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.

Few sentences can have global significance.

Ethan Zuckerman invented the pop-up commercial in 1997.

He was working for Tripod.com, an online service that let people make little web pages for free. Tripod offered advertising to make money. Advertisers didn't enjoy seeing their advertising next to filthy content, like a user's anal sex website.

Zuckerman's boss wanted a solution. Wasn't there a way to move the ads away from user-generated content?

When you visited a Tripod page, a pop-up ad page appeared. So, the ad isn't officially tied to any user page. It'd float onscreen.

Here’s the thing, though: Zuckerman’s bit of Javascript, that created the popup ad? It was incredibly short — a single line of code:

window.open('http://tripod.com/navbar.html'
"width=200, height=400, toolbar=no, scrollbars=no, resizable=no, target=_top");

Javascript tells the browser to open a 200-by-400-pixel window on top of any other open web pages, without a scrollbar or toolbar.

Simple yet harmful! Soon, commercial websites mimicked Zuckerman's concept, infesting the Internet with pop-up advertising. In the early 2000s, a coder for a download site told me that most of their revenue came from porn pop-up ads.

Pop-up advertising are everywhere. You despise them. Hopefully, your browser blocks them.

Zuckerman wrote a single line of code that made the world worse.

I read Zuckerman's story in How 26 Lines of Code Changed the World. Torie Bosch compiled a humorous anthology of short writings about code that tipped the world.

Most of these samples are quite short. Pop-cultural preconceptions about coding say that important code is vast and expansive. Hollywood depicts programmers as blurs spouting out Niagaras of code. Google's success was formerly attributed to its 2 billion lines of code.

It's usually not true. Google's original breakthrough, the piece of code that propelled Google above its search-engine counterparts, was its PageRank algorithm, which determined a web page's value based on how many other pages connected to it and the quality of those connecting pages. People have written their own Python versions; it's only a few dozen lines.

Google's operations, like any large tech company's, comprise thousands of procedures. So their code base grows. The most impactful code can be brief.

The examples are fascinating and wide-ranging, so read the whole book (or give it to nerds as a present). Charlton McIlwain wrote a chapter on the police beat algorithm developed in the late 1960s to anticipate crime hotspots so law enforcement could dispatch more officers there. It created a racial feedback loop. Since poor Black neighborhoods were already overpoliced compared to white ones, the algorithm directed more policing there, resulting in more arrests, which convinced it to send more police; rinse and repeat.

Kelly Chudler's You Are Not Expected To Understand This depicts the police-beat algorithm.

Even shorter code changed the world: the tracking pixel.

Lily Hay Newman's chapter on monitoring pixels says you probably interact with this code every day. It's a snippet of HTML that embeds a single tiny pixel in an email. Getting an email with a tracking code spies on me. As follows: My browser requests the single-pixel image as soon as I open the mail. My email sender checks to see if Clives browser has requested that pixel. My email sender can tell when I open it.

Adding a tracking pixel to an email is easy:

<img src="URL LINKING TO THE PIXEL ONLINE" width="0" height="0">

An older example: Ellen R. Stofan and Nick Partridge wrote a chapter on Apollo 11's lunar module bailout code. This bailout code operated on the lunar module's tiny on-board computer and was designed to prioritize: If the computer grew overloaded, it would discard all but the most vital work.

When the lunar module approached the moon, the computer became overloaded. The bailout code shut down anything non-essential to landing the module. It shut down certain lunar module display systems, scaring the astronauts. Module landed safely.

22-line code

POODOO    INHINT
    CA  Q
    TS  ALMCADR

    TC  BANKCALL
    CADR  VAC5STOR  # STORE ERASABLES FOR DEBUGGING PURPOSES.

    INDEX  ALMCADR
    CAF  0
ABORT2    TC  BORTENT

OCT77770  OCT  77770    # DONT MOVE
    CA  V37FLBIT  # IS AVERAGE G ON
    MASK  FLAGWRD7
    CCS  A
    TC  WHIMPER -1  # YES.  DONT DO POODOO.  DO BAILOUT.

