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.

The company's epidemic gains are gone.

Mark Zuckerberg was in line behind Jeff Bezos and Bill Gates less than two years ago. His wealth soared to $142 billion. Facebook's shares reached $382 in September 2021.

What comes next is either the start of something truly innovative or the beginning of an epic rise and fall story.

In order to start over (and avoid Facebook's PR issues), he renamed the firm Meta. Along with the new logo, he announced a turn into unexplored territory, the Metaverse, as the next chapter for the internet after mobile. Or, Zuckerberg believed Facebook's death was near, so he decided to build a bigger, better, cooler ship. Then we saw his vision (read: dystopian nightmare) in a polished demo that showed Zuckerberg in a luxury home and on a spaceship with aliens. Initially, it looked entertaining. A problem was obvious, though. He might claim this was the future and show us using the Metaverse for business, play, and more, but when I took off my headset, I'd realize none of it was genuine.

The stock price is almost as low as January 2019, when Facebook was dealing with the aftermath of the Cambridge Analytica crisis.

Irony surrounded the technology's aim. Zuckerberg says the Metaverse connects people. Despite some potential uses, this is another step away from physical touch with people. Metaverse worlds can cause melancholy, addiction, and mental illness. But forget all the cool stuff you can't afford. (It may be too expensive online, too.)

Metaverse activity slowed for a while. In early February 2022, we got an earnings call update. Not good. Reality Labs lost $10 billion on Oculus and Zuckerberg's Metaverse. Zuckerberg expects losses to rise. Meta's value dropped 20% in 11 minutes after markets closed.

It was a sign of things to come.

The corporation has failed to create interest in Metaverse, and there is evidence the public has lost interest. Meta still relies on Facebook's ad revenue machine, which is also struggling. In July, the company announced a decrease in revenue and missed practically all its forecasts, ending a decade of exceptional growth and relentless revenue. They blamed a dismal advertising demand climate, and Apple's monitoring changes smashed Meta's ad model. Throw in whistleblowers, leaked data revealing the firm knows Instagram negatively affects teens' mental health, the current Capital Hill probe, and the fact TikTok is eating its breakfast, lunch, and dinner, and 2022 might be the corporation's worst year ever.

After a rocky start, tech saw unprecedented growth during the pandemic. It was a tech bubble and then some.

The gains reversed after the dust settled and stock markets adjusted. Meta's year-to-date decline is 60%. Apple Inc is down 14%, Amazon is down 26%, and Alphabet Inc is down 29%. At the time of writing, Facebook's stock price is almost as low as January 2019, when the Cambridge Analytica scandal broke. Zuckerberg owns 350 million Meta shares. This drop costs him $71 billion.

The company's problems are growing, and solutions won't be easy.

• Facebook's period of unabated expansion and exorbitant ad revenue is ended, and the company's impact is dwindling as it continues to be the program that only your parents use. Because of the decreased ad spending and stagnant user growth, Zuckerberg will have less time to create his vision for the Metaverse because of the declining stock value and decreasing ad spending.

• Instagram is progressively dying in its attempt to resemble TikTok, alienating its user base and further driving users away from Meta-products.

• And now that the corporation has shifted its focus to the Metaverse, it is clear that, in its eagerness to improve its image, it fired the launch gun too early. You're fighting a lost battle when you announce an idea and then claim it won't happen for 10-15 years. When the idea is still years away from becoming a reality, the public is already starting to lose interest.

So, as I questioned earlier, is it the beginning of a technological revolution that will take this firm to stratospheric growth and success, or are we witnessing the end of Meta and Zuckerberg himself?

Dharmesh Shah co-founded HubSpot. WordPlay reached $90,000 per month in revenue without utilizing any of his wealth.

His method:

Take Advantage Of An Established Trend

Remember Wordle? Dharmesh was instantly hooked. As was the tech world.

HubSpot's co-founder noted inefficiencies in a recent My First Million episode. He wanted to play daily. Dharmesh, a tinkerer and software engineer, decided to design a word game.

He's a billionaire. How could he?

1. Wordle had limitations in his opinion;

2. Dharmesh is fundamentally a developer. He desired to start something new and increase his programming knowledge;

3. This project may serve as an excellent illustration for his son, who had begun learning about software development.

Better It Up

Building a new Wordle wasn't successful.

WordPlay lets you play with friends and family. You could challenge them and compare the results. It is a built-in growth tool.

WordPlay features:

• the capacity to follow sophisticated statistics after creating an account;

• continuous feedback on your performance;

• Outstanding domain name (wordplay.com).

Project Development

WordPlay has 9.5 million visitors and 45 million games played since February.

HubSpot co-founder credits tremendous growth to flywheel marketing, pushing the game through his own following.

Choosing an exploding specialty and making sharing easy also helped.

Shah enabled Google Ads on the website to test earning potential. Monthly revenue was $90,000.

That's just Google Ads. If monetization was the goal, a specialized ad network like Ezoic could double or triple the amount.

Wordle was a great buy for The New York Times at $1 million.

There’s a funny part of ‘Steve Jobs’ by Walter Isaacson where Jobs says that Bill Gates was more a copier than an innovator:

“Bill is basically unimaginative and has never invented anything, which is why I think he’s more comfortable now in philanthropy than technology. He just shamelessly ripped off other people’s ideas….He’d be a broader guy if he had dropped acid once or gone off to an ashram when he was younger.”

Gates lacked flavor. Nobody ever got excited about a Microsoft launch, despite their good products. Jobs had the world's best product taste. Apple vs. Microsoft.

A CEO's core job functions are all driven by taste: recruiting, vision, and company culture all require good taste. Depending on the type of company you want to build, know where you stand between Microsoft and Apple.

How can you improve your product judgment? How to acquire taste?

Test and refine

Product development follows two parallel paths: the ‘customer obsession’ path and the ‘taste and iterate’ path.

The customer obsession path involves solving customer problems. Lean Startup frameworks show you what to build at each step.

Taste-and-iterate doesn't involve the customer. You iterate internally and rely on product leaders' taste and judgment.

Creative Selection by Ken Kocienda explains this method. In Creative Selection, demos are iterated and presented to product leaders. Your boss presents to their boss, and so on up to Steve Jobs. If you have good product taste, you can be a panelist.

The iPhone follows this path. Before seeing an iPhone, consumers couldn't want one. Customer obsession wouldn't have gotten you far because iPhone buyers didn't know they wanted one.

In The Hard Thing About Hard Things, Ben Horowitz writes:

“It turns out that is exactly what product strategy is all about — figuring out the right product is the innovator’s job, not the customer’s job. The customer only knows what she thinks she wants based on her experience with the current product. The innovator can take into account everything that’s possible, but often must go against what she knows to be true. As a result, innovation requires a combination of knowledge, skill, and courage.“

One path solves a problem the customer knows they have, and the other doesn't. Instead of asking a person what they want, observe them and give them something they didn't know they needed.

It's much harder. Apple is the world's most valuable company because it's more valuable. It changes industries permanently.

If you want to build superior products, use the iPhone of your industry.

How to Improve Your Taste

I. Work for a company that has taste.

People with the best taste in products, markets, and people are rewarded for building great companies. Tasteful people know quality even when they can't describe it. Taste isn't writable. It's feel-based.

Moving into a community that's already doing what you want to do may be the best way to develop entrepreneurial taste. Most company-building knowledge is tacit.

Joining a company you want to emulate allows you to learn its inner workings. It reveals internal patterns intuitively. Many successful founders come from successful companies.

Consumption determines taste. Excellence will refine you. This is why restauranteurs visit the world's best restaurants and serious painters visit Paris or New York. Joining a company with good taste is beneficial.

2. Possess a wide range of interests

“Edwin Land of Polaroid talked about the intersection of the humanities and science. I like that intersection. There’s something magical about that place… The reason Apple resonates with people is that there’s a deep current of humanity in our innovation. I think great artists and great engineers are similar, in that they both have a desire to express themselves.” — Steve Jobs

I recently discovered Edwin Land. Jobs modeled much of his career after Land's. It makes sense that Apple was inspired by Land.

A Triumph of Genius: Edwin Land, Polaroid, and the Kodak Patent War notes:

“Land was introverted in person, but supremely confident when he came to his ideas… Alongside his scientific passions, lay knowledge of art, music, and literature. He was a cultured person growing even more so as he got older, and his interests filtered into the ethos of Polaroid.”

Founders' philosophies shape companies. Jobs and Land were invested. It showed in the products their companies made. Different. His obsession was spreading Microsoft software worldwide. Microsoft's success is why their products are bland and boring.

Experience is important. It's probably why startups are built by generalists and scaled by specialists.

Jobs combined design, typography, storytelling, and product taste at Apple. Some of the best original Mac developers were poets and musicians. Edwin Land liked broad-minded people, according to his biography. Physicist-musicians or physicist-photographers.

Da Vinci was a master of art, engineering, architecture, anatomy, and more. He wrote and drew at the same desk. His genius is remembered centuries after his death. Da Vinci's statue would stand at the intersection of humanities and science.

We find incredibly creative people here. Superhumans. Designers, creators, and world-improvers. These are the people we need to navigate technology and lead world-changing companies. Generalists lead.

Traditional sales organizations used the concept of a sales “funnel” to describe the process through which potential customers move, ending up with sales at the end. Winners today have abandoned that way of thinking in favor of building flywheels — business models in which every element reinforces every other.

Ah, the marketing funnel…

Prospective clients go through a predictable set of experiences, students learn in business school marketing classes. It looks like this:

Understanding the funnel helps evaluate sales success indicators. Gail Goodwin, former CEO of small business direct mail provider Constant Contact, said managing the pipeline was key to escaping the sluggish SaaS ramp of death.

Like the funnel concept. To predict how well your business will do, measure how many potential clients are aware of it (awareness) and how many take the next step. If 1,000 people heard about your offering and 10% showed interest, you'd have 100 at that point. If 50% of these people made buyer-like noises, you'd know how many were, etc. It helped model buying trends.

TV, magazine, and radio advertising are pricey for B2C enterprises. Traditional B2B marketing involved armies of sales reps, which was expensive and a barrier to entry.

Cracks in the funnel model

Digital has exposed the funnel's limitations. Hubspot was born at a time when buyers and sellers had huge knowledge asymmetries, according to co-founder Brian Halligan. Those selling a product could use the buyer's lack of information to become a trusted partner.

As the world went digital, getting information and comparing offerings became faster, easier, and cheaper. Buyers didn't need a seller to move through a funnel. Interactions replaced transactions, and the relationship didn't end with a sale.

Instead, buyers and sellers interacted in a constant flow. In many modern models, the sale is midway through the process (particularly true with subscription and software-as-a-service models). Example:

You're creating a winding journey with many touch points, not a funnel (and lots of opportunities for customers to get lost).

From winding journey to flywheel

Beyond this revised view of an interactive customer journey, a company can create what Jim Collins famously called a flywheel. Imagine rolling a heavy disc on its axis. The first few times you roll it, you put in a lot of effort for a small response. The same effort yields faster turns as it gains speed. Over time, the flywheel gains momentum and turns without your help.

Modern digital organizations have created flywheel business models, in which any additional force multiplies throughout the business. The flywheel becomes a force multiplier, according to Collins.

Amazon is a famous flywheel example. Collins explained the concept to Amazon CEO Jeff Bezos at a corporate retreat in 2001. In The Everything Store, Brad Stone describes in his book The Everything Store how he immediately understood Amazon's levers.

The result (drawn on a napkin):

Low prices and a large selection of products attracted customers, while a focus on customer service kept them coming back, increasing traffic. Third-party sellers then increased selection. Low-cost structure supports low-price commitment. It's brilliant! Every wheel turn creates acceleration.

Where from here?

Flywheel over sales funnel! Consider these business terms.