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
3 years ago
👏 Awesome post! When is part 3 coming?
Trent Lapinski
3 years ago
Very complex topic, great explanation
More on Web3 & Crypto
Chris
2 years ago
What the World's Most Intelligent Investor Recently Said About Crypto
Cryptoshit. This thing is crazy to buy.
Charlie Munger is revered and powerful in finance.
Munger, vice chairman of Berkshire Hathaway, is noted for his wit, no-nonsense attitude to investment, and ability to spot promising firms and markets.
Munger's crypto views have upset some despite his reputation as a straight shooter.
“There’s only one correct answer for intelligent people, just totally avoid all the people that are promoting it.” — Charlie Munger
The Munger Interview on CNBC (4:48 secs)
This Monday, CNBC co-anchor Rebecca Quick interviewed Munger and brought up his 2007 statement, "I'm not allowed to have an opinion on this subject until I can present the arguments against my viewpoint better than the folks who are supporting it."
Great investing and life advice!
If you can't explain the opposing reasons, you're not informed enough to have an opinion.
In today's world, it's important to grasp both sides of a debate before supporting one.
Rebecca inquired:
Does your Wall Street Journal article on banning cryptocurrency apply? If so, would you like to present the counterarguments?
Mungers reply:
I don't see any viable counterarguments. I think my opponents are idiots, hence there is no sensible argument against my position.
Consider his words.
Do you believe Munger has studied both sides?
He said, "I assume my opponents are idiots, thus there is no sensible argument against my position."
This is worrisome, especially from a guy who once encouraged studying both sides before forming an opinion.
Munger said:
National currencies have benefitted humanity more than almost anything else.
Hang on, I think we located the perpetrator.
Munger thinks crypto will replace currencies.
False.
I doubt he studied cryptocurrencies because the name is deceptive.
He misread a headline as a Dollar destroyer.
Cryptocurrencies are speculations.
Like Tesla, Amazon, Apple, Google, Microsoft, etc.
Crypto won't replace dollars.
In the interview with CNBC, Munger continued:
“I’m not proud of my country for allowing this crap, what I call the cryptoshit. It’s worthless, it’s no good, it’s crazy, it’ll do nothing but harm, it’s anti-social to allow it.” — Charlie Munger
Not entirely inaccurate.
Daily cryptos are established solely to pump and dump regular investors.
Let's get into Munger's crypto aversion.
Rat poison is bitcoin.
Munger famously dubbed Bitcoin rat poison and a speculative bubble that would implode.
Partially.
But the bubble broke. Since 2021, the market has fallen.
Scam currencies and NFTs are being eliminated, which I like.
Whoa.
Why does Munger doubt crypto?
Mungers thinks cryptocurrencies has no intrinsic value.
He worries about crypto fraud and money laundering.
Both are valid issues.
Yet grouping crypto is intellectually dishonest.
Ethereum, Bitcoin, Solana, Chainlink, Flow, and Dogecoin have different purposes and values (not saying they’re all good investments).
Fraudsters who hurt innocents will be punished.
Therefore, complaining is useless.
Why not stop it? Repair rather than complain.
Regrettably, individuals today don't offer solutions.
Blind Areas for Mungers
As with everyone, Mungers' bitcoin views may be impacted by his biases and experiences.
OK.
But Munger has always advocated classic value investing and may be wary of investing in an asset outside his expertise.
Mungers' banking and insurance investments may influence his bitcoin views.
Could a coworker or acquaintance have told him crypto is bad and goes against traditional finance?
Right?
Takeaways
Do you respect Charlie Mungers?
Yes and no, like any investor or individual.
To understand Mungers' bitcoin beliefs, you must be critical.
Mungers is a successful investor, but his views about bitcoin should be considered alongside other viewpoints.
Munger’s success as an investor has made him an influencer in the space.
Influence gives power.
He controls people's thoughts.
Munger's ok. He will always be heard.
I'll do so cautiously.
Alex Bentley
3 years ago
Why Bill Gates thinks Bitcoin, crypto, and NFTs are foolish
Microsoft co-founder Bill Gates assesses digital assets while the bull is caged.
Bill Gates is well-respected.
Reasonably. He co-founded and led Microsoft during its 1980s and 1990s revolution.
After leaving Microsoft, Bill Gates pursued other interests. He and his wife founded one of the world's largest philanthropic organizations, Bill & Melinda Gates Foundation. He also supports immunizations, population control, and other global health programs.
