How We Just Raised $6M At An $80M Valuation From 100+ Investors Using A Link (Without Pitching)
Lawtrades nearly failed three years ago.
We couldn't raise Series A or enthusiasm from VCs.
We raised $6M (at a $80M valuation) from 100 customers and investors using a link and no pitching.
Step-by-step:
We refocused our business first.
Lawtrades raised $3.7M while Atrium raised $75M. By comparison, we seemed unimportant.
We had to close the company or try something new.
As I've written previously, a pivot saved us. Our initial focus on SMBs attracted many unprofitable customers. SMBs needed one-off legal services, meaning low fees and high turnover.
Tech startups were different. Their General Councels (GCs) needed near-daily support, resulting in higher fees and lower churn than SMBs.
We stopped unprofitable customers and focused on power users. To avoid dilution, we borrowed against receivables. We scaled our revenue 10x, from $70k/mo to $700k/mo.
Then, we reconsidered fundraising (and do it differently)
This time was different. Lawtrades was cash flow positive for most of last year, so we could dictate our own terms. VCs were still wary of legaltech after Atrium's shutdown (though they were thinking about the space).
We neither wanted to rely on VCs nor dilute more than 10% equity. So we didn't compete for in-person pitch meetings.
AngelList Roll-Up Vehicle (RUV). Up to 250 accredited investors can invest in a single RUV. First, we emailed customers the RUV. Why? Because I wanted to help the platform's users.
Imagine if Uber or Airbnb let all drivers or Superhosts invest in an RUV. Humans make the platform, theirs and ours. Giving people a chance to invest increases their loyalty.
We expanded after initial interest.
We created a Journey link, containing everything that would normally go in an investor pitch:
- Slides
- Trailer (from me)
- Testimonials
- Product demo
- Financials
We could also link to our AngelList RUV and send the pitch to an unlimited number of people. Instead of 1:1, we had 1:10,000 pitches-to-investors.
We posted Journey's link in RUV Alliance Discord. 600 accredited investors noticed it immediately. Within days, we raised $250,000 from customers-turned-investors.
Stonks, which live-streamed our pitch to thousands of viewers, was interested in our grassroots enthusiasm. We got $1.4M from people I've never met.
These updates on Pump generated more interest. Facebook, Uber, Netflix, and Robinhood executives all wanted to invest. Sahil Lavingia, who had rejected us, gave us $100k.
We closed the round with public support.
Without a single pitch meeting, we'd raised $2.3M. It was a result of natural enthusiasm: taking care of the people who made us who we are, letting them move first, and leveraging their enthusiasm with VCs, who were interested.
We used network effects to raise $3.7M from a founder-turned-VC, bringing the total to $6M at a $80M valuation (which, by the way, I set myself).
What flipping the fundraising script allowed us to do:
We started with private investors instead of 2–3 VCs to show VCs what we were worth. This gave Lawtrades the ability to:
- Without meetings, share our vision. Many people saw our Journey link. I ended up taking meetings with people who planned to contribute $50k+, but still, the ratio of views-to-meetings was outrageously good for us.
- Leverage ourselves. Instead of us selling ourselves to VCs, they did. Some people with large checks or late arrivals were turned away.
- Maintain voting power. No board seats were lost.
- Utilize viral network effects. People-powered.
- Preemptively halt churn by turning our users into owners. People are more loyal and respectful to things they own. Our users make us who we are — no matter how good our tech is, we need human beings to use it. They deserve to be owners.
I don't blame founders for being hesitant about this approach. Pump and RUVs are new and scary. But it won’t be that way for long. Our approach redistributed some of the power that normally lies entirely with VCs, putting it into our hands and our network’s hands.
This is the future — another way power is shifting from centralized to decentralized.
More on Entrepreneurship/Creators

Alana Rister, Ph.D.
3 years ago
Don't rely on lessons you learned with a small audience.
My growth-killing mistake
When you initially start developing your audience, you need guidance.
What does my audience like? What do they not like? How can I grow more?
