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SAHIL SAPRU

SAHIL SAPRU

3 years ago

How I grew my business to a $5 million annual recurring revenue

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cdixon

cdixon

3 years ago

2000s Toys, Secrets, and Cycles

During the dot-com bust, I started my internet career. People used the internet intermittently to check email, plan travel, and do research. The average internet user spent 30 minutes online a day, compared to 7 today. To use the internet, you had to "log on" (most people still used dial-up), unlike today's always-on, high-speed mobile internet. In 2001, Amazon's market cap was $2.2B, 1/500th of what it is today. A study asked Americans if they'd adopt broadband, and most said no. They didn't see a need to speed up email, the most popular internet use. The National Academy of Sciences ranked the internet 13th among the 100 greatest inventions, below radio and phones. The internet was a cool invention, but it had limited uses and wasn't a good place to build a business. 

A small but growing movement of developers and founders believed the internet could be more than a read-only medium, allowing anyone to create and publish. This is web 2. The runner up name was read-write web. (These terms were used in prominent publications and conferences.) 

Web 2 concepts included letting users publish whatever they want ("user generated content" was a buzzword), social graphs, APIs and mashups (what we call composability today), and tagging over hierarchical navigation. Technical innovations occurred. A seemingly simple but important one was dynamically updating web pages without reloading. This is now how people expect web apps to work. Mobile devices that could access the web were niche (I was an avid Sidekick user). 

The contrast between what smart founders and engineers discussed over dinner and on weekends and what the mainstream tech world took seriously during the week was striking. Enterprise security appliances, essentially preloaded servers with security software, were a popular trend. Many of the same people would talk about "serious" products at work, then talk about consumer internet products and web 2. It was tech's biggest news. Web 2 products were seen as toys, not real businesses. They were hobbies, not work-related. 

There's a strong correlation between rich product design spaces and what smart people find interesting, which took me some time to learn and led to blog posts like "The next big thing will start out looking like a toy" Web 2's novel product design possibilities sparked dinner and weekend conversations. Imagine combining these features. What if you used this pattern elsewhere? What new product ideas are next? This excited people. "Serious stuff" like security appliances seemed more limited. 

The small and passionate web 2 community also stood out. I attended the first New York Tech meetup in 2004. Everyone fit in Meetup's small conference room. Late at night, people demoed their software and chatted. I have old friends. Sometimes I get asked how I first met old friends like Fred Wilson and Alexis Ohanian. These topics didn't interest many people, especially on the east coast. We were friends. Real community. Alex Rampell, who now works with me at a16z, is someone I met in 2003 when a friend said, "Hey, I met someone else interested in consumer internet." Rare. People were focused and enthusiastic. Revolution seemed imminent. We knew a secret nobody else did. 

My web 2 startup was called SiteAdvisor. When my co-founders and I started developing the idea in 2003, web security was out of control. Phishing and spyware were common on Internet Explorer PCs. SiteAdvisor was designed to warn users about security threats like phishing and spyware, and then, using web 2 concepts like user-generated reviews, add more subjective judgments (similar to what TrustPilot seems to do today). This staged approach was common at the time; I called it "Come for the tool, stay for the network." We built APIs, encouraged mashups, and did SEO marketing. 

Yahoo's 2005 acquisitions of Flickr and Delicious boosted web 2 in 2005. By today's standards, the amounts were small, around $30M each, but it was a signal. Web 2 was assumed to be a fun hobby, a way to build cool stuff, but not a business. Yahoo was a savvy company that said it would make web 2 a priority. 

As I recall, that's when web 2 started becoming mainstream tech. Early web 2 founders transitioned successfully. Other entrepreneurs built on the early enthusiasts' work. Competition shifted from ideation to execution. You had to decide if you wanted to be an idealistic indie bar band or a pragmatic stadium band. 

Web 2 was booming in 2007 Facebook passed 10M users, Twitter grew and got VC funding, and Google bought YouTube. The 2008 financial crisis tested entrepreneurs' resolve. Smart people predicted another great depression as tech funding dried up. 