    TC  DOWNFLAG
    ADRES  STATEFLG

    TC  DOWNFLAG
    ADRES  REINTFLG

    TC  DOWNFLAG
    ADRES  NODOFLAG

    TC  BANKCALL
    CADR  MR.KLEAN
    TC  WHIMPER

This fun book is worth reading.

I'm a contributor to the New York Times Magazine, Wired, and Mother Jones. I've also written Coders: The Making of a New Tribe and the Remaking of the World and Smarter Than You Think: How Technology is Changing Our Minds. Twitter and Instagram: @pomeranian99; Mastodon: @clive@saturation.social.

The web adds millions or billions of pages per day.

How much Google traffic does this content get?

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

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

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

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

How many pages get search traffic?

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

90% of google pages get no organic traffic

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

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

Reason #1: No backlinks

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

Backlinks boost Google rankings.

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

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

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

Did you notice the trend already?

Most pages lack search traffic and backlinks.

But are these the same pages?

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

More backlinks equals more Google organic traffic.

Referring domains and keyword rankings are correlated.

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

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

Is organic traffic possible without links?

Here are the numbers:

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

Most get 300 or fewer organic visits per month.

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

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

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

Internal links likely pass PageRank to new pages.

Two other reasons:

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

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

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

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

Some pages with many backlinks get no Google traffic.

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

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

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

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

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

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

#2 is self-explanatory:

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

Consider one of our blog posts' metrics:

No organic traffic despite 337 backlinks from 132 sites.

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

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

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

Note:

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

The Guardian's top "Donald Trump" story:

Ahrefs caught on quickly:

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

I bet traffic has dropped if you check now.

One of the quickest and most effective SEO wins is:

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

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

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

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

We see:

The guide has 67 backlinks but no organic traffic.

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

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

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

Reason #3: Search intent isn't met

Google returns the most relevant search results.

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

Google knows that most searchers aren't buying.

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

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

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

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

We realized the problem after analyzing search intent.

People wanted a free tool, not a landing page.

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

Reason #4: Unindexed page

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

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

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

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

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

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

Surprisingly common.

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

One dead page has 131 backlinks:

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

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

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

This post is a summary. See the original post here

One great idea might make you rich.

Imagine having a million-dollar concept in college that made a million.

2005 precisely.

Alex Tew, 21, from Wiltshire, England, created The Million Dollar Homepage in August 2005. The idea is basic but beyond the ordinary, which is why it worked.

Alex built a 1,000,000-pixel webpage.

Each website pixel would cost $1. Since pixels are hard to discern, he sold 10x10 squares for $100.

He'd make a million if all the spots sold.

He may have thought about NFTs and the Metaverse decades ago.

MillionDollarHomepage.com launched in 2005.

Businesses and individuals could buy a website spot and add their logo, website link, and tagline. You bought an ad, but nobody visited the website.

If a few thousand people visited the website, it could drive traffic to your business's site.

Alex promised buyers the website would be up for 5 years, so it was a safe bet.

Alex's friend with a music website was the first to buy real estate on the site. Within two weeks, 4,700 pixels sold, and a tracker showed how many were sold and available.

Word-of-mouth marketing got the press's attention quickly. Everyone loves reading about new ways to make money, so it was a good news story.

By September, over 250,000 pixels had been sold, according to a BBC press release.

Alex and the website gained more media and public attention, so traffic skyrocketed. Two months after the site launched, 1,400 customers bought more than 500,000 pixels.

Businesses bought online real estate. They heard thousands visited the site, so they could get attention cheaply.

Unless you bought a few squares, I'm not sure how many people would notice your ad or click your link.

A sponge website owner emailed Alex:

“We tried Million Dollar Homepage because we were impressed at the level of ingenuity and the sheer simplicity of it. If we’re honest, we didn’t expect too much from it. Now, as a direct result, we are pitching for £18,000 GBP worth of new clients and have seen our site traffic increase over a hundred-fold. We’re even going to have to upgrade our hosting facility! It’s been exceptional.”

