Never design a product without explicit metrics and tracking tools.

Imagine driving cross-country without a dashboard. How do you know your school zone speed? Low gas? Without a dashboard, you can't monitor your car. You can't improve what you don't measure, as Peter Drucker said. Product managers must constantly enhance their understanding of their users, how they use their product, and how to improve it for optimum value. Customers will only pay if they consistently acquire value from your product.

I’m Solomon Ayanlakin. I’m a product manager at CredPal, a financial business that offers credit cards and Buy Now Pay Later services. Before falling into product management (like most PMs lol), I self-trained as a data analyst, using Alex the Analyst's YouTube playlists and DannyMas' virtual data internship. This article aims to help product managers, owners, and CXOs understand product metrics, give a methodology for creating them, and execute product experiments to enhance them.

☝🏽Introduction

Product metrics assist companies track product performance from the user's perspective. Metrics help firms decide what to construct (feature priority), how to build it, and the outcome's success or failure. To give the best value to new and existing users, track product metrics.

Why should a product manager monitor metrics?

• to assist your users in having a "aha" moment

• To inform you of which features are frequently used by users and which are not

• To assess the effectiveness of a product feature

• To aid in enhancing client onboarding and retention

• To assist you in identifying areas throughout the user journey where customers are satisfied or dissatisfied

• to determine the percentage of returning users and determine the reasons for their return

📈 What Metrics Ought a Product Manager to Monitor?

What indicators should a product manager watch to monitor product health? The metrics to follow change based on the industry, business stage (early, growth, late), consumer needs, and company goals. A startup should focus more on conversion, activation, and active user engagement than revenue growth and retention. The company hasn't found product-market fit or discovered what features drive customer value.

Depending on your use case, company goals, or business stage, here are some important product metric buckets:

All measurements shouldn't be used simultaneously. It depends on your business goals and what value means for your users, then selecting what metrics to track to see if they get it.

Some KPIs are more beneficial to track, independent of industry or customer type. To prevent recording vanity metrics, product managers must clearly specify the types of metrics they should track. Here's how to segment metrics:

1. The North Star Metric, also known as the Focus Metric, is the indicator and aid in keeping track of the top value you provide to users.

2. Primary/Level 1 Metrics: These metrics should either add to the north star metric or be used to determine whether it is moving in the appropriate direction. They are metrics that support the north star metric.

3. These measures serve as leading indications for your north star and Level 2 metrics. You ought to have been aware of certain problems with your L2 measurements prior to the North star metric modifications.

North Star Metric

This is the key metric. A good north star metric measures customer value. It emphasizes your product's longevity. Many organizations fail to grow because they confuse north star measures with other indicators. A good focus metric should touch all company teams and be tracked forever. If a company gives its customers outstanding value, growth and success are inevitable. How do we measure this value?

A north star metric has these benefits:

• Customer Obsession: It promotes a culture of customer value throughout the entire organization.

• Consensus: Everyone can quickly understand where the business is at and can promptly make improvements, according to consensus.

• Growth: It provides a tool to measure the company's long-term success. Do you think your company will last for a long time?

How can I pick a reliable North Star Metric?

Some fear a single metric. Ensure product leaders can objectively determine a north star metric. Your company's focus metric should meet certain conditions. Here are a few:

1. A good focus metric should reflect value and, as such, should be closely related to the point at which customers obtain the desired value from your product. For instance, the quick delivery to your home is a value proposition of UberEats. The value received from a delivery would be a suitable focal metric to use. While counting orders is alluring, the quantity of successfully completed positive review orders would make a superior north star statistic. This is due to the fact that a client who placed an order but received a defective or erratic delivery is not benefiting from Uber Eats. By tracking core value gain, which is the number of purchases that resulted in satisfied customers, we are able to track not only the total number of orders placed during a specific time period but also the core value proposition.
   

2. Focus metrics need to be quantifiable; they shouldn't only be feelings or states; they need to be actionable. A smart place to start is by counting how many times an activity has been completed.
   