When I started writing two years ago, I inquired daily. Taking cues from your audience to develop more valuable content is a good concept, but it's simple to let them destroy your growth.
A small audience doesn't represent the full picture.
When I had fewer than 100 YouTube subscribers, I tried several video styles and topics. I looked to my audience for what to preserve and what to change.
If my views, click-through rate, or average view % dropped, that topic or style was awful. Avoiding that style helped me grow.
Vlogs, talking head videos on writing, and long-form tutorials didn't fare well.
Since I was small, I've limited the types of films I make. I have decided to make my own videos.
Surprisingly, the videos I avoided making meet or exceed my views, CTR, and audience retention.
A limited audience can't tell you what your tribe wants. Therefore, limiting your innovation will prohibit you from reaching the right audience. Finding them may take longer.
Large Creators Experience The Same Issue
In the last two years, I've heard Vanessa Lau and Cathrin Manning say they felt pigeonholed into generating videos they didn't want to do.
Why does this happen over and over again?
Once you have a popular piece of content, your audience will grow. So when you publish inconsistent material, fewer of your new audience will view it. You interpret the drop in views as a sign that your audience doesn't want the content, so you stop making it.
Repeat this procedure a few times, and you'll create stuff you're not passionate about because you're frightened to publish it.
How to Manage Your Creativity and Audience Development
I'm not recommending you generate random content.
Instead of feeling trapped by your audience, you can cultivate a diverse audience.
Create quality material on a range of topics and styles as you improve. Be creative until you get 100 followers. Look for comments on how to improve your article.
If you observe trends in the types of content that expand your audience, focus 50-75% of your material on those trends. Allow yourself to develop 25% non-performing material.
This method can help you expand your audience faster with your primary trends and like all your stuff. Slowly, people will find 25% of your material, which will boost its performance.
How to Expand Your Audience Without Having More Limited Content
Follow these techniques to build your audience without feeling confined.
Don't think that you need restrict yourself to what your limited audience prefers.
Don't let the poor performance of your desired material demotivate you.
You shouldn't restrict the type of content you publish or the themes you cover when you have less than 100 followers.
When your audience expands, save 25% of your content for your personal interests, regardless of how well it does.

Greg Lim
3 years ago
How I made $160,000 from non-fiction books
I've sold over 40,000 non-fiction books on Amazon and made over $160,000 in six years while writing on the side.
I have a full-time job and three young sons; I can't spend 40 hours a week writing. This article describes my journey.
I write mainly tech books:
Thanks to my readers, many wrote positive evaluations. Several are bestsellers.
A few have been adopted by universities as textbooks:
My books' passive income allows me more time with my family.
Knowing I could quit my job and write full time gave me more confidence. And I find purpose in my work (i am in christian ministry).
I'm always eager to write. When work is a dread or something bad happens, writing gives me energy. Writing isn't scary. In fact, I can’t stop myself from writing!
Writing has also established my tech authority. Universities use my books, as I've said. Traditional publishers have asked me to write books.
These mindsets helped me become a successful nonfiction author:
1. You don’t have to be an Authority
Yes, I have computer science experience. But I'm no expert on my topics. Before authoring "Beginning Node.js, Express & MongoDB," my most profitable book, I had no experience with those topics. Node was a new server-side technology for me. Would that stop me from writing a book? It can. I liked learning a new technology. So I read the top three Node books, took the top online courses, and put them into my own book (which makes me know more than 90 percent of people already).
I didn't have to worry about using too much jargon because I was learning as I wrote. An expert forgets a beginner's hardship.
"The fellow learner can aid more than the master since he knows less," says C.S. Lewis. The problem he must explain is recent. The expert has forgotten.”
2. Solve a micro-problem (Niching down)
I didn't set out to write a definitive handbook. I found a market with several challenges and wrote one book. Ex:
- Instead of web development, what about web development using Angular?
- Instead of Blockchain, what about Blockchain using Solidity and React?
- Instead of cooking recipes, how about a recipe for a specific kind of diet?
- Instead of Learning math, what about Learning Singapore Math?
