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.

Canonicalization and canonical URLs are essential for SEO, and improper implementation can negatively impact your site's performance.

Canonical tags were introduced in 2009 to help webmasters with duplicate or similar content on multiple URLs.

To use canonical tags properly, you must understand their purpose, operation, and implementation.

Canonical URLs and Tags

Canonical tags tell search engines that a certain URL is a page's master copy. They specify a page's canonical URL. Webmasters can avoid duplicate content by linking to the "canonical" or "preferred" version of a page.

How are canonical tags and URLs different? Can these be specified differently?

Tags

Canonical tags are found in an HTML page's head></head> section.

<link rel="canonical" href="https://www.website.com/page/" />

These can be self-referencing or reference another page's URL to consolidate signals.

Canonical tags and URLs are often used interchangeably, which is incorrect.

The rel="canonical" tag is the most common way to set canonical URLs, but it's not the only way.

Canonical URLs

What's a canonical link? Canonical link is the'master' URL for duplicate pages.

In Google's own words:

A canonical URL is the page Google thinks is most representative of duplicate pages on your site.

— Google Search Console Help

You can indicate your preferred canonical URL. For various reasons, Google may choose a different page than you.

When set correctly, the canonical URL is usually your specified URL.

Canonical URLs determine which page will be shown in search results (unless a duplicate is explicitly better for a user, like a mobile version).

Canonical URLs can be on different domains.

Other ways to specify canonical URLs

Canonical tags are the most common way to specify a canonical URL.

You can also set canonicals by:

• Setting the HTTP header rel=canonical.

• All pages listed in a sitemap are suggested as canonicals, but Google decides which pages are duplicates.

• Redirects 301.

Google recommends these methods, but they aren't all appropriate for every situation, as we'll see below. Each has its own recommended uses.

Setting canonical URLs isn't required; if you don't, Google will use other signals to determine the best page version.

To control how your site appears in search engines and to avoid duplicate content issues, you should use canonicalization effectively.

Why Duplicate Content Exists

Before we discuss why you should use canonical URLs and how to specify them in popular CMSs, we must first explain why duplicate content exists. Nobody intentionally duplicates website content.

Content management systems create multiple URLs when you launch a page, have indexable versions of your site, or use dynamic URLs.

Assume the following URLs display the same content to a user:

1. https://www.website.com/category/product-a/

2. https://www.website.com/product-a/

3. https://website.com/product-a/

4. http://www.website.com/product-a/

5. http://website.com/product-a/

6. https://m.website.com/product-a/

7. https://www.website.com/product-a

8. https://www.website.com/product-A/

A search engine sees eight duplicate pages, not one.

• URLs #1 and #2: the CMS saves product URLs with and without the category name.

• #3, #4, and #5 result from the site being accessible via HTTP, HTTPS, www, and non-www.

• #6 is a subdomain mobile-friendly URL.

• URL #7 lacks URL #2's trailing slash.

• URL #8 uses a capital "A" instead of a lowercase one.

Duplicate content may also exist in URLs like:

https://www.website.com
https://www.website.com/index.php

Duplicate content is easy to create.

Canonical URLs help search engines identify different page variations as a single URL on many sites.

SEO Canonical URLs

Canonical URLs help you manage duplicate content that could affect site performance.

Canonical URLs are a technical SEO focus area for many reasons.

Specify URL for search results

When you set a canonical URL, you tell Google which page version to display.

Which would you click?

https://www.domain.com/page-1/

https://www.domain.com/index.php?id=2

First, probably.

Canonicals tell search engines which URL to rank.

Consolidate link signals on similar pages

When you have duplicate or nearly identical pages on your site, the URLs may get external links.

Canonical URLs consolidate multiple pages' link signals into a single URL.

This helps your site rank because signals from multiple URLs are consolidated into one.

Syndication management

Content is often syndicated to reach new audiences.