3. A great focus metric is one that can be measured within predetermined time limits; otherwise, you are not measuring at all. The company can improve that measure more quickly by having time-bound focus metrics. Measuring and accounting for progress over set time periods is the only method to determine whether or not you are moving in the right path. You can then evaluate your metrics for today and yesterday. It's generally not a good idea to use a year as a time frame. Ideally, depending on the nature of your organization and the measure you are focusing on, you want to take into account on a daily, weekly, or monthly basis.
   

4. Everyone in the firm has the potential to affect it: A short glance at the well-known AAARRR funnel, also known as the Pirate Metrics, reveals that various teams inside the organization have an impact on the funnel. Ideally, the NSM should be impacted if changes are made to one portion of the funnel. Consider how the growth team in your firm is enhancing customer retention. This would have a good effect on the north star indicator because at this stage, a repeat client is probably being satisfied on a regular basis. Additionally, if the opposite were true and a client churned, it would have a negative effect on the focus metric.
   

5. It ought to be connected to the business's long-term success: The direction of sustainability would be indicated by a good north star metric. A company's lifeblood is product demand and revenue, so it's critical that your NSM points in the direction of sustainability. If UberEats can effectively increase the monthly total of happy client orders, it will remain in operation indefinitely.

Many product teams make the mistake of focusing on revenue. When the bottom line is emphasized, a company's goal moves from giving value to extracting money from customers. A happy consumer will stay and pay for your service. Customer lifetime value always exceeds initial daily, monthly, or weekly revenue.

Great North Star Metrics Examples

🥇 Basic/L1 Metrics:

The NSM is broad and focuses on providing value for users, while the primary metric is product/feature focused and utilized to drive the focus metric or signal its health. The primary statistic is team-specific, whereas the north star metric is company-wide. For UberEats' NSM, the marketing team may measure the amount of quality food vendors who sign up using email marketing. With quality vendors, more orders will be satisfied. Shorter feedback loops and unambiguous team assignments make L1 metrics more actionable and significant in the immediate term.

🥈 Supporting L2 metrics:

These are supporting metrics to the L1 and focus metrics. Location, demographics, or features are examples of L1 metrics. UberEats' supporting metrics might be the number of sales emails sent to food vendors, the number of opens, and the click-through rate. Secondary metrics are low-level and evident, and they relate into primary and north star measurements. UberEats needs a high email open rate to attract high-quality food vendors. L2 is a leading sign for L1.

Where can I find product metrics?

How can I measure in-app usage and activity now that I know what metrics to track? Enter product analytics. Product analytics tools evaluate and improve product management parameters that indicate a product's health from a user's perspective.

Various analytics tools on the market supply product insight. From page views and user flows through A/B testing, in-app walkthroughs, and surveys. Depending on your use case and necessity, you may combine tools to see how users engage with your product. Gainsight, MixPanel, Amplitude, Google Analytics, FullStory, Heap, and Pendo are product tools.

This article isn't sponsored and doesn't market product analytics tools. When choosing an analytics tool, consider the following:

• Tools for tracking your Focus, L1, and L2 measurements

• Pricing

• Adaptations to include external data sources and other products

• Usability and the interface

• Scalability

• Security

An investment in the appropriate tool pays off. To choose the correct metrics to track, you must first understand your business need and what value means to your users. Metrics and analytics are crucial for any tech product's growth. It shows how your business is doing and how to best serve users.

Product management wizardry.

Early adopters buy a product even if it hasn't hit the market or has flaws.

Who are the early adopters?

Early adopters try a new technology or product first. Early adopters are interested in trying or buying new technologies and products before others. They're risk-tolerant and can provide initial cash flow and product reviews. They help a company's new product or technology gain social proof.

Early adopters are most common in the technology industry, but they're in every industry. They don't follow the crowd. They seek innovation and report product flaws before mass production. If the product works well, the first users become loyal customers, and colleagues value their opinion.

What to do with early adopters?

They can be used to collect feedback and initial product promotion, first sales, and product value validation.