3. Piggy Backing Trends
The above topics may still be a competitive market. E.g. Angular, React. To stand out, include the latest technologies or trends in your book. Learn iOS 15 instead of iOS programming. Instead of personal finance, what about personal finance with NFTs.
Even though you're a newbie author, your topic is well-known.
4. Publish short books
My books are known for being direct. Many people like this:
Your reader will appreciate you cutting out the fluff and getting to the good stuff. A reader can finish and review your book.
Second, short books are easier to write. Instead of creating a 500-page book for $50 (which few will buy), write a 100-page book that answers a subset of the problem and sell it for less. (You make less, but that's another subject). At least it got published instead of languishing. Less time spent creating a book means less time wasted if it fails. Write a small-bets book portfolio like Daniel Vassallo!
Third, it's $2.99-$9.99 on Amazon (gets 70 percent royalties for ebooks). Anything less receives 35% royalties. $9.99 books have 20,000–30,000 words. If you write more and charge more over $9.99, you get 35% royalties. Why not make it a $9.99 book?
(This is the ebook version.) Paperbacks cost more. Higher royalties allow for higher prices.
5. Validate book idea
Amazon will tell you if your book concept, title, and related phrases are popular. See? Check its best-sellers list.
150,000 is preferable. It sells 2–3 copies daily. Consider your rivals. Profitable niches have high demand and low competition.
Don't be afraid of competitive niches. First, it shows high demand. Secondly, what are the ways you can undercut the completion? Better book? Or cheaper option? There was lots of competition in my NodeJS book's area. None received 4.5 stars or more. I wrote a NodeJS book. Today, it's a best-selling Node book.
What’s Next
So long. Part II follows. Meanwhile, I will continue to write more books!
Follow my journey on Twitter.
This post is a summary. Read full article here

Aaron Dinin, PhD
2 years ago
The Advantages and Disadvantages of Having Investors Sign Your NDA
Startup entrepreneurs assume what risks when pitching?
Last week I signed four NDAs.
Four!
NDA stands for non-disclosure agreement. A legal document given to someone receiving confidential information. By signing, the person pledges not to share the information for a certain time. If they do, they may be in breach of contract and face legal action.
Companies use NDAs to protect trade secrets and confidential internal information from employees and contractors. Appropriate. If you manage a huge, successful firm, you don't want your employees selling their information to your competitors. To be true, business NDAs don't always prevent corporate espionage, but they usually make employees and contractors think twice before sharing.
I understand employee and contractor NDAs, but I wasn't asked to sign one. I counsel entrepreneurs, thus the NDAs I signed last week were from startups that wanted my feedback on their concepts.
I’m not a startup investor. I give startup guidance online. Despite that, four entrepreneurs thought their company ideas were so important they wanted me to sign a generically written legal form they probably acquired from a shady, spam-filled legal templates website before we could chat.
False. One company tried to get me to sign their NDA a few days after our conversation. I gently rejected, but their tenacity encouraged me. I considered sending retroactive NDAs to everyone I've ever talked to about one of my startups in case they establish a successful company based on something I said.
Two of the other three NDAs were from nearly identical companies. Good thing I didn't sign an NDA for the first one, else they may have sued me for talking to the second one as though I control the firms people pitch me.
I wasn't talking to the fourth NDA company. Instead, I received an unsolicited email from someone who wanted comments on their fundraising pitch deck but required me to sign an NDA before sending it.
That's right, before I could read a random Internet stranger's unsolicited pitch deck, I had to sign his NDA, potentially limiting my ability to discuss what was in it.
You should understand. Advisors, mentors, investors, etc. talk to hundreds of businesses each year. They cannot manage all the companies they deal with, thus they cannot risk legal trouble by talking to someone. Well, if I signed NDAs for all the startups I spoke with, half of the 300+ articles I've written on Medium over the past several years could get me sued into the next century because I've undoubtedly addressed topics in my articles that I discussed with them.
The four NDAs I received last week are part of a recent trend of entrepreneurs sending out NDAs before meetings, despite the practical and legal issues. They act like asking someone to sign away their right to talk about all they see and hear in a day is as straightforward as asking for a glass of water.