Web.archive.org screenshots show how the website changed.

“The idea is to create something of an internet time capsule: a homepage that is unique and permanent. Everything on the internet keeps changing so fast, it will be nice to have something that stays solid and permanent for many years. You can be a part of that!” Alex Tew, 2005

The last 1,000 pixels were sold on January 1, 2006.

By then, the homepage had hundreds of thousands of monthly visitors. Alex put the last space on eBay due to high demand.

MillionDollarWeightLoss.com won the last pixels for $38,100, bringing revenue to $1,037,100 in 4 months.

Many have tried to replicate this website's success. They've all failed.

This idea only worked because no one had seen this website before.

This winner won't be repeated, but it should inspire you to try something new and creative.

Still popular, you could buy one of the linked domains. You can't buy pixels, but you can buy an expired domain.

One link I clicked costs $59,888.

You'd own a piece of internet history if you spent that much on a domain.

Someone bought stablesgallery.co.uk after the domain expired and restored it.

Many of the linked websites have expired or been redirected, but some still link to the original. I couldn't find sponge's website. Can you?

This is a great example of how a simple creative idea can go viral.

Comment on this amazing success story.

The internet is a massive information resource. With so much stuff, it's easy to forget about useful websites. Here are five essential websites you may not have known about.

1. Companies.tools

Companies.tools are what successful startups employ. This website offers a curated selection of design, research, coding, support, and feedback resources. Ct has the latest app development platform and greatest client feedback method.

2. Excel Formula Bot

Excel Formula Bot can help if you forget a formula. Formula Bot uses AI to convert text instructions into Excel formulas, so you don't have to remember them.

Just tell the Bot what to do, and it will do it. Excel Formula Bot can calculate sales tax and vacation days. When you're stuck, let the Bot help.

3.TypeLit

TypeLit helps you improve your typing abilities while reading great literature.

TypeLit.io lets you type any book or dozens of preset classics. TypeLit provides real-time feedback on accuracy and speed.

Goals and progress can be tracked. Why not improve your typing and learn great literature with TypeLit?

4. Calm Schedule

Finding a meeting time that works for everyone is difficult. Personal and business calendars might be difficult to coordinate.

Synchronize your two calendars to save time and avoid problems. You may avoid searching through many calendars for conflicts and keep your personal information secret. Having one source of truth for personal and work occasions will help you never miss another appointment.
https://calmcalendar.com/

5. myNoise

myNoise makes the outside world quieter. myNoise is the right noise for a noisy office or busy street.

If you can't locate the right noise, make it. MyNoise unlocks the world. Shut out distractions. Thank your ears.

6. Synthesia

Professional videos require directors, filmmakers, editors, and animators. Now, thanks to AI, you can generate high-quality videos without video editing experience.

AI avatars are crucial. You can design a personalized avatar using a web-based software like synthesia.io. Our avatars can lip-sync in over 60 languages, so you can make worldwide videos. There's an AI avatar for every video goal.

Not free. Amazing service, though.

7. Cleaning-up-images

Have you shot a wonderful photo just to notice something in the background? You may have a beautiful headshot but wish to erase an imperfection.

Cleanup.pictures removes undesirable objects from photos. Our algorithms will eliminate the selected object.

Cleanup.pictures can help you obtain the ideal shot every time. Next time you take images, let Cleanup.pictures fix any flaws.

8. PDF24 Tools

Editing a PDF can be a pain. Most of us don't know Adobe Acrobat's functionalities. Why buy something you'll rarely use? Better options exist.

PDF24 is an online PDF editor that's free and subscription-free. Rotate, merge, split, compress, and convert PDFs in your browser. PDF24 makes document signing easy.

Upload your document, sign it (or generate a digital signature), and download it. It's easy and free. PDF24 is a free alternative to pricey PDF editing software.