When Gates criticized Bitcoin, cryptocurrencies, and NFTs, it made news.
Bill Gates said at the 58th Munich Security Conference...
“You have an asset class that’s 100% based on some sort of greater fool theory that somebody’s going to pay more for it than I do.”
Gates means digital assets. Like many bitcoin critics, he says digital coins and tokens are speculative.
And he's not alone. Financial experts have dubbed Bitcoin and other digital assets a "bubble" for a decade.
Gates also made fun of Bored Ape Yacht Club and NFTs, saying, "Obviously pricey digital photographs of monkeys will help the world."
Why does Bill Gates dislike digital assets?
According to Gates' latest comments, Bitcoin, cryptos, and NFTs aren't good ways to hold value.
Bill Gates is a better investor than Elon Musk.
“I’m used to asset classes, like a farm where they have output, or like a company where they make products,” Gates said.
The Guardian claimed in April 2021 that Bill and Melinda Gates owned the most U.S. farms. Over 242,000 acres of farmland.
The Gates couple has enough farmland to cover Hong Kong.
Bill Gates is a classic investor. He wants companies with an excellent track record, strong fundamentals, and good management. Or tangible assets like land and property.
Gates prefers the "old economy" over the "new economy"
Gates' criticism of Bitcoin and cryptocurrency ventures isn't surprising. These digital assets lack all of Gates's investing criteria.
Volatile digital assets include Bitcoin. Their costs might change dramatically in a day. Volatility scares risk-averse investors like Gates.
Gates has a stake in the old financial system. As Microsoft's co-founder, Gates helped develop a dominant tech company.
Because of his business, he's one of the world's richest men.
Bill Gates is invested in protecting the current paradigm.
He won't invest in anything that could destroy the global economy.
When Gates criticizes Bitcoin, cryptocurrencies, and NFTs, he's suggesting they're a hoax. These soapbox speeches are one way he protects his interests.
Digital assets aren't a bad investment, though. Many think they're the future.
Changpeng Zhao and Brian Armstrong are two digital asset billionaires. Two crypto exchange CEOs. Binance/Coinbase.
Digital asset revolution won't end soon.
If you disagree with Bill Gates and plan to invest in Bitcoin, cryptocurrencies, or NFTs, do your own research and understand the risks.
But don’t take Bill Gates’ word for it.
He’s just an old rich guy with a lot of farmland.
He has a lot to lose if Bitcoin and other digital assets gain global popularity.
This post is a summary. Read the full article here.
Jayden Levitt
3 years ago
The country of El Salvador's Bitcoin-obsessed president lost $61.6 million.
It’s only a loss if you sell, right?
Nayib Bukele proclaimed himself “the world’s coolest dictator”.
His jokes aren't clear.
El Salvador's 43rd president self-proclaimed “CEO of El Salvador” couldn't be less presidential.
His thin jeans, aviator sunglasses, and baseball caps like a cartel lord.
He's popular, though.
Bukele won 53% of the vote by fighting violent crime and opposition party corruption.
El Salvador's 6.4 million inhabitants are riding the cryptocurrency volatility wave.
They were powerless.
Their autocratic leader, a former Yamaha Motors salesperson and Bitcoin believer, wants to help 70% unbanked locals.
He intended to give the citizens a way to save money and cut the country's $200 million remittance cost.
Transfer and deposit costs.
This makes logical sense when the president’s theatrics don’t blind you.
El Salvador's Bukele revealed plans to make bitcoin legal tender.
Remittances total $5.9 billion (23%) of the country's expenses.
Anything that reduces costs could boost the economy.
The country’s unbanked population is staggering. Here’s the data by % of people who either have a bank account (Blue) or a mobile money account (Black).
According to Bukele, 46% of the population has downloaded the Chivo Bitcoin Wallet.
In 2021, 36% of El Salvadorans had bank accounts.
Large rural countries like Kenya seem to have resolved their unbanked dilemma.
An economy surfaced where village locals would sell, trade and store network minutes and data as a store of value.
Kenyan phone networks realized unbanked people needed a safe way to accumulate wealth and have an emergency fund.
96% of Kenyans utilize M-PESA, which doesn't require a bank account.
The software involves human agents who hang out with cash and a phone.
These people are like ATMs.
You offer them cash to deposit money in your mobile money account or withdraw cash.
In a country with a faulty banking system, cash availability and a safe place to deposit it are important.