Many people struggled during the recession. 2008-2011 was a golden age for startups. By 2009, talented founders were flooding Apple's iPhone app store. Mobile apps were booming. Uber, Venmo, Snap, and Instagram were all founded between 2009 and 2011. Social media (which had replaced web 2), cloud computing (which enabled apps to scale server side), and smartphones converged. Even if social, cloud, and mobile improve linearly, the combination could improve exponentially. 

This chart shows how I view product and financial cycles. Product and financial cycles evolve separately. The Nasdaq index is a proxy for the financial sentiment. Financial sentiment wildly fluctuates. 

Next row shows iconic startup or product years. Bottom-row product cycles dictate timing. Product cycles are more predictable than financial cycles because they follow internal logic. In the incubation phase, enthusiasts build products for other enthusiasts on nights and weekends. When the right mix of technology, talent, and community knowledge arrives, products go mainstream. (I show the biggest tech cycles in the chart, but smaller ones happen, like web 2 in the 2000s and fintech and SaaS in the 2010s.) 

Tech has changed since the 2000s. Few tech giants dominate the internet, exerting economic and cultural influence. In the 2000s, web 2 was ignored or dismissed as trivial. Entrenched interests respond aggressively to new movements that could threaten them. Creative patterns from the 2000s continue today, driven by enthusiasts who see possibilities where others don't. Know where to look. Crypto and web 3 are where I'd start. 

Today's negative financial sentiment reminds me of 2008. If we face a prolonged downturn, we can learn from 2008 by preserving capital and focusing on the long term. Keep an eye on the product cycle. Smart people are interested in things with product potential. This becomes true. Toys become necessities. Hobbies become mainstream. Optimists build the future, not cynics.


Full article is available here

Owolabi Judah

Owolabi Judah

3 years ago

How much did YouTube pay for 10 million views?

Ali's $1,054,053.74 YouTube Adsense haul.

How Much YouTube Paid Ali Abdaal For 10,000,000 views

YouTuber, entrepreneur, and former doctor Ali Abdaal. He began filming productivity and financial videos in 2017. Ali Abdaal has 3 million YouTube subscribers and has crossed $1 million in AdSense revenue. Crazy, no?

Ali will share the revenue of his top 5 youtube videos, things he's learned that you can apply to your side hustle, and how many views it takes to make a livelihood off youtube.

First, "The Long Game."

All good things take time to bear fruit. Compounding improves everything. Long-term work yields better returns. Ali made his first dollar after nine months and 85 videos.

Second, "One piece of content can transform your life, but you never know which one."

This video transformed Ali's life.

Had he abandoned YouTube at 84 videos without making any money, he wouldn't have filmed the 85th video that altered everything.

Third Lesson: Your Industry Choice Can Multiply.

The industry or niche you target as a business owner or side hustler can have a major impact on how much money you make.

Here are the top 5 videos.

1) 9.8m views: $191,258.16 for 9 passive income ideas

9.8m views: $191,258.16 for 9 passive income ideas

Ali made 2 points.

We should consider YouTube videos digital assets. They're investments, which make us money. His investments are yielding passive income.

Investing extra time and effort in your films can pay off.

2) How to Invest for Beginners — 5.2m Views: $87,200.08.

How to Invest for Beginners — 5.2m Views: $87,200.08.

This video did poorly in the first several weeks after it was published; it was his tenth poorest performer. Don't worry about things you can't control. This applies to life, not just YouTube videos.

He stated we constantly have anxieties, fears, and concerns about things outside our control, but if we can find that line, life is easier and more pleasurable.

3) How to Build a Website in 2022— 866.3k views: $42,132.72.

How to Build a Website in 2022— 866.3k views: $42,132.72.

The RPM was $48.86 per thousand views, making it his highest-earning video. Squarespace, Wix, and other website builders are trying to put ads on it and competing against one other, so ad rates go up.

Because it was beyond his niche, Ali almost didn't make the video. He made the video because he wanted to help at least one person.

4) How I take notes on my iPad in medical school — 5.9m views: $24,479.80

How I take notes on my iPad in medical school — 5.9m views: $24,479.80

85th video. It's the video that affected Ali's YouTube channel and his life the most. The video's success wasn't certain.