Given this inflow of NDAs, I wanted to briefly remind entrepreneurs reading this blog about the merits and cons of requesting investors (or others in the startup ecosystem) to sign your NDA.
Benefits of having investors sign your NDA include:
None. Zero. Nothing.
Disadvantages of requesting investor NDAs:
You'll come off as an amateur who has no idea what it takes to launch a successful firm.
Investors won't trust you with their money since you appear to be a complete amateur.
Printing NDAs will be a waste of paper because no genuine entrepreneur will ever sign one.
I apologize for missing any cons. Please leave your remarks.
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Michelle Teheux
3 years ago
Get Real, All You Grateful Laid-Off LinkedIn Users
WTF is wrong with you people?
When I was laid off as editor of my town's daily newspaper, I went silent on social media. I knew it was coming and had been quietly removing personal items each day, but the pain was intense.
I posted a day later. I didn't bad-mouth GateHouse Media but expressed my sadness at leaving the newspaper industry, pride in my accomplishments, and hope for success in another industry.
Normal job-loss response.
What do you recognize as abnormal?
The bullshit I’ve been reading from laid-off folks on LinkedIn.
If you're there, you know. Many Twitter or Facebook/Meta employees recently lost their jobs.
Well, many of them did not “lose their job,” actually. They were “impacted by the layoffs” at their former employer. I keep seeing that phrase.
Why don’t they want to actually say it? Why the euphemism?
Many are excited about the opportunities ahead. The jobless deny being sad.
They're ecstatic! They have big plans.
Hope so. Sincerely! Being laid off stinks, especially if, like me, your skills are obsolete. It's worse if, like me, you're too old to start a new career. Ageism exists despite denials.
Nowadays, professionalism seems to demand psychotic levels of fake optimism.
Why? Life is unpredictable. That's indisputable. You shouldn't constantly complain or cry in public, but you also shouldn't pretend everything's great.
It makes you look psychotic, not positive. It's like saying at work:
“I was impacted by the death of my spouse of 20 years this week, and many of you have reached out to me, expressing your sympathy. However, I’m choosing to remember the amazing things we shared. I feel confident that there is another marriage out there for me, and after taking a quiet weekend trip to reset myself, I’ll be out there looking for the next great marital adventure! #staypositive #available #opentolove
Also:
“Now looking for our next #dreamhome after our entire neighborhood was demolished by a wildfire last night. We feel so lucky to have lived near so many amazing and inspirational neighbors, all of whom we will miss as we go on our next housing adventure. The best house for us is yet to come! If you have a great neighborhood you’d recommend, please feel free to reach out and touch base with us! #newhouse #newneighborhood #newlife
Admit it. That’s creepy.
The constant optimism makes me feel sick to my stomach.
Viscerally.
I hate fakes.
Imagine a fake wood grain desk. Wouldn't it be better if the designer accepted that it's plastic and went with that?
Real is better but not always nice. When something isn't nice, you don't have to go into detail, but you also shouldn't pretend it's great.
How to announce your job loss to the world.
Do not pretend to be happy, but don't cry and drink vodka all afternoon.
Say you loved your job, and that you're looking for new opportunities.
Yes, if you'll miss your coworkers. Otherwise, don't badmouth. No bridge-burning!
Please specify the job you want. You may want to pivot.
Alternatively, try this.
You could always flame out.
If you've pushed yourself too far into toxic positivity, you may be ready to burn it all down. If so, make it worthwhile by writing something like this:
Well, I was shitcanned by the losers at #Acme today. That bitch Linda in HR threw me under the bus just because she saw that one of my “friends” tagged me in some beach pics on social media after I called in sick with Covid. The good thing is I will no longer have to watch my ass around that #asspincher Ron in accounting, but I’m sad that I will no longer have a cushy job with high pay or access to the primo office supplies I’ve been sneaking home for the last five years. (Those gel pens were the best!) I am going to be taking some time off to enjoy my unemployment and hammer down shots of Jägermeister but in about five months I’ll be looking for anything easy with high pay and great benefits. Reach out if you can help! #officesupplies #unemploymentrocks #drinkinglikeagirlboss #acmesucks
It beats the fake positivity.

