5 Ways You Do Complex Math in Your Head Without Realizing It


The one thing that people who love math and people who hate math tend to agree on is this: You're only really doing math if you sit down and write formal equations. This idea is so widely embraced that to suggest otherwise is "to start a fight," says Maria Droujkova, math educator and founder of Natural Math, a site for kids and parents who want to incorporate math into their daily lives. Mathematicians cherish their formal proofs, considering them the best expression of their profession, while the anti-math don't believe that much of the math they studied in school applies to "real life."

But in reality, "we do an awful lot of things in our daily lives that are profoundly mathematical, but that may not look that way on the surface," Christopher Danielson, a Minnesota-based math educator and author of a number of books, including Common Core Math for Parents for Dummies, tells Mental Floss. Our mathematical thinking includes not just algebra or geometry, but trigonometry, calculus, probability, statistics, and any of the at least 60 types [PDF] of math out there. Here are five examples.


Of all the maths, algebra seems to draw the most ire, with some people even writing entire books on why college students shouldn't have to endure it because, they claim, it holds the students back from graduating. But if you cook, you're likely doing algebra. When preparing a meal, you often have to think proportionally, and "reasoning with proportions is one of the cornerstones of algebraic thinking," Droujkova tells Mental Floss.

You're also thinking algebraically whenever you're adjusting a recipe, whether for a larger crowd or because you have to substitute or reduce ingredients. Say, for example, you want to make pancakes, but you only have two eggs left and the recipe calls for three. How much flour should you use when the original recipe calls for one cup? Since one cup is 8 ounces, you can figure this out using the following algebra equation: n/8 : 2/3.

algebraic equation illustrates adjustment of a recipe
Lucy Quintanilla

However, when thinking proportionally, you can just reason that since you have one-third less eggs, you should just use one-third less flour.

You're also doing that proportional thinking when you consider the cooking times of the various courses of your meal and plan accordingly so all the elements of your dinner are ready at the same time. For example, it will usually take three times as long to cook rice as it will a flattened chicken breast, so starting the rice first makes sense.

"People do mathematics in their own way," Droujkova says, "even if they cannot do it in a very formalized way."


woman enjoys listening to music in headphones

The making of music involves many different types of math, from algebra and geometry to group theory and pattern theory and beyond, and a number of mathematicians (including Pythagoras and Galileo) and musicians have connected the two disciplines (Stravinsky claimed that music is "something like mathematical thinking").

But simply listening to music can make you think mathematically too. When you recognize a piece of music, you are identifying a pattern of sound. Patterns are a fundamental part of math; the branch known as pattern theory is applied to everything from statistics to machine learning.

Danielson, who teaches kids about patterns in his math classes, says figuring out the structure of a pattern is vital for understanding math at higher levels, so music is a great gateway: "If you're thinking about how two songs have similar beats, or time signatures, or you're creating harmonies, you're working on the structure of a pattern and doing some really important mathematical thinking along the way."

So maybe you weren't doing math on paper if you were debating with your friends about whether Tom Petty was right to sue Sam Smith in 2015 over "Stay With Me" sounding a lot like "I Won't Back Down," but you were still thinking mathematically when you compared the songs. And that earworm you can't get out of your head? It follows a pattern: intro, verse, chorus, bridge, end.

When you recognize these kinds of patterns, you're also recognizing symmetry (which in a pop song tends to involve the chorus and the hook, because both repeat). Symmetry [PDF] is the focus of group theory, but it's also key to geometry, algebra, and many other maths.


six steps of crocheting a hyperbolic plane
Cheryl, Flickr // CC BY-SA 2.0

Droujkova, an avid crocheter, she says she is often intrigued by the very mathematical discussions fellow crafters have online about the best patterns for their projects, even if they will often insist they are awful at math or uninterested in it. And yet, such crafts cannot be done without geometric thinking: When you knit or crochet a hat, you're creating a half sphere, which follows a geometric formula.

Droujkova isn't the only math lover who has made the connection between geometry and crocheting. Cornell mathematician Daina Taimina found crocheting to be the perfect way to illustrate the geometry of a hyperbolic plane, or a surface that has a constant negative curvature, like a lettuce leaf. Hyperbolic geometry is also used in navigation apps, and explains why flat maps distort the size of landforms, making Greenland, for example, look far larger on most maps than it actually is.


people playing pool

If you play billiards, pool, or snooker, it's very likely that you are using trigonometric reasoning. Sinking a ball into a pocket by using another ball involves understanding not just how to measure angles by sight but triangulation, which is the cornerstone of trigonometry. (Triangulation is a surprisingly accurate way to measure distance. Long before powered flight was possible, surveyors used triangulation to measure the heights of mountains from their bases and were off by only a matter of feet.)