William Jack and Tavneet Suri found that M-PESA brought 194,000 Kenyan households out of poverty by making transactions cheaper and creating a safe store of value.
Mobile money, a service that allows monetary value to be stored on a mobile phone and sent to other users via text messages, has been adopted by most Kenyan households. We estimate that access to the Kenyan mobile money system M-PESA increased per capita consumption levels and lifted 194,000 households, or 2% of Kenyan households, out of poverty.
The impacts, which are more pronounced for female-headed households, appear to be driven by changes in financial behaviour — in particular, increased financial resilience and saving. Mobile money has therefore increased the efficiency of the allocation of consumption over time while allowing a more efficient allocation of labour, resulting in a meaningful reduction of poverty in Kenya.
Currently, El Salvador has 2,301 Bitcoin.
At publication, it's worth $44 million. That remains 41% of Bukele's original $105.6 million.
Unknown if the country has sold Bitcoin, but Bukeles keeps purchasing the dip.
It's still falling.
This might be a fantastic move for the impoverished country over the next five years, if they can live economically till Bitcoin's price recovers.
The evidence demonstrates that a store of value pulls individuals out of poverty, but others say Bitcoin is premature.
You may regard it as an aggressive endeavor to front run the next wave of adoption, offering El Salvador a financial upside.
You might also like
Mangu Solutions
3 years ago
Growing a New App to $15K/mo in 6 Months [SaaS Case Study]
Discover How We Used Facebook Ads to Grow a New Mobile App from $0 to $15K MRR in Just 6 Months and Our Strategy to Hit $100K a Month.
Our client introduced a mobile app for Poshmark resellers in December and wanted as many to experience it and subscribe to the monthly plan.
An Error We Committed
We initiated a Facebook ad campaign with a "awareness" goal, not "installs." This sent them to a landing page that linked to the iPhone App Store and Android Play Store. Smart, right?
We got some installs, but we couldn't tell how many came from the ad versus organic/other channels because the objective we chose only reported landing page clicks, not app installs.
We didn't know which interest groups/audiences had the best cost per install (CPI) to optimize and scale our budget.
After spending $700 without adequate data (installs and trials report), we stopped the campaign and worked with our client's app developer to set up app events tracking.
This allowed us to create an installs campaign and track installs, trials, and purchases (in some cases).
Finding a Successful Audience
Once we knew what ad sets brought in what installs at what cost, we began optimizing and testing other interest groups and audiences, growing the profitable low CPI ones and eliminating the high CPI ones.
We did all our audience testing using an ABO campaign (Ad Set Budget Optimization), spending $10 to $30 on each ad set for three days and optimizing afterward. All ad sets under $30 were moved to a CBO campaign (Campaign Budget Optimization).
We let Facebook's AI decide how much to spend on each ad set, usually the one most likely to convert at the lowest cost.
If the CBO campaign maintains a nice CPI, we keep increasing the budget by $50 every few days or duplicating it sometimes in order to double the budget. This is how we've scaled to $400/day profitably.
Finding Successful Creatives
Per campaign, we tested 2-6 images/videos. Same ad copy and CTA. There was no clear winner because some images did better with some interest groups.
The image above with mail packages, for example, got us a cheap CPI of $9.71 from our Goodwill Stores interest group but, a high $48 CPI from our lookalike audience. Once we had statistically significant data, we turned off the high-cost ad.
New marketers who are just discovering A/B testing may assume it's black and white — winner and loser. However, Facebook ads' machine learning and reporting has gotten so sophisticated that it's hard to call a creative a flat-out loser, but rather a 'bad fit' for some audiences, and perfect for others.
You can see how each creative performs across age groups and optimize.
How Many Installs Did It Take Us to Earn $15K Per Month?
Six months after paying $25K, we got 1,940 app installs, 681 free trials, and 522 $30 monthly subscriptions. 522 * $30 gives us $15,660 in monthly recurring revenue (MRR).
Next, what? $100K per month
The conversation above is with the app's owner. We got on a 30-minute call where I shared how I plan to get the app to be making $100K a month like I’ve done for other businesses.
Reverse Engineering $100K
Formula:
For $100K/month, we need 3,334 people to pay $30/month. 522 people pay that. We need 2,812 more paid users.
522 paid users from 1,940 installs is a 27% conversion rate. To hit $100K/month, we need 10,415 more installs. Assuming...