9. Class Central

Finding online classes is much easier. Class Central has classes from Harvard, Stanford, Coursera, Udemy, and Google, Amazon, etc. in one spot.

Whether you want to acquire a new skill or increase your knowledge, you'll find something. New courses bring variety.

10. Rome2rio

Foreign travel offers countless transport alternatives. How do you get from A to B? It’s easy!

Rome2rio will show you the best method to get there, including which mode of transport is ideal.

• Plane

• Car

• Train

• Bus

• Ferry

• Driving

• Shared bikes

• Walking

Do you know any free, useful websites?

I'm trying to understand why Bitcoin NFTs aren't ready.

Ordinals, a new Bitcoin protocol, has been controversial. NFTs can be added to Bitcoin transactions using the protocol. They are not tokens or fungible. Bitcoin NFTs are transaction metadata. Yes. They're not owned.

In January, the Ordinals protocol allowed data like photos to be directly encoded onto sats, the smallest units of Bitcoin worth 0.00000001 BTC, on the Bitcoin blockchain. Ordinals does not need a sidechain or token like other techniques. The Ordinals protocol has encoded JPEG photos, digital art, new profile picture (PFP) projects, and even 1993 DOOM onto the Bitcoin network.

Ordinals inscriptions are permanent digital artifacts preserved on the Bitcoin blockchain. It differs from Ethereum, Solana, and Stacks NFT technologies that allow smart contract creators to change information. Ordinals store the whole image or content on the blockchain, not just a link to an external server, unlike centralized databases, which can change the linked image, description, category, or contract identifier.

So far, more than 50,000 ordinals have been produced on the Bitcoin blockchain, and some of them have already been sold for astronomical amounts. The Ethereum-based CryptoPunks NFT collection spawned Ordinal Punk. Inscription 620 sold for 9.5 BTC, or $218,000, the most.

Segwit and Taproot, two important Bitcoin blockchain updates, enabled this. These protocols store transaction metadata, unlike Ethereum, where the NFT is the token. Bitcoin's NFT is a sat's transaction details.

What effects do ordinary values and NFTs have on the Bitcoin blockchain?

Ordinals will likely have long-term effects on the Bitcoin Ecosystem since they store, transact, and compute more data.

Charges Ordinals introduce scalability challenges. The Bitcoin network has limited transaction throughput and increased fees during peak demand. NFTs could make network transactions harder and more expensive. Ordinals currently occupy over 50% of block space, according to Glassnode.

One of the protocols that supported Ordinals Taproot has also seen a huge uptick:

Taproot use increases block size and transaction costs.

This could cause network congestion but also support more L2s with Ordinals-specific use cases. Dune info here.

Storage Needs The Bitcoin blockchain would need to store more data to store NFT data directly. Since ordinals were introduced, blocksize has tripled from 0.7mb to over 2.2mb, which could increase storage costs and make it harder for nodes to join the network.

Use Case Diversity On the other hand, NFTs on the Bitcoin blockchain could broaden Bitcoin's use cases beyond storage and payment. This could expand Bitcoin's user base. This is two-sided. Bitcoin was designed to be trustless, decentralized, peer-to-peer money.

Chain to permanently store NFTs as ordinals will change everything.

Popularity rise This new use case will boost Bitcoin appeal, according to some. This argument fails since Bitcoin is the most popular cryptocurrency. Popularity doesn't require a new use case. Cryptocurrency adoption boosts Bitcoin. It need not compete with Ethereum or provide extra benefits to crypto investors. If there was a need for another chain that supports NFTs (there isn't), why would anyone choose the slowest and most expensive network? It appears contradictory and unproductive.

Nonetheless, holding an NFT on the Bitcoin blockchain is more secure than any other blockchain, but this has little utility.

Bitcoin NFTs are undoubtedly controversial. NFTs are strange and perhaps harmful to Bitcoin's mission. If Bitcoin NFTs are here to stay, I hope a sidechain or rollup solution will take over and leave the base chain alone.