How to find early followers?

Start with your immediate environment and target audience. Communicate with them to see if they're interested in your value proposition.

1) Innovators (2.5% of the population) are risk-takers seeking novelty. These people are the first to buy new and trendy items and drive social innovation. However, these people are usually elite;

Early adopters (13.5%) are inclined to accept innovations but are more cautious than innovators; they start using novelties when innovators or famous people do;

3) The early majority (34%) is conservative; they start using new products when many people have mastered them. When the early majority accepted the innovation, it became ingrained in people's minds.

4) Attracting 34% of the population later means the novelty has become a mass-market product. Innovators are using newer products;

5) Laggards (16%) are the most conservative, usually elderly people who use the same products.

Stages of new information acceptance

1. The information is strange and rejected by most. Accepted only by innovators;

2. When early adopters join, more people believe it's not so bad; when a critical mass is reached, the novelty becomes fashionable and most people use it.

3. Fascination with a novelty peaks, then declines; the majority and laggards start using it later; novelty becomes obsolete; innovators master something new.

Problems with early implementation

Early adopter sales have disadvantages.

Higher risk of defects

Selling to first-time users increases the risk of defects. Early adopters are often influential, so this can affect the brand's and its products' long-term perception.

Not what was expected

First-time buyers may be disappointed by the product. Marketing messages can mislead consumers, and if the first users believe the company misrepresented the product, this will affect future sales.

Compatibility issues

Some technological advances cause compatibility issues. Consumers may be disappointed if new technology is incompatible with their electronics.

Method 5 WHY

Let's talk about 5 why, a good tool for finding project problems' root causes. This method is also known as the five why rule, method, or questions.

The 5 why technique came from Toyota's lean manufacturing and helps quickly determine a problem's root cause.

On one, two, and three, you simply do this:

1. We identify and frame the issue for which a solution is sought.

2. We frequently ponder this question. The first 2-3 responses are frequently very dull, making you want to give up on this pointless exercise. However, after that, things get interesting. And occasionally it's so fascinating that you question whether you really needed to know.

3. We consider the final response, ponder it, and choose a course of action.

Always do the 5 whys with the customer or team to have a reasonable discussion and better understand what's happening.

And the “five whys” is a wonderful and simplest tool for introspection. With the accumulated practice, it is used almost automatically in any situation like “I can’t force myself to work, the mood is bad in the morning” or “why did I decide that I have no life without this food processor for 20,000 rubles, which will take half of my rather big kitchen.”

An illustration of the five whys

A simple, but real example from my work practice that I think is very indicative, given the participants' low IT skills.  Anonymized, of course.

Users spend too long looking for tender documents.

Why? Because they must search through many company tender documents.

Why? Because the system can't filter department-specific bids.

Why? Because our contract management system requirements didn't include a department-tender link. That's it, right? We'll add a filter and be happy. but still…

why? Because we based the system's requirements on regulations for working with paper tender documents (when they still had envelopes and autopsies), not electronic ones, and there was no search mechanism.

Why? We didn't consider how our work would change when switching from paper to electronic tenders when drafting the requirements.

Now I know what to do in the future. We add a filter, enter department data, and teach users to use it. This is tactical, but strategically we review the same forgotten requirements to make all the necessary changes in a package, plus we include it in the checklist for the acceptance of final requirements for the future.

Errors when using 5 why

Five whys seems simple, but it can be misused.

Popular ones:

1. The accusation of everyone and everything is then introduced. After all, the 5 why method focuses on identifying the underlying causes rather than criticizing others. As a result, at the third step, it is not a good idea to conclude that the system is ineffective because users are stupid and that we can therefore do nothing about it.

2. to fight with all my might so that the outcome would be exactly 5 reasons, neither more nor less. 5 questions is a typical number (it sounds nice, yes), but there could be 3 or 7 in actuality.

3. Do not capture in-between responses. It is difficult to overestimate the power of the written or printed word, so the result is so-so when the focus is lost. That's it, I suppose. Simple, quick, and brilliant, like other project management tools.