5) How I Type Fast 156 Words Per Minute — 8.2M views: $25,143.17

How I Type Fast 156 Words Per Minute — 8.2M views: $25,143.17

Ali didn't know this video would perform well; he made it because he can type fast and has been practicing for 10 years. So he made a video with his best advice.

How many views to different wealth levels?

It depends on geography, niche, and other monetization sources. To keep things simple, he would solely utilize AdSense.

How many views to generate money?

To generate money on Youtube, you need 1,000 subscribers and 4,000 hours of view time. How much work do you need to make pocket money?

Ali's first 1,000 subscribers took 52 videos and 6 months. The typical channel with 1,000 subscribers contains 152 videos, according to Tubebuddy. It's time-consuming.

After monetizing, you'll need 15,000 views/month to make $5-$10/day.

How many views to go part-time?

Say you make $35,000/year at your day job. If you work 5 days/week, you make $7,000/year each day. If you want to drop down from 5 days to 4 days/week, you need to make an extra $7,000/year from YouTube, or $600/month.

What's the quit-your-job budget?

Silicon Valley Girl is in a highly successful niche targeting tech-focused folks in the west. When her channel had 500k views/month, she made roughly $3,000/month or $47,000/year, enough to quit your work.

Marina has another 1.5m subscriber channel in Russia, which has a lower rpm because fewer corporations advertise there than in the west. 2.3 million views/month is $4,000/month or $50,000/year, enough to quit your employment.

Marina is an intriguing example because she has three YouTube channels with the same skills, but one is 16x more profitable due to the niche she chose.

In Ali's case, he made 100+ videos when his channel was producing enough money to quit his job, roughly $4,000/month.

How many views make you rich?

How many views make you rich?

Depending on how you define rich. Ali felt prosperous with over $100,000/year and 3–5m views/month.

Conclusion

YouTubers and artists don't treat their work like a company, which is a mistake. Businesses have been attempting to figure this out for decades, if not centuries.

We can learn from the business world how to monetize YouTube, Instagram, and Tiktok and make them into sustainable enterprises where we can hire people and delegate tasks.

Bonus

Watch Ali's video explaining all this:


This post is a summary. Read the full article here

Pat Vieljeux

Pat Vieljeux

3 years ago

In 5 minutes, you can tell if a startup will succeed.

Or the “lie to me” method.

I can predict a startup's success in minutes.

Just interview its founder.

Ask "why?"

I question "why" till I sense him.

I need to feel the person I have in front of me. I need to know if he or she can deliver. Startups aren't easy. Without abilities, a brilliant idea will fail.

Good entrepreneurs have these qualities: He's a leader, determined, and resilient.

For me, they can be split in two categories.

The first entrepreneur aspires to live meaningfully. The second wants to get rich. The second is communicative. He wants to wow the crowd. He's motivated by the thought of one day sailing a boat past palm trees and sunny beaches.

What drives the first entrepreneur is evident in his speech, face, and voice. He will not speak about his product. He's (nearly) uninterested. He's not selling anything. He's not a salesman. He wants to succeed. The product is his fuel.

He'll explain his decision. He'll share his motivations. His desire. And he'll use meaningful words.

Paul Ekman has shown that face expressions aren't cultural. His study influenced the American TV series "lie to me" about body language and speech.

Passionate entrepreneurs are obvious. It's palpable. Faking passion is tough. Someone who wants your favor and money will expose his actual motives through his expressions and language.

The good liar will be able to fool you for a while, but not for long if you pay attention to his body language and how he expresses himself.

And also, if you look at his business plan.

His business plan reveals his goals. Read between the lines.

Entrepreneur 1 will focus on his "why", whereas Entrepreneur 2 will focus on the "how".

Entrepreneur 1 will develop a vision-driven culture.

The second, on the other hand, will focus on his EBITDA.

Why is the culture so critical? Because it will allow entrepreneur 1 to develop a solid team that can tackle his problems and trials. His team's "why" will keep them together in tough times.

"Give me a terrific start-up team with a mediocre idea over a weak one any day." Because a great team knows when to pivot and trusts each other. Weak teams fail.” — Bernhard Schroeder

Closings thoughts

Every VC must ask Why. Entrepreneur's motivations. This "why" will create the team's culture. This culture will help the team adjust to any setback.