In a 2010 paper [PDF], Louisiana mathematician Rick Mabry studied the trigonometry (and basic calculus) of pool, focusing on the straight-in shot. In a bar in Shreveport, Louisiana, he scribbled equations on napkins for each shot, and he calculated the most difficult straight-in shot of all. Most experienced pool players would say it’s one where the target ball is halfway between the pocket and the cue ball. But that, according to Mabry’s equations, turned out not to be true. The hardest shot of all had a surprising feature: The distance from the cue ball to the pocket was exactly 1.618 times the distance from the target ball to the pocket. That number is the golden ratio, which is found everywhere in nature—and, apparently, on pool tables.

Do you need to consider the golden ratio when deciding where to place the cue ball? Nope, unless you want to prove a point, or set someone else up to lose. You're doing the trig automatically. The pool sharks at the bar must have known this, because someone threw away Mabry's math napkins.


tiled bathroom with shower stall

Many students don't get to calculus in high school, or even in college, but a cornerstone of that branch of math is optimization—or figuring out how to get the most precise use of a space or chunk of time.

Consider a home improvement project where you're confronted with tiling around something whose shape doesn't fit a geometric formula like a circle or rectangle, such as the asymmetric base of a toilet or freestanding sink. This is where the fundamental theorem of calculus—which can be used to calculate the precise area of an irregular object—comes in handy. When thinking about how those tiles will best fit around the curve of that sink or toilet, and how much of each tile needs to be cut off or added, you're employing the kind of reasoning done in a Riemann sum.

Riemann sums (named after a 19th-century German mathematician) are crucial to explaining integration in calculus, as tangible introductions to the more precise fundamental theorem. A graph of a Riemann sum shows how the area of a curve can be found by building rectangles along the x, or horizontal axis, first up to the curve, and then over it, and then averaging the distance between the over- and underlap to get a more precise measurement. 

How Coronavirus and 31 Other Infectious Diseases and Viruses Got Their Names

Rotavirus—from the Latin rota, for "wheel"—is named for the wheel-like appearance of its particles.
Rotavirus—from the Latin rota, for "wheel"—is named for the wheel-like appearance of its particles.
Dr_Microbe/iStock via Getty Images

As you may already know, the corona in coronavirus has no relation to a certain refreshing beer often served with a slice of lime. Corōna means “crown” in Latin—and Spanish and Italian, too—and virologists chose it in 1968 to describe the group of viruses characterized by crown-like spikes that protrude from their surfaces.

So how do other viruses and diseases get their names? Based on the infographic below, created by Adam Aleksic for his website, The Etymology Nerd, there isn’t just one way. Some, like the coronavirus, are named for how they look under a microscope. The rota in rotavirus, for example, which means “wheel” in Latin, reflects the virus’s wheel-like appearance when viewed beneath an electron microscope.

Others are named after the locations where they were discovered or studied. In 1947, scientists named a newly identified mosquito-borne virus after Uganda’s Zika Forest. In 1977, Yale researchers investigating a string of pediatric arthritis cases in the town of Lyme, Connecticut, started referring to the illness as “Lyme arthritis.” Later, the name was modified to “Lyme disease” when scientists realized patients were exhibiting other symptoms, too.

Still others are characterized by the symptoms they cause. People with tetanus—from the Greek tetanos, for “tension”—usually experience muscle stiffness, and the skin of yellow fever sufferers often takes on a yellow tint due to jaundice.

Find out the origins of malaria, measles, and more below. And follow The Etymology Nerd on Instagram for more fascinating etymological explanations.

etymology nerd infectious disease names infographic
Unsurprisingly, there's a lot of Latin in this infographic.

These 8 MasterClass Courses Will Get You Out of Your Netflix Funk

Chef Gordon Ramsay is just one of the professionals lending their knowledge to a MasterClass course.
Chef Gordon Ramsay is just one of the professionals lending their knowledge to a MasterClass course.
Franco Origlia/Getty Images

Although binge-watching Netflix is always a great way to kill a few hours, you might want to shake it up and do something a little more substantial with your free time. That’s where MasterClass comes in. A subscription package that features over 80 virtual courses in a wide range of subjects, MasterClass can help you explore exciting new subjects or improve your knowledge in an area you’re already familiar with. And all the classes are taught by highly recognizable experts in their fields, so you can be confident that the lessons you’re learning are solid (that Martin Scorsese probably knows a thing or two about filmmaking, after all).