Dr Mehmet Yildiz
3 years ago
How I train my brain daily for clarity and productivity.
I use a conceptual and practical system I developed decades ago as an example.
Since childhood, I've been interested in the brain-mind connection, so I developed a system using scientific breakthroughs, experiments, and the experiences of successful people in my circles.
This story provides a high-level overview of a custom system to inform and inspire readers. Creating a mind gym was one of my best personal and professional investments.
Such a complex system may not be possible for everyone or appear luxurious at first. However, the process and approach may help you find more accessible and viable solutions.
Visualizing the brain as a muscle, I learned to stimulate it with physical and mental exercises, applying a new mindset and behavioral changes.
My methods and practices may not work for others because we're all different. I focus on the approach's principles and highlights so you can create your own program.
Some create a conceptual and practical system intuitively, and others intellectually. Both worked. I see intellect and intuition as higher selves.
The mental tools I introduce are based on lifestyle changes and can be personalized by anyone, barring physical constraints or underlying health conditions.
Some people can't meditate despite wanting to due to mental constraints. This story lacks exceptions.
People's systems may vary. Many have used my tools successfully. All have scientific backing because their benefits attracted scientists. None are unethical or controversial.
My focus is cognition, which is the neocortex's ability. These practices and tools can affect the limbic and reptilian brain regions.
A previous article discussed brain health's biological aspects. This article focuses on psychology.
Thinking, learning, and remembering are cognitive abilities. Cognitive abilities determine our health and performance.
Cognitive health is the ability to think, concentrate, learn, and remember. Cognitive performance boosting involves various tools and processes. My system and protocols address cognitive health and performance.
As a biological organ, the brain's abilities decline with age, especially if not used regularly. Older people have more neurodegenerative disorders like dementia.
As aging is inevitable, I focus on creating cognitive reserves to remain mentally functional as we age and face mental decline or cognitive impairment.
My protocols focus on neurogenesis, or brain growth and maintenance. Neurons and connections can grow at any age.
Metacognition refers to knowing our cognitive abilities, like thinking about thinking and learning how to learn.
In the following sections, I provide an overview of my system, mental tools, and protocols.
This system summarizes my 50-year career. Some may find it too abstract, so I give examples.
First, explain the system. Section 2 introduces activities. Third, how to measure and maintain mental growth.
1 — Developed a practical mental gym.
The mental gym is a metaphor for the physical fitness gym to improve our mental muscles.
This concept covers brain and mind functionality. Integrated biological and psychological components.
I'll describe my mental gym so my other points make sense. My mental gym has physical and mental tools.
Mindfulness, meditation, visualization, self-conversations, breathing exercises, expressive writing, working in a flow state, reading, music, dance, isometric training, barefoot walking, cold/heat exposure, CBT, and social engagements are regular tools.
Dancing, walking, and thermogenesis are body-related tools. As the brain is part of the body and houses the mind, these tools can affect mental abilities such as attention, focus, memory, task switching, and problem-solving.
Different people may like different tools. I chose these tools based on my needs, goals, and lifestyle. They're just examples. You can choose tools that fit your goals and personality.
2 — Performed tasks regularly.
These tools gave me clarity. They became daily hobbies. Some I did alone, others with others.
Some examples: I meditate daily. Even though my overactive mind made daily meditation difficult at first, I now enjoy it. Meditation three times a day sharpens my mind.
Self-talk is used for self-therapy and creativity. Self-talk was initially difficult, but neurogenesis rewired my brain to make it a habit.
Cold showers, warm baths with Epsom salts, fasting, barefoot walks on the beach or grass, dancing, calisthenics, trampoline hopping, and breathing exercises increase my mental clarity, creativity, and productivity.
These exercises can increase BDNF, which promotes nervous system growth. They improve mental capacity and performance by increasing blood flow and brain oxygenation.
I use weekly and occasional activities like dry saunas, talking with others, and community activities.