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Tim Soulo

Tim Soulo

3 years ago

Here is why 90.63% of Pages Get No Traffic From Google. 

The web adds millions or billions of pages per day.

How much Google traffic does this content get?

In 2017, we studied 2 million randomly-published pages to answer this question. Only 5.7% of them ranked in Google's top 10 search results within a year of being published.

94.3 percent of roughly two million pages got no Google traffic.

Two million pages is a small sample compared to the entire web. We did another study.

We analyzed over a billion pages to see how many get organic search traffic and why.

How many pages get search traffic?

90% of pages in our index get no Google traffic, and 5.2% get ten visits or less.

90% of google pages get no organic traffic

How can you join the minority that gets Google organic search traffic?

There are hundreds of SEO problems that can hurt your Google rankings. If we only consider common scenarios, there are only four.

Reason #1: No backlinks

I hate to repeat what most SEO articles say, but it's true:

Backlinks boost Google rankings.

Google's "top 3 ranking factors" include them.

Why don't we divide our studied pages by the number of referring domains?

66.31 percent of pages have no backlinks, and 26.29 percent have three or fewer.

Did you notice the trend already?

Most pages lack search traffic and backlinks.

But are these the same pages?

Let's compare monthly organic search traffic to backlinks from unique websites (referring domains):

More backlinks equals more Google organic traffic.

Referring domains and keyword rankings are correlated.

It's important to note that correlation does not imply causation, and none of these graphs prove backlinks boost Google rankings. Most SEO professionals agree that it's nearly impossible to rank on the first page without backlinks.

You'll need high-quality backlinks to rank in Google and get search traffic. 

Is organic traffic possible without links?

Here are the numbers:

Four million pages get organic search traffic without backlinks. Only one in 20 pages without backlinks has traffic, which is 5% of our sample.

Most get 300 or fewer organic visits per month.

What happens if we exclude high-Domain-Rating pages?

The numbers worsen. Less than 4% of our sample (1.4 million pages) receive organic traffic. Only 320,000 get over 300 monthly organic visits, or 0.1% of our sample.

This suggests high-authority pages without backlinks are more likely to get organic traffic than low-authority pages.

Internal links likely pass PageRank to new pages.

Two other reasons:

  1. Our crawler's blocked. Most shady SEOs block backlinks from us. This prevents competitors from seeing (and reporting) PBNs.

  2. They choose low-competition subjects. Low-volume queries are less competitive, requiring fewer backlinks to rank.

If the idea of getting search traffic without building backlinks excites you, learn about Keyword Difficulty and how to find keywords/topics with decent traffic potential and low competition.

Reason #2: The page has no long-term traffic potential.

Some pages with many backlinks get no Google traffic.

Why? I filtered Content Explorer for pages with no organic search traffic and divided them into four buckets by linking domains.

Almost 70k pages have backlinks from over 200 domains, but no search traffic.

By manually reviewing these (and other) pages, I noticed two general trends that explain why they get no traffic:

  1. They overdid "shady link building" and got penalized by Google;

  2. They're not targeting a Google-searched topic.

I won't elaborate on point one because I hope you don't engage in "shady link building"

#2 is self-explanatory:

If nobody searches for what you write, you won't get search traffic.

Consider one of our blog posts' metrics:

No organic traffic despite 337 backlinks from 132 sites.

The page is about "organic traffic research," which nobody searches for.

News articles often have this. They get many links from around the web but little Google traffic.

People can't search for things they don't know about, and most don't care about old events and don't search for them.


Note:

Some news articles rank in the "Top stories" block for relevant, high-volume search queries, generating short-term organic search traffic.

The Guardian's top "Donald Trump" story:

Ahrefs caught on quickly:

"Donald Trump" gets 5.6M monthly searches, so this page got a lot of "Top stories" traffic.

I bet traffic has dropped if you check now.


One of the quickest and most effective SEO wins is:

  1. Find your website's pages with the most referring domains;

  2. Do keyword research to re-optimize them for relevant topics with good search traffic potential.

Bryan Harris shared this "quick SEO win" during a course interview:

He suggested using Ahrefs' Site Explorer's "Best by links" report to find your site's most-linked pages and analyzing their search traffic. This finds pages with lots of links but little organic search traffic.