With a $400 daily ad spend, we average 40 installs per day. This means that if everything stays the same, it would take us 260 days (around 9 months) to get to $100K a month (MRR).
Conclusion
You must market your goods to reach your income objective (without waiting forever). Paid ads is the way to go if you hate knocking on doors or irritating friends and family (who aren’t scalable anyways).
You must also test and optimize different angles, audiences, interest groups, and creatives.
Joseph Mavericks
3 years ago
The world's 36th richest man uses a 5-step system to get what he wants.
Ray Dalio's super-effective roadmap
Ray Dalio's $22 billion net worth ranks him 36th globally. From 1975 to 2011, he built the world's most successful hedge fund, never losing more than 4% from 1991 to 2020. (and only doing so during 3 calendar years).
Dalio describes a 5-step process in his best-selling book Principles. It's the playbook he's used to build his hedge fund, beat the markets, and face personal challenges.
This 5-step system is so valuable and well-explained that I didn't edit or change anything; I only added my own insights in the parts I found most relevant and/or relatable as a young entrepreneur. The system's overview:
Have clear goals
Identify and don’t tolerate problems
Diagnose problems to get at their root causes
Design plans that will get you around those problems
Do what is necessary to push through the plans to get results
If you follow these 5 steps in a virtuous loop, you'll almost always see results. Repeat the process for each goal you have.
1. Have clear goals
a) Prioritize: You can have almost anything, but not everything.
I started and never launched dozens of projects for 10 years because I was scattered. I opened a t-shirt store, traded algorithms, sold art on Instagram, painted skateboards, and tinkered with electronics. I decided to try blogging for 6 months to see where it took me. Still going after 3 years.
b) Don’t confuse goals with desires.
A goal inspires you to act. Unreasonable desires prevent you from achieving your goals.
c) Reconcile your goals and desires to decide what you want.
d) Don't confuse success with its trappings.
e) Never dismiss a goal as unattainable.
Always one path is best. Your perception of what's possible depends on what you know now. I never thought I'd make money writing online so quickly, and now I see a whole new horizon of business opportunities I didn't know about before.
f) Expectations create abilities.
Don't limit your abilities. More you strive, the more you'll achieve.
g) Flexibility and self-accountability can almost guarantee success.
Flexible people accept what reality or others teach them. Self-accountability is the ability to recognize your mistakes and be more creative, flexible, and determined.
h) Handling setbacks well is as important as moving forward.
Learn when to minimize losses and when to let go and move on.
2. Don't ignore problems
a) See painful problems as improvement opportunities.
Every problem, painful situation, and challenge is an opportunity. Read The Art of Happiness for more.
b) Don't avoid problems because of harsh realities.
Recognizing your weaknesses isn't the same as giving in. It's the first step in overcoming them.
c) Specify your issues.
There is no "one-size-fits-all" solution.
d) Don’t mistake a cause of a problem with the real problem.
"I can't sleep" is a cause, not a problem. "I'm underperforming" could be a problem.
e) Separate big from small problems.
You have limited time and energy, so focus on the biggest problems.
f) Don't ignore a problem.
Identifying a problem and tolerating it is like not identifying it.
3. Identify problems' root causes
a) Decide "what to do" after assessing "what is."
"A good diagnosis takes 15 to 60 minutes, depending on its accuracy and complexity. [...] Like principles, root causes recur in different situations.
b) Separate proximate and root causes.
"You can only solve problems by removing their root causes, and to do that, you must distinguish symptoms from disease."
c) Knowing someone's (or your own) personality can help you predict their behavior.
4. Design plans that will get you around the problems
a) Retrace your steps.
Analyze your past to determine your future.
b) Consider your problem a machine's output.
Consider how to improve your machine. It's a game then.
c) There are many ways to reach your goals.
Find a solution.
d) Visualize who will do what in your plan like a movie script.
Consider your movie's actors and script's turning points, then act accordingly. The game continues.
e) Document your plan so others can judge your progress.
Accountability boosts success.
f) Know that a good plan doesn't take much time.
The execution is usually the hardest part, but most people either don't have a plan or keep changing it. Don't drive while building the car. Build it first, because it'll be bumpy.
5. Do what is necessary to push through the plans to get results
a) Great planners without execution fail.
Life is won with more than just planning. Similarly, practice without talent beats talent without practice.
b) Work ethic is undervalued.