These activities stimulate the brain and mind, improving performance and cognitive capacity.
3 — Measured progress, set growth goals.
Measuring progress helps us stay on track. Without data, it's hard to stay motivated. When we face inevitable setbacks, we may abandon our dreams.
I created a daily checklist for a spreadsheet with macros. I tracked how often and long I did each activity.
I measured my progress objectively and subjectively. In the progress spreadsheet, I noted my meditation hours and subjective feelings.
In another column, I used good, moderate, and excellent to get qualitative data. It took time and effort. Later, I started benefiting from this automated structure.
Creating a page for each activity, such as meditation, self-talk, cold showers, walking, expressive writing, personal interactions, etc., gave me empirical data I could analyze, modify, and graph to show progress.
Colored charts showed each area's strengths and weaknesses.
Strengths motivate me to continue them. Identifying weaknesses helped me improve them.
As the system matured, data recording became a habit and took less time. I saw the result immediately because I automated the charts when I entered daily data. Early time investment paid off later.
Mind Gym Benefits, Effective Use, and Progress Measuring
This concept helped me move from comfort to risk. I accept things as they are.
Turnarounds were made. I stopped feeling "Fight-Flight-Freeze" and maintained self-control.
I tamed my overactive amygdala by strengthening my brain. Stress and anxiety decreased. With these shifts, I accepted criticism and turned envy into admiration. Clarity improved.
When the cognitive part of the brain became stronger and the primitive part was tamed, managing thoughts and emotions became easier. My AQ increased. I learned to tolerate people, physical, mental, and emotional obstacles.
Accessing vast information sources in my subconscious mind through an improved RAS allowed me to easily tap into my higher self and recognize flaws in my lower self.
Summary
The brain loves patterns and routines, so habits help. Observing, developing, and monitoring habits mindfully can be beneficial. Mindfulness helps us achieve this goal systematically.
As body and mind are connected, we must consider both when building habits. Consistent and joyful practices can strengthen neurons and neural connections.
Habits help us accomplish more with less effort. Regularly using mental tools and processes can improve our cognitive health and performance as we age.
Creating daily habits to improve cognitive abilities can sharpen our minds and boost our well-being.
Some apps monitor our activities and behavior to help build habits. If you can't replicate my system, try these apps. Some smartwatches and fitness devices include them.
Set aside time each day for mental activities you enjoy. Regular scheduling and practice can strengthen brain regions and form habits. Once you form habits, tasks become easy.
Improving our minds is a lifelong journey. It's easier and more sustainable to increase our efforts daily, weekly, monthly, or annually.
Despite life's ups and downs, many want to remain calm and cheerful.
This valuable skill is unrelated to wealth or fame. It's about our mindset, fueled by our biological and psychological needs.
Here are some lessons I've learned about staying calm and composed despite challenges and setbacks.
1 — Tranquillity starts with observing thoughts and feelings.
2 — Clear the mental clutter and emotional entanglements with conscious breathing and gentle movements.
3 — Accept situations and events as they are with no resistance.
4 — Self-love can lead to loving others and increasing compassion.
5 — Count your blessings and cultivate gratitude.
Clear thinking can bring joy and satisfaction. It's a privilege to wake up with a healthy body and clear mind, ready to connect with others and serve them.
Thank you for reading my perspectives. I wish you a healthy and happy life.

Shalitha Suranga
3 years ago
The Top 5 Mathematical Concepts Every Programmer Needs to Know
Using math to write efficient code in any language
Programmers design, build, test, and maintain software. Employ cases and personal preferences determine the programming languages we use throughout development. Mobile app developers use JavaScript or Dart. Some programmers design performance-first software in C/C++.
A generic source code includes language-specific grammar, pre-implemented function calls, mathematical operators, and control statements. Some mathematical principles assist us enhance our programming and problem-solving skills.
We all use basic mathematical concepts like formulas and relational operators (aka comparison operators) in programming in our daily lives. Beyond these mathematical syntaxes, we'll see discrete math topics. This narrative explains key math topics programmers must know. Master these ideas to produce clean and efficient software code.