We see:

The guide has 67 backlinks but no organic traffic.

We could fix this by re-optimizing the page for "SERP"

A similar guide with 26 backlinks gets 3,400 monthly organic visits, so we should easily increase our traffic.

Don't do this with all low-traffic pages with backlinks. Choose your battles wisely; some pages shouldn't be ranked.

Reason #3: Search intent isn't met

Google returns the most relevant search results.

That's why blog posts with recommendations rank highest for "best yoga mat."

Google knows that most searchers aren't buying.

It's also why this yoga mats page doesn't rank, despite having seven times more backlinks than the top 10 pages:

The page ranks for thousands of other keywords and gets tens of thousands of monthly organic visits. Not being the "best yoga mat" isn't a big deal.

If you have pages with lots of backlinks but no organic traffic, re-optimizing them for search intent can be a quick SEO win.

It was originally a boring landing page describing our product's benefits and offering a 7-day trial.

We realized the problem after analyzing search intent.

People wanted a free tool, not a landing page.

In September 2018, we published a free tool at the same URL. Organic traffic and rankings skyrocketed.

Reason #4: Unindexed page

Google can’t rank pages that aren’t indexed.

If you think this is the case, search Google for site:[url]. You should see at least one result; otherwise, it’s not indexed.

A rogue noindex meta tag is usually to blame. This tells search engines not to index a URL.

Rogue canonicals, redirects, and robots.txt blocks prevent indexing.

Check the "Excluded" tab in Google Search Console's "Coverage" report to see excluded pages.

Google doesn't index broken pages, even with backlinks.

Surprisingly common.

In Ahrefs' Site Explorer, the Best by Links report for a popular content marketing blog shows many broken pages.

One dead page has 131 backlinks:

According to the URL, the page defined content marketing. —a keyword with a monthly search volume of 5,900 in the US.

Luckily, another page ranks for this keyword. Not a huge loss.

At least redirect the dead page's backlinks to a working page on the same topic. This may increase long-tail keyword traffic.


This post is a summary. See the original post here

ANDREW SINGER

ANDREW SINGER

3 years ago

Crypto seen as the ‘future of money’ in inflation-mired countries

Crypto as the ‘future of money' in inflation-stricken nations

Citizens of devalued currencies “need” crypto. “Nice to have” in the developed world.

According to Gemini's 2022 Global State of Crypto report, cryptocurrencies “evolved from what many considered a niche investment into an established asset class” last year.

More than half of crypto owners in Brazil (51%), Hong Kong (51%), and India (54%), according to the report, bought cryptocurrency for the first time in 2021.

The study found that inflation and currency devaluation are powerful drivers of crypto adoption, especially in emerging market (EM) countries:

“Respondents in countries that have seen a 50% or greater devaluation of their currency against the USD over the last decade were more than 5 times as likely to plan to purchase crypto in the coming year.”

Between 2011 and 2021, the real lost 218 percent of its value against the dollar, and 45 percent of Brazilians surveyed by Gemini said they planned to buy crypto in 2019.

The rand (South Africa's currency) has fallen 103 percent in value over the last decade, second only to the Brazilian real, and 32 percent of South Africans expect to own crypto in the coming year. Mexico and India, the third and fourth highest devaluation countries, followed suit.

Compared to the US dollar, Hong Kong and the UK currencies have not devalued in the last decade. Meanwhile, only 5% and 8% of those surveyed in those countries expressed interest in buying crypto.

What can be concluded? Noah Perlman, COO of Gemini, sees various crypto use cases depending on one's location. 

‘Need to have' investment in countries where the local currency has devalued against the dollar, whereas in the developed world it is still seen as a ‘nice to have'.

Crypto as money substitute

As an adjunct professor at New York University School of Law, Winston Ma distinguishes between an asset used as an inflation hedge and one used as a currency replacement.

Unlike gold, he believes Bitcoin (BTC) is not a “inflation hedge”. They acted more like growth stocks in 2022. “Bitcoin correlated more closely with the S&P 500 index — and Ether with the NASDAQ — than gold,” he told Cointelegraph. But in the developing world, things are different:

“Inflation may be a primary driver of cryptocurrency adoption in emerging markets like Brazil, India, and Mexico.”

