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

It's cheaper, faster charging, longer lasting, safer, and better for the environment.

Batteries are the future. Next-gen and planet-saving technology, including solar power and EVs, require batteries. As these smart technologies become more popular, we find that our batteries can't keep up. Lithium-ion batteries are expensive, slow to charge, big, fast to decay, flammable, and not environmentally friendly. MIT just created a new battery that eliminates all of these problems.  So, is this the battery of the future? Or is there a catch?

When I say entirely new, I mean it. This battery employs no currently available materials. Its electrodes are constructed of aluminium and pure sulfur instead of lithium-complicated ion's metals and graphite. Its electrolyte is formed of molten chloro-aluminate salts, not an organic solution with lithium salts like lithium-ion batteries.

How does this change in materials help?

Aluminum, sulfur, and chloro-aluminate salts are abundant, easy to acquire, and cheap. This battery might be six times cheaper than a lithium-ion battery and use less hazardous mining. The world and our wallets will benefit.

But don’t go thinking this means it lacks performance.

This battery charged in under a minute in tests. At 25 degrees Celsius, the battery will charge 25 times slower than at 110 degrees Celsius. This is because the salt, which has a very low melting point, is in an ideal state at 110 degrees and can carry a charge incredibly quickly. Unlike lithium-ion, this battery self-heats when charging and discharging, therefore no external heating is needed.

Anyone who's seen a lithium-ion battery burst might be surprised. Unlike lithium-ion batteries, none of the components in this new battery can catch fire. Thus, high-temperature charging and discharging speeds pose no concern.

These batteries are long-lasting. Lithium-ion batteries don't last long, as any iPhone owner can attest. During charging, metal forms a dendrite on the electrode. This metal spike will keep growing until it reaches the other end of the battery, short-circuiting it. This is why phone batteries only last a few years and why electric car range decreases over time. This new battery's molten salt slows deposition, extending its life. This helps the environment and our wallets.

These batteries are also energy dense. Some lithium-ion batteries have 270 Wh/kg energy density (volume and mass). Aluminum-sulfur batteries could have 1392 Wh/kg, according to calculations. They'd be 5x more energy dense. Tesla's Model 3 battery would weigh 96 kg instead of 480 kg if this battery were used. This would improve the car's efficiency and handling.

These calculations were for batteries without molten salt electrolyte. Because they don't reflect the exact battery chemistry, they aren't a surefire prediction.

This battery seems great. It will take years, maybe decades, before it reaches the market and makes a difference. Right?

Nope. The project's scientists founded Avanti to develop and market this technology.

So we'll soon be driving cheap, durable, eco-friendly, lightweight, and ultra-safe EVs? Nope.

This battery must be kept hot to keep the salt molten; otherwise, it won't work and will expand and contract, causing damage. This issue could be solved by packs that can rapidly pre-heat, but that project is far off.

Rapid and constant charge-discharge cycles make these batteries ideal for solar farms, homes, and EV charging stations. The battery is constantly being charged or discharged, allowing it to self-heat and maintain an ideal temperature.

These batteries aren't as sexy as those making EVs faster, more efficient, and cheaper. Grid batteries are crucial to our net-zero transition because they allow us to use more low-carbon energy. As we move away from fossil fuels, we'll need millions of these batteries, so the fact that they're cheap, safe, long-lasting, and environmentally friendly will be huge. Who knows, maybe EVs will use this technology one day. MIT has created another world-changing technology.

WRITE FOR THE WEB3

Why web3’s messaging is lost and how crypto winter is growing growth seeds

People interested in crypto, blockchain, and web3 typically read Bitcoin and Ethereum's white papers. It's a good idea. Documents produced for developers and academia aren't always the ideal resource for beginners.

Given the surge of extremely technical material and the number of fly-by-nights, rug pulls, and other scams, it's little wonder mainstream audiences regard the blockchain sector as an expensive sideshow act.

What's the solution?

Web3 needs more than just builders.

After joining TikTok, I followed Amy Suto of SutoScience. Amy switched from TV scriptwriting to IT copywriting years ago. She concentrates on web3 now. Decentralized autonomous organizations (DAOs) are seeking skilled copywriters for web3.

Amy has found that web3's basics are easy to grasp; you don't need technical knowledge. There's a paradigm shift in knowing the basics; be persistent and patient.

Apple is positioning itself as a data privacy advocate, leveraging web3's zero-trust ethos on data ownership.

Finn Lobsien, who writes about web3 copywriting for the Mirror and Twitter, agrees: acronyms and abstractions won't do.

Web3 preached to the choir. Curious newcomers have only found whitepapers and scams when trying to learn why the community loves it. No wonder people resist education and buy-in.

Due to the gender gap in crypto (Crypto Bro is not just a stereotype), it attracts people singing to the choir or trying to cash in on the next big thing.

Last year, the industry was booming, so writing wasn't necessary. Now that the bear market has returned (for everyone, but especially web3), holding readers' attention is a valuable skill.

White papers and the Web3

Why does web3 rely so much on non-growth content?

Businesses must polish and improve their messaging moving into the 2022 recession. The 2021 tech boom provided such a sense of affluence and (unsustainable) growth that no one needed great marketing material. The market found them.

This was especially true for web3 and the first-time crypto believers. Obviously. If they knew which was good.

White papers help. White papers are highly technical texts that walk a reader through a product's details. How Does a White Paper Help Your Business and That White Paper Guy discuss them.