Expressions in mathematics and built-in mathematical functions
A source code can only contain a mathematical algorithm or prebuilt API functions. We develop source code between these two ends. If you create code to fetch JSON data from a RESTful service, you'll invoke an HTTP client and won't conduct any math. If you write a function to compute the circle's area, you conduct the math there.
When your source code gets more mathematical, you'll need to use mathematical functions. Every programming language has a math module and syntactical operators. Good programmers always consider code readability, so we should learn to write readable mathematical expressions.
Linux utilizes clear math expressions.
Inbuilt max and min functions can minimize verbose if statements.
How can we compute the number of pages needed to display known data? In such instances, the ceil function is often utilized.
import math as m
results = 102
items_per_page = 10
pages = m.ceil(results / items_per_page)
print(pages)Learn to write clear, concise math expressions.
Combinatorics in Algorithm Design
Combinatorics theory counts, selects, and arranges numbers or objects. First, consider these programming-related questions. Four-digit PIN security? what options exist? What if the PIN has a prefix? How to locate all decimal number pairs?
Combinatorics questions. Software engineering jobs often require counting items. Combinatorics counts elements without counting them one by one or through other verbose approaches, therefore it enables us to offer minimum and efficient solutions to real-world situations. Combinatorics helps us make reliable decision tests without missing edge cases. Write a program to see if three inputs form a triangle. This is a question I commonly ask in software engineering interviews.
Graph theory is a subfield of combinatorics. Graph theory is used in computerized road maps and social media apps.
Logarithms and Geometry Understanding
Geometry studies shapes, angles, and sizes. Cartesian geometry involves representing geometric objects in multidimensional planes. Geometry is useful for programming. Cartesian geometry is useful for vector graphics, game development, and low-level computer graphics. We can simply work with 2D and 3D arrays as plane axes.
GetWindowRect is a Windows GUI SDK geometric object.
High-level GUI SDKs and libraries use geometric notions like coordinates, dimensions, and forms, therefore knowing geometry speeds up work with computer graphics APIs.
How does exponentiation's inverse function work? Logarithm is exponentiation's inverse function. Logarithm helps programmers find efficient algorithms and solve calculations. Writing efficient code involves finding algorithms with logarithmic temporal complexity. Programmers prefer binary search (O(log n)) over linear search (O(n)). Git source specifies O(log n):
Logarithms aid with programming math. Metas Watchman uses a logarithmic utility function to find the next power of two.
Employing Mathematical Data Structures
Programmers must know data structures to develop clean, efficient code. Stack, queue, and hashmap are computer science basics. Sets and graphs are discrete arithmetic data structures. Most computer languages include a set structure to hold distinct data entries. In most computer languages, graphs can be represented using neighboring lists or objects.
Using sets as deduped lists is powerful because set implementations allow iterators. Instead of a list (or array), store WebSocket connections in a set.
Most interviewers ask graph theory questions, yet current software engineers don't practice algorithms. Graph theory challenges become obligatory in IT firm interviews.
Recognizing Applications of Recursion
A function in programming isolates input(s) and output(s) (s). Programming functions may have originated from mathematical function theories. Programming and math functions are different but similar. Both function types accept input and return value.
Recursion involves calling the same function inside another function. In its implementation, you'll call the Fibonacci sequence. Recursion solves divide-and-conquer software engineering difficulties and avoids code repetition. I recently built the following recursive Dart code to render a Flutter multi-depth expanding list UI:
Recursion is not the natural linear way to solve problems, hence thinking recursively is difficult. Everything becomes clear when a mathematical function definition includes a base case and recursive call.
Conclusion
Every codebase uses arithmetic operators, relational operators, and expressions. To build mathematical expressions, we typically employ log, ceil, floor, min, max, etc. Combinatorics, geometry, data structures, and recursion help implement algorithms. Unless you operate in a pure mathematical domain, you may not use calculus, limits, and other complex math in daily programming (i.e., a game engine). These principles are fundamental for daily programming activities.
Master the above math fundamentals to build clean, efficient code.