The courses themselves are broken up into individual lessons that are only around 10 minutes long, so fitting them into your schedule is as easy as becoming a professional chess player (or it will be, once you’ve finished the course). MasterClass is priced at $15 a month for unlimited classes or at $90 per course, and you can sign up here.

So whether you want to become the next great young adult novelist or an expert bartender, MasterClass has something for everyone. Check out a few highlights from the course list.

1. Cooking with Gordon Ramsay

Gordon Ramsay, known for his sharp demeanor and high expectations in the kitchen, is an international chef, restaurateur, and television host who’s nothing short of legendary. And in his MasterClass series, he’ll teach you to become a legend, too. This series features the softer side of Ramsay, who teaches you knife techniques, seasoning tricks, kitchen layout, and much more. And, for anyone ready to level up, he also offers Cooking II: Restaurant Recipes in the Kitchen.

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2. Mixology with Lynnette Marrero & Ryan Chetiyawardana

If you’re ready to become an award-winning bartender—or just make a decent martini at home—look no further than this 17-lesson course with master mixologists Lynnette Marrero and Ryan Chetiyawardana. You’ll not only learn to craft the perfect cocktail, but also how to safely incorporate raw eggs into drinks, make complementary drink “seasonings,” and discover the best liquor to pair with food (who knew that whiskey and blue cheese were a match made in heaven?). A good drink has the power to bring people together, and after this course, you’ll be the go-to guru for any dinner party.

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3. Writing for Young Adults With R.L. Stine

After you’re through with R.L. Stine’s class on writing for young adults, you may just become the next sultan of the Scholastic Book Fair. According to his website, Stine has written over 330 books over the course of his career, and he’s provided thrills and chills to millions of readers with his beloved Goosebumps and Fear Street series. Now, he’ll teach you some of his favorite tricks of the trade, like why you should always start with the ending (so you can focus on fooling your reader for the entire book) and how writing from personal experience makes for a more sincere scare. This masterclass will help you perfect the art of scary storytelling and overcome any fears you might have about putting your own experiences on the page.

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4. Conservation with Dr. Jane Goodall

Dr. Jane Goodall is best known for her groundbreaking work with chimpanzees and gorillas, but she’s more than just a scientist—she’s an environmental activist and conservationist who wants to ensure that animal habitats are preserved for years to come. In this course, Dr. Goodall will share some of the conservation lessons she’s learned as a scientist, identify the central problems facing our planet today, and share effective methods for creating change. As she says, one of the best ways to confront environmental issues is by “telling stories, meeting with people, listening to them, and then finding a way to reach the heart.” This course will show you how.

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5. Chess with Garry Kasparov

Garry Kasparov, a chess grandmaster and world chess champion, will help you dust off your old chessboard and learn to play the game like it’s an art form. This 29-lesson class starts with the basics and gets more complex the further you get into the course. Using the tactics he’s curated throughout his career, Kasparov will show you how to approach chess with a strategist's mindset, including the basics of openings, interference plans, and endgames. This class even features other “students” so you’re not just studying the techniques, but seeing how they play out in real time, too.

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6. Comedy with Steve Martin

One of comedy’s greatest names—and one of film’s most beloved actors—is Steve Martin. The star of film classics such as Father of the Bride and Planes, Trains, and Automobiles, Martin will teach you how to find your comedic voice, develop your persona, and work a crowd so that you’ll never get booed off the stage. And he’ll share his comedic process, which involves more writing and editing than it does practicing in the mirror. This is the perfect class for anyone looking to practice their improv skills or sign up for their first open mic night.

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7. Space Exploration with Chris Hadfield

Make your childhood dreams come true with this course from retired astronaut Chris Hadfield, who’s flown two Space Shuttle missions and served as commander of the International Space Station during his 21 years as an astronaut. And in his course, he’ll teach you the intricacies of space exploration, from learning how rockets work to preparing your body for liftoff. Even if you're not planning on leaving the atmosphere (or your couch) anytime soon, this class teaches you what it's like to be an astronaut. And as Hadfield talks about his unique journey to the stars, there's the chance to learn plenty of life and career lessons that you can apply on Earth.

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8. Adventure Photography with Jimmy Chin

Even if your biggest adventure is going on a walk in your local park, award-winning adventure photographer Jimmy Chin (the face behind documentaries such as Free Solo and Meru) will still teach you how to turn your photographs into works of art. Chin’s class acts as part photography course and part adventure guide, teaching you every step from choosing the right location to editing the final product. You’ll learn how to plan shoots, pack the right gear, and even find clients if you want to go professional. Whether you want to make photography a career or just take the perfect photo of your dog at golden hour, this masterclass will get you feeling comfortable behind the camera and bring some more adventure into your everyday life.

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