletter mentioned a similar and cheaper service (Graphically).

I Googled other unlimited design companies. Many provide different costs and services. Some supplied solely graphic design, web development, or copywriting.

Step 2: Brainstorming (Day 1)

I did something simple.

  • What benefits and services to provide

  • Price to charge

Since it's a one-person performance (for now), I'm focusing on graphic design. I can charge less.

So I don't overwhelm myself and can accommodate budget-conscious clientele.

Step 3: Construction (Day 1 & 2)

This project includes a management tool, a website, and a team procedure.

I built a project management tool and flow first. Once I had the flow and a Notion board, I tested it with design volunteers. They fake-designed while I built the website.

Tool for Project Management

I modified a Notion template. My goal is to keep clients and designers happy.

Screenshot of project management board in Notion

Team Approach

My sister, my partner, and I kept this business lean. I tweaked the Notion board to make the process smooth. By the end of Sunday, I’d say it’s perfect!

Website

I created the website after they finished the fake design demands. EditorX's drag-and-drop builder attracted me. I didn't need to learn code, and there are templates.

I used a template wireframe.

This project's hardest aspect is developing the site. It's my first time using EditorX and I'm no developer.

People answer all your inquiries in a large community forum.

As a first-time user developing a site in two days, I think I performed OK. Here's the site for feedback.

Screenshot of deuxcreators.com homepage

4th step: testing (Day 2)

Testing is frustrating because it works or doesn't. My testing day was split in two.

  • testing the workflow from payment to onboarding to the website

  • the demand being tested

It's working so far. If someone gets the trial, they can request design work.

I've gotten a couple of inquiries about demand. I’ll be working with them as a start.

Completion

Finally! I built my side project in one weekend. It's too early to tell if this is successful. I liked that I didn't squander months of resources testing out an idea.

Tim Denning

Tim Denning

3 years ago

Read These Books on Personal Finance to Boost Your Net Worth

And retire sooner.

Photo by Karlie Mitchell on Unsplash

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:

Image Credit-RayDalio/OptimizedPortfolio

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