Hyper-productivity is praised in corporate America, even if it leads nowhere. To get things done, use checklists, fewer emails, and more desk time.
c) Set clear metrics to ensure plan adherence.
I've written about the OKR strategy for organizations with multiple people here. If you're on your own, I recommend the Wheel of Life approach. Both systems start with goals and tasks to achieve them. Then start executing on a realistic timeline.
If you find solutions, weaknesses don't matter.
Everyone's weak. You, me, Gates, Dalio, even Musk. Nobody will be great at all 5 steps of the system because no one can think in all the ways required. Some are good at analyzing and diagnosing but bad at executing. Some are good planners but poor communicators. Others lack self-discipline.
Stay humble and ask for help when needed. Nobody has ever succeeded 100% on their own, without anyone else's help. That's the paradox of individual success: teamwork is the only way to get there.
Most people won't have the skills to execute even the best plan. You can get missing skills in two ways:
Self-taught (time-consuming)
Others' (requires humility) light
On knowing what to do with your life
“Some people have good mental maps and know what to do on their own. Maybe they learned them or were blessed with common sense. They have more answers than others. Others are more humble and open-minded. […] Open-mindedness and mental maps are most powerful.” — Ray Dalio
I've always known what I wanted to do, so I'm lucky. I'm almost 30 and have always had trouble executing. Good thing I never stopped experimenting, but I never committed to anything long-term. I jumped between projects. I decided 3 years ago to stick to one project for at least 6 months and haven't looked back.
Maybe you're good at staying focused and executing, but you don't know what to do. Maybe you have none of these because you haven't found your purpose. Always try new projects and talk to as many people as possible. It will give you inspiration and ideas and set you up for success.
There is almost always a way to achieve a crazy goal or idea.
Enjoy the journey, whichever path you take.
Katrine Tjoelsen
3 years ago
8 Communication Hacks I Use as a Young Employee
Learn these subtle cues to gain influence.
Hate being ignored?
As a 24-year-old, I struggled at work. Attention-getting tips How to avoid being judged by my size, gender, and lack of wrinkles or gray hair?
I've learned seniority hacks. Influence. Within two years as a product manager, I led a team. I'm a Stanford MBA student.
These communication hacks can make you look senior and influential.
1. Slowly speak
We speak quickly because we're afraid of being interrupted.
When I doubt my ideas, I speak quickly. How can we slow down? Jamie Chapman says speaking slowly saps our energy.
Chapman suggests emphasizing certain words and pausing.
2. Interrupted? Stop the stopper
Someone interrupt your speech?
Don't wait. "May I finish?" No pause needed. Stop interrupting. I first tried this in Leadership Laboratory at Stanford. How quickly I gained influence amazed me.
Next time, try “May I finish?” If that’s not enough, try these other tips from Wendy R.S. O’Connor.
3. Context
Others don't always see what's obvious to you.
Through explanation, you help others see the big picture. If a senior knows it, you help them see where your work fits.
4. Don't ask questions in statements
“Your statement lost its effect when you ended it on a high pitch,” a group member told me. Upspeak, it’s called. I do it when I feel uncertain.
Upspeak loses influence and credibility. Unneeded. When unsure, we can say "I think." We can even ask a proper question.
Someone else's boasting is no reason to be dismissive. As leaders and colleagues, we should listen to our colleagues even if they use this speech pattern.
Give your words impact.
5. Signpost structure
Signposts improve clarity by providing structure and transitions.
Communication coach Alexander Lyon explains how to use "first," "second," and "third" He explains classic and summary transitions to help the listener switch topics.
Signs clarify. Clarity matters.
6. Eliminate email fluff
“Fine. When will the report be ready? — Jeff.”
Notice how senior leaders write short, direct emails? I often use formalities like "dear," "hope you're well," and "kind regards"
Formality is (usually) unnecessary.
7. Replace exclamation marks with periods
See how junior an exclamation-filled email looks:
Hi, all!
Hope you’re as excited as I am for tomorrow! We’re celebrating our accomplishments with cake! Join us tomorrow at 2 pm!
See you soon!
Why the exclamation points? Why not just one?
Hi, all.
Hope you’re as excited as I am for tomorrow. We’re celebrating our accomplishments with cake. Join us tomorrow at 2 pm!
See you soon.
8. Take space
"Playing high" means having an open, relaxed body, says Stanford professor and author Deborah Gruenfield.
Crossed legs or looking small? Relax. Get bigger.