Canonical URLs consolidate ranking signals to prevent duplicate pages from ranking and ensure the original content ranks.

Avoid Googlebot duplicate page crawling

Canonical URLs ensure that Googlebot crawls your new pages rather than duplicated versions of the same one across mobile and desktop versions, for example.

Crawl budgets aren't an issue for most sites unless they have 100,000+ pages.

How to Correctly Implement the rel=canonical Tag

Using the header tag rel="canonical" is the most common way to specify canonical URLs.

Adding tags and HTML code may seem daunting if you're not a developer, but most CMS platforms allow canonicals out-of-the-box.

These URLs each have one product.

How to Correctly Implement a rel="canonical" HTTP Header

A rel="canonical" HTTP header can replace canonical tags.

This is how to implement a canonical URL for PDFs or non-HTML documents.

You can specify a canonical URL in your site's.htaccess file using the code below.

<Files "file-to-canonicalize.pdf"> Header add Link "< http://www.website.com/canonical-page/>; rel=\"canonical\"" </Files>

301 redirects for canonical URLs

Google says 301 redirects can specify canonical URLs.

Only the canonical URL will exist if you use 301 redirects. This will redirect duplicates.

This is the best way to fix duplicate content across:

• HTTPS and HTTP

• Non-WWW and WWW

• Trailing-Slash and Non-Trailing Slash URLs

On a single page, you should use canonical tags unless you can confidently delete and redirect the page.

Sitemaps' canonical URLs

Google assumes sitemap URLs are canonical, so don't include non-canonical URLs.

This does not guarantee canonical URLs, but is a best practice for sitemaps.

Best-practice Canonical Tag

Once you understand a few simple best practices for canonical tags, spotting and cleaning up duplicate content becomes much easier.

Always include:

One canonical URL per page

If you specify multiple canonical URLs per page, they will likely be ignored.

Correct Domain Protocol

If your site uses HTTPS, use this as the canonical URL. It's easy to reference the wrong protocol, so check for it to catch it early.

Trailing slash or non-trailing slash URLs

Be sure to include trailing slashes in your canonical URL if your site uses them.

Specify URLs other than WWW

Search engines see non-WWW and WWW URLs as duplicate pages, so use the correct one.

Absolute URLs

To ensure proper interpretation, canonical tags should use absolute URLs.

So use:

<link rel="canonical" href="https://www.website.com/page-a/" />

And not:

<link rel="canonical" href="/page-a/" />

If not canonicalizing, use self-referential canonical URLs.

When a page isn't canonicalizing to another URL, use self-referencing canonical URLs.

Canonical tags refer to themselves here.

Common Canonical Tags Mistakes

Here are some common canonical tag mistakes.

301 Canonicalization

Set the canonical URL as the redirect target, not a redirected URL.

Incorrect Domain Canonicalization

If your site uses HTTPS, don't set canonical URLs to HTTP.

Irrelevant Canonicalization

Canonicalize URLs to duplicate or near-identical content only.

SEOs sometimes try to pass link signals via canonical tags from unrelated content to increase rank. This isn't how canonicalization should be used and should be avoided.

Multiple Canonical URLs

Only use one canonical tag or URL per page; otherwise, they may all be ignored.

When overriding defaults in some CMSs, you may accidentally include two canonical tags in your page's <head>.

Pagination vs. Canonicalization

Incorrect pagination can cause duplicate content. Canonicalizing URLs to the first page isn't always the best solution.

Canonicalize to a 'view all' page.

How to Audit Canonical Tags (and Fix Issues)

Audit your site's canonical tags to find canonicalization issues.

SEMrush Site Audit can help. You'll find canonical tag checks in your website's site audit report.

Let's examine these issues and their solutions.

No Canonical Tag on AMP

Site Audit will flag AMP pages without canonical tags.

Canonicalization between AMP and non-AMP pages is important.

Add a rel="canonical" tag to each AMP page's head>.

