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

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iStock

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.

1. COOKING // ALGEBRA

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."

2. LISTENING TO MUSIC // PATTERN THEORY AND SYMMETRY

woman enjoys listening to music in headphones
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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.

3. KNITTING AND CROCHETING // GEOMETRIC THINKING

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.

4. PLAYING POOL // TRIGONOMETRY

people playing pool
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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.

5. RE-TILING THE BATHROOM // CALCULUS

tiled bathroom with shower stall
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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. 

Wayfair’s Fourth of July Clearance Sale Takes Up to 60 Percent Off Grills and Outdoor Furniture

Wayfair/Weber
Wayfair/Weber

This Fourth of July, Wayfair is making sure you can turn your backyard into an oasis while keeping your bank account intact with a clearance sale that features savings of up to 60 percent on essentials like chairs, hammocks, games, and grills. Take a look at some of the highlights below.

Outdoor Furniture

Brisbane bench from Wayfair
Brisbane/Wayfair

- Jericho 9-Foot Market Umbrella $92 (Save 15 percent)
- Woodstock Patio Chairs (Set of Two) $310 (Save 54 percent)
- Brisbane Wooden Storage Bench $243 (Save 62 percent)
- Kordell Nine-Piece Rattan Sectional Seating Group with Cushions $1800 (Save 27 percent)
- Nelsonville 12-Piece Multiple Chairs Seating Group $1860 (Save 56 percent)
- Collingswood Three-Piece Seating Group with Cushions $410 (Save 33 percent)

Grills and Accessories

Dyna-Glo electric smoker.
Dyna-Glo/Wayfair

- Spirit® II E-310 Gas Grill $479 (Save 17 percent)
- Portable Three-Burner Propane Gas Grill $104 (Save 20 percent)
- Digital Bluetooth Electric Smoker $224 (Save 25 percent)
- Cuisinart Grilling Tool Set $38 (Save 5 percent)

Outdoor games

American flag cornhole game.
GoSports

- American Flag Cornhole Board $57 (Save 19 percent)
- Giant Four in a Row Game $30 (Save 6 percent)
- Giant Jenga Game $119 (Save 30 percent)

This article contains affiliate links to products selected by our editors. Mental Floss may receive a commission for purchases made through these links.

How the Scientist Who Invented Ibuprofen Accidentally Discovered It Was Great for Hangovers

This man had too many dry martinis at a business lunch.
This man had too many dry martinis at a business lunch.
George Marks/Retrofile/Getty Images

When British pharmacologist Stewart Adams and his colleague John Nicholson began tinkering with various drug compounds in the 1950s, they were hoping to come up with a cure for rheumatoid arthritis—something with the anti-inflammatory effects of aspirin, but without the risk of allergic reaction or internal bleeding.

Though they never exactly cured rheumatoid arthritis, they did succeed in developing a nonsteroidal anti-inflammatory drug (NSAID) that greatly reduced pain of all kinds. In 1966, they patented their creation, which was first known as 2-(4-isobutylphenyl) propionic acid and later renamed ibuprofen. While originally approved as a prescription drug in the UK, it soon became clear ibuprofen was safer and more effective than other pain relievers. It eventually hit the market as an over-the-counter medication.

During that time, Adams conducted one last impromptu experiment with the drug, which took place far outside the lab and involved only a single participant: himself.

In 1971, Adams arrived in Moscow to speak at a pharmacology conference and spent the night before his scheduled appearance tossing back shots of vodka at a reception with the other attendees. When he awoke the next morning, he was greeted with a hammering headache. So, as Smithsonian.com reports, Adams tossed back 600 milligrams of ibuprofen.