They're meant for knowledgeable readers. Investors and the technical (academic/developer) community read web3 white papers. White papers are used when a product is extremely technical or difficult to assist an informed reader to a conclusion. Web3 uses them most often for ICOs (initial coin offerings).

White papers for web3 education help newcomers learn about the web3 industry's components. It's like sending a first-grader to the Annotated Oxford English Dictionary to learn to read. It's a reference, not a learning tool, for words.

Newcomers can use platforms that teach the basics. These included Coinbase's Crypto Basics tutorials or Cryptochicks Academy, founded by the mother of Ethereum's inventor to get more women utilizing and working in crypto.

Discord and Web3 communities

Discord communities are web3's opposite. Discord communities involve personal communications and group involvement.

Online audience growth begins with community building. User personas prefer 1000 dedicated admirers over 1 million lukewarm followers, and the language is much more easygoing. Discord groups are renowned for phishing scams, compromised wallets, and incorrect information, especially since the crypto crisis.

White papers and Discord increase industry insularity. White papers are complicated, and Discord has a high risk threshold.

Web3 and writing ads

Copywriting is emotional, but white papers are logical. It uses the brain's quick-decision centers. It's meant to make the reader invest immediately.

Not bad. People think sales are sleazy, but they can spot the poor things.

Ethical copywriting helps you reach the correct audience. People who gain a following on Medium are likely to have copywriting training and a readership (or three) in mind when they publish. Tim Denning and Sinem Günel know how to identify a target audience and make them want to learn more.

In a fast-moving market, copywriting is less about long-form content like sales pages or blogs, but many organizations do. Instead, the copy is concise, individualized, and high-value. Tweets, email marketing, and IM apps (Discord, Telegram, Slack to a lesser extent) keep engagement high.

What does web3's messaging lack? As DAOs add stricter copyrighting, narrative and connecting tales seem to be missing.

Web3 is passionate about constructing the next internet. Now, they can connect their passion to a specific audience so newcomers understand why.

What Is That?

Berkus is a pre-revenue valuation method based exclusively on qualitative criteria, like Scorecard.

Few firms match their financial estimates, especially in the early stages, so valuation methodologies like the Berkus method are a good way to establish a valuation when the economic measures are not reliable.

How does it work?

This technique evaluates five key success factors.

• Fundamental principle

• Technology

• Execution

• Strategic alliances in its primary market

• Production, followed by sales

The Berkus technique values the business idea and four success factors. As seen in the matrix below, each of these dimensions poses a danger to the startup's success.

It assigns $0-$500,000 to each of these beginning regions. This approach enables a maximum $2.5M pre-money valuation.

This approach relies significantly on geography and uses the US as a baseline, as it differs in every country in Europe.

A set of standards for analyzing each dimension individually

Fundamental principle (or strength of the idea)

Ideas are worthless; execution matters. Most of us can relate to seeing a new business open in our area or a startup get funded and thinking, "I had this concept years ago!" Someone did it.

The concept remains. To assess the idea's viability, we must consider several criteria.

• The concept's exclusivity It is necessary to protect a product or service's concept using patents and copyrights. Additionally, it must be capable of generating large profits.

• Planned growth and growth that goes in a specific direction have a lot of potential, therefore incorporating them into a business is really advantageous.

• The ability of a concept to grow A venture's ability to generate scalable revenue is a key factor in its emergence and continuation. A startup needs a scalable idea in order to compete successfully in the market.

• The attraction of a business idea to a broad spectrum of people is significantly influenced by the current socio-political climate. Thus, the requirement for the assumption of conformity.

• Concept Validation Ideas must go through rigorous testing with a variety of audiences in order to lower risk during the implementation phase.

Technology (Prototype)

This aspect reduces startup's technological risk. How good is the startup prototype when facing cyber threats, GDPR compliance (in Europe), tech stack replication difficulty, etc.?

Execution

Check the management team's efficacy. A potential angel investor must verify the founders' experience and track record with previous ventures. Good leadership is needed to chart a ship's course.

Strategic alliances in its primary market

Existing and new relationships will play a vital role in the development of both B2B and B2C startups. What are the startup's synergies? potential ones?

Production, followed by sales (product rollout)

Startup success depends on its manufacturing and product rollout. It depends on the overall addressable market, the startup's ability to market and sell their product, and their capacity to provide consistent, high-quality support.

Example

We're now founders of EyeCaramba, a machine vision-assisted streaming platform. My imagination always goes to poor puns when naming a startup.

Since we're first-time founders and the Berkus technique depends exclusively on qualitative methods and the evaluator's skill, we ask our angel-investor acquaintance for a pre-money appraisal of EyeCaramba.

Our friend offers us the following table:

Because we're first-time founders, our pal lowered our Execution score. He knows the idea's value and that the gaming industry is red-hot, with worse startup ideas getting funded, therefore he gave the Basic value the highest value (idea).

EyeCaramba's pre-money valuation is $400,000 + $250,000 + $75,000 + $275,000 + $164,000 (1.16M). Good.

References

• https://medium.com/humble-ventures/how-angel-investors-value-pre-revenue-startups-part-iii-8271405f0774#:~:text=pre%2Drevenue%20startups.-,Berkus%20Method,potential%20of%20the%20idea%20itself.%E2%80%9D

• https://eqvista.com/berkus-valuation-method-for-startups/

• https://www.venionaire.com/early-stage-startup-valuation-part-2-the-berkus-method/