No HTTPS redirect or canonical from HTTP homepage

Duplicate content issues will be flagged in the Site Audit if your site is accessible via HTTPS and HTTP.

You can fix this by 301 redirecting or adding a canonical tag to HTTP pages that references HTTPS.

Broken canonical links

Broken canonical links won't be considered canonical URLs.

This error could mean your canonical links point to non-existent pages, complicating crawling and indexing.

Update broken canonical links to the correct URLs.

Multiple canonical URLs

This error occurs when a page has multiple canonical URLs.

Remove duplicate tags and leave one.

Canonicalization is a key SEO concept, and using it incorrectly can hurt your site's performance.

Once you understand how it works, what it does, and how to find and fix issues, you can use it effectively to remove duplicate content from your site.

Canonicalization SEO Myths

After viewing its insanity, I had to leave this platform.

I joined LinkedIn recently. That's how I aim to increase my readership and gain recognition. LinkedIn's premise appealed to me: a Facebook-like platform for professional networking.

I don't use Facebook since it's full of propaganda. It seems like a professional, apolitical space, right?

I expected people to:

• be more formal and respectful than on Facebook.

• Talk about the inclusiveness of the workplace. Studies consistently demonstrate that inclusive, progressive workplaces outperform those that adhere to established practices.

• Talk about business in their industry. Yep. I wanted to read articles with advice on how to write better and reach a wider audience.

Oh, sh*t. I hadn't anticipated that.

After posting and reading about inclusivity and pro-choice, I was startled by how many professionals acted unprofessionally. I've seen:

• Men have approached me in the DMs in a really aggressive manner. Yikes. huge yikes Not at all professional.

• I've heard pro-choice women referred to as infant killers by many people. If I were the CEO of a company and I witnessed one of my employees acting that poorly, I would immediately fire them.

• Many posts are anti-LGBTQIA+, as I've noticed. a lot, like, a lot. Some are subtly stating that the world doesn't need to know, while others are openly making fun of transgender persons like myself.

• Several medical professionals were posting explicitly racist comments. Even if you are as white as a sheet like me, you should be alarmed by this. Who's to guarantee a patient who is black won't unintentionally die?

• I won't even get into how many men in STEM I observed pushing for the exclusion of women from their fields. I shouldn't be surprised considering the majority of those men I've encountered have a passionate dislike for women, but goddamn, dude.

Many people appear entirely too at ease displaying their bigotry on their professional profiles.

As a white female, I'm always shocked by people's open hostility. Professional environments are very important.

I don't know if this is still true (people seem too politicized to care), but if I heard many of these statements in person, I'd suppose they feel ashamed. Really.

Are you not ashamed of being so mean? Are you so weak that competing with others terrifies you? Isn't this embarrassing?

LinkedIn isn't great at censoring offensive comments. These people aren't getting warnings. So they were safe while others were unsafe.

The CEO in me would want to know if I had placed a bigot on my staff.

I always wondered if people's employers knew about their online behavior. If they know how horrible they appear, they don't care.

As a manager, I was picky about hiring. Obviously. In most industries, it costs $1,000 or more to hire a full-time employee, so be sure it pays off.

Companies that embrace diversity and tolerance (and are intolerant of intolerance) are more profitable, likely to recruit top personnel, and successful.

People avoid businesses that alienate them. That's why I don't eat at Chic-Fil-A and why folks avoid MyPillow. Being inclusive is good business.

CEOs are harmed by online bigots. Image is an issue. If you're a business owner, you can fire staff who don't help you.

On the one hand, I'm delighted it makes it simpler to identify those with whom not to do business.

Don’t get me wrong. I'm glad I know who to avoid when hiring, getting references, or searching for a job. When people are bad, it saves me time.

What's up with professionalism?

Really. I need to know. I've crossed the boundary between acceptable and unacceptable behavior, but never on a professional platform. I got in trouble for not wearing bras even though it's not part of my gender expression.