According to Justin d'Anethan, institutional sales director at the Amber Group, a Singapore-based digital asset firm, early adoption was driven by countries where currency stability and/or access to proper banking services were issues. Simply put, he said, developing countries want alternatives to easily debased fiat currencies.

“The larger flows may still come from institutions and developed countries, but the actual users may come from places like Lebanon, Turkey, Venezuela, and Indonesia.”

“Inflation is one of the factors that has and continues to drive adoption of Bitcoin and other crypto assets globally,” said Sean Stein Smith, assistant professor of economics and business at Lehman College.

But it's only one factor, and different regions have different factors, says Stein Smith. As a “instantaneously accessible, traceable, and cost-effective transaction option,” investors and entrepreneurs increasingly recognize the benefits of crypto assets. Other places promote crypto adoption due to “potential capital gains and returns”.

According to the report, “legal uncertainty around cryptocurrency,” tax questions, and a general education deficit could hinder adoption in Asia Pacific and Latin America. In Africa, 56% of respondents said more educational resources were needed to explain cryptocurrencies.

Not only inflation, but empowering our youth to live better than their parents without fear of failure or allegiance to legacy financial markets or products, said Monica Singer, ConsenSys South Africa lead. Also, “the issue of cash and remittances is huge in Africa, as is the issue of social grants.”

Money's future?

The survey found that Brazil and Indonesia had the most cryptocurrency ownership. In each country, 41% of those polled said they owned crypto. Only 20% of Americans surveyed said they owned cryptocurrency.

These markets are more likely to see cryptocurrencies as the future of money. The survey found:

“The majority of respondents in Latin America (59%) and Africa (58%) say crypto is the future of money.”
Brazil (66%), Nigeria (63%), Indonesia (61%), and South Africa (57%). Europe and Australia had the fewest believers, with Denmark at 12%, Norway at 15%, and Australia at 17%.

Will the Ukraine conflict impact adoption?

The poll was taken before the war. Will the devastating conflict slow global crypto adoption growth?

With over $100 million in crypto donations directly requested by the Ukrainian government since the war began, Stein Smith says the war has certainly brought crypto into the mainstream conversation.

“This real-world demonstration of decentralized money's power could spur wider adoption, policy debate, and increased use of crypto as a medium of exchange.”
But the war may not affect all developing nations. “The Ukraine war has no impact on African demand for crypto,” Others loom larger. “Yes, inflation, but also a lack of trust in government in many African countries, and a young demographic very familiar with mobile phones and the internet.”

A major success story like Mpesa in Kenya has influenced the continent and may help accelerate crypto adoption. Creating a plan when everyone you trust fails you is directly related to the African spirit, she said.

On the other hand, Ma views the Ukraine conflict as a sort of crisis check for cryptocurrencies. For those in emerging markets, the Ukraine-Russia war has served as a “stress test” for the cryptocurrency payment rail, he told Cointelegraph.

“These emerging markets may see the greatest future gains in crypto adoption.”
Inflation and currency devaluation are persistent global concerns. In such places, Bitcoin and other cryptocurrencies are now seen as the “future of money.” Not in the developed world, but that could change with better regulation and education. Inflation and its impact on cash holdings are waking up even Western nations.

Read original post here.

Shalitha Suranga

Shalitha Suranga

3 years ago

The Top 5 Mathematical Concepts Every Programmer Needs to Know

Using math to write efficient code in any language

Photo by Emile Perron on Unsplash, edited with Canva

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.

A mathematical expression/formula in the Linux codebase, a screenshot by the author

Inbuilt max and min functions can minimize verbose if statements.

Reducing a verbose nested-if with the min function in Neutralinojs, a screenshot by the author

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.

GetWindowRect outputs an LPRECT geometric object, a screenshot by the author

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):

The Git codebase defines a function with logarithmic time complexity, a screenshot by the author

Logarithms aid with programming math. Metas Watchman uses a logarithmic utility function to find the next power of two.

A utility function that uses ceil, a screenshot by the author

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.