Conclusion

Today we analyzed important study elements:

Early adopters and 5 WHY We've analyzed cases and live examples of how these methods help with product research and growth point identification. Next, consider the HADI cycle.

10-sketch explanation

Choosing Tasks

Bringing News

carrying out 1:1s

providing critique

Managing Turbulence

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

I've worked in a People role at seven early-stage firms for over 15 years (I enjoy chasing a dream!). Few of the seven achieved profitability, including unicorn status or acquisition.

Did early-stage startups share anything? Was there a difference between winners and losers? YES.

I support founders and entrepreneurs building financially sustainable enterprises with a compelling cause. This isn't something everyone would do. A company's success demands more than guts. Founders drive startup success.

Six Qualities of Successful Startups

Successful startup founders either innately grasped the correlation between strong team engagement and a well-executed business model, or they knew how to ask and listen to others (executive coaches, other company leaders, the team itself) to learn about it.

Successful startups:

1. Co-founders agreed and got along personally.

Multi-founder startups are common. When co-founders agree on strategic decisions and are buddies, there's less friction and politics at work.

As a co-founder, ask your team if you're aligned. They'll explain.

I've seen C-level leaders harbor personal resentments over disagreements. A co-departure founder's caused volatile leadership and work disruptions that the team struggled to manage during and after.

2. Team stayed.

Successful startups have low turnover. Nobody is leaving. There may be a termination for performance, but other team members will have observed the issues and agreed with the decision.

You don't want organizational turnover of 30%+, with leaders citing performance issues but the team not believing them. This breeds suspicion.

Something is wrong if many employees leave voluntarily or involuntarily. You may hear about lack of empowerment, support, or toxic leadership in exit interviews and from the existing team. Intellectual capital loss and resource instability harm success.

3. Team momentum.

A successful startup's team is excited about its progress. Consistently achieving goals and having trackable performance metrics. Some describe this period of productivity as magical, with great talents joining the team and the right people in the right places. Increasing momentum.

I've also seen short-sighted decisions where only some departments, like sales and engineering, had goals. Lack of a unified goals system created silos and miscommunication. Some employees felt apathetic because they didn't know how they contributed to team goals.

4. Employees advanced in their careers.

Even if you haven't created career pathing or professional development programs, early-stage employees will grow and move into next-level roles. If you hire more experienced talent and leaders, expect them to mentor existing team members. Growing companies need good performers.

New talent shouldn't replace and discard existing talent. This creates animosity and makes existing employees feel unappreciated for their early contributions to the company.

5. The company lived its values.

Culture and identity are built on lived values. A company's values affect hiring, performance management, rewards, and other processes. Identify, practice, and believe in company values. Starting with team values instead of management or consultants helps achieve this. When a company's words and actions match, it builds trust.

When company values are beautifully displayed on a wall but few employees understand them, the opposite is true. If an employee can't name the company values, they're useless.

6. Communication was clear.

When necessary information is shared with the team, they feel included, trusted, and like owners. Transparency means employees have the needed information to do their jobs. Disclosure builds trust. The founders answer employees' questions honestly.

Information accessibility decreases office politics. Without transparency, even basic information is guarded and many decisions are made in secret. I've seen founders who don't share financial, board meeting, or compensation and equity information. The founders' lack of trust in the team wasn't surprising, so it was reciprocated.

The Choices

Finally. All six of the above traits (leadership alignment, minimal turnover, momentum, professional advancement, values, and transparency) were high in the profitable startups I've worked at, including unicorn status or acquisition.

I've seen these as the most common and constant signals of startup success or failure.

These characteristics are the product of founders' choices. These decisions lead to increased team engagement and business execution.

Here's something to consider for startup employees and want-to-bes. 90% of startups fail, despite the allure of building something new and gaining ownership. With the emotional and time investment in startup formation, look for startups with these traits to reduce your risk.

Both you and the startup will thrive in these workplaces.

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.