If I behaved like that at my last two office jobs, my supervisors would have fired me immediately. Some of the behavior I've seen is so outrageous, I can't believe these people have employment. Some are even leaders.

Like…how? Is hatred now normalized?

Please pay attention whether you're seeking for a job or even simply a side gig.

Do not add to the tragedy that LinkedIn comments can be, or at least don't make uninformed comments. Even if you weren't banned, the site may still bite you.

Recruiters can and do look at your activity. Your writing goes on your résumé. The wrong comment might lose you a job.

Recruiters and CEOs might reject candidates whose principles contradict with their corporate culture. Bigotry will get you banned from many companies, especially if others report you.

If you want a high-paying job, avoid being a LinkedIn asshole. People care even if you think no one does. Before speaking, ponder. Is this how you want to be perceived?

Better advice:

If your politics might turn off an employer, stop posting about them online and ask yourself why you hold such objectionable ideas.

After 10 years of working in scale-ups, I've embraced a few concepts for scaling Tech and Product teams.

First, cross-functionalize teams. Product Managers represent the business, Product Designers the consumer, and Engineers build.

I organize teams of 5-10 individuals, following AWS's two pizza teams guidelines, with a Product Trio guiding each.

If more individuals are needed to reach a goal, I group teams under a Product Trio.

With Engineering being the biggest group, Staff/Principal Engineers often support the Trio on cross-team technical decisions.

Product Managers, Engineering Managers, or Engineers in the team may manage projects (depending on the project or aim), but the trio is collectively responsible for the team's output and outcome.

Once the Product Trio model is created, roles, duties, team ceremonies, and cooperation models must be clarified.

Keep reporting lines by discipline. Line managers are accountable for each individual's advancement, thus it's crucial that they know the work in detail.

Cross-team collaboration becomes more important after 3 teams (15-30 people). Teams can easily diverge in how they write code, run ceremonies, and build products.

Establishing groups of people that are cross-team, but grouped by discipline and skills, sharing and agreeing on working practices becomes critical.

The “Spotify Guild” model has been where I’ve taken a lot of my inspiration from.

Last, establish a taxonomy for communication channels.

In Slack, I create one channel per team and one per guild (and one for me to have discussions with the team leads).

These are just some of the basic principles I follow to organize teams.

A book I particularly like about team types and how they interact with each other is https://teamtopologies.com/.

It's ok and even desired to be unprofitable if you're gaining revenue at a reasonable cost and have 100%+ net dollar retention, meaning you never lose customers and expand them. To estimate the acceptable cost of new SaaS revenue, we compare new revenue to operating loss and payback period. If you pay back the customer acquisition cost in 1.5 years and never lose them (100%+ NDR), you're doing well.

To evaluate payback period, we compared new revenue to net operating loss for the last 73 SaaS companies to IPO since October 2017. (55 out of 73). Here's the data. 1/(new revenue/operating loss) equals payback period. New revenue/operating loss equals cost of new revenue.

Payback averages a year. 55 SaaS companies that weren't profitable at IPO got a 1-year payback. Outstanding. If you pay for a customer in a year and never lose them (100%+ NDR), you're establishing a valuable business. The average was 1.3 years, which is within the 1.5-year range.

New revenue costs $0.96 on average. These SaaS companies lost $0.96 every $1 of new revenue last year. Again, impressive. Average new revenue per operating loss was $1.59.

Loss-in-operations definition. Operating loss revenue COGS S&M R&D G&A (technical point: be sure to use the absolute value of operating loss). It's wrong to only consider S&M costs and ignore other business costs. Operating loss and new revenue are measured over one year to eliminate seasonality.

Operating losses are desirable if you never lose a customer and have a quick payback period, especially when SaaS enterprises are valued on ARR. The payback period should be under 1.5 years, the cost of new income < $1, and net dollar retention 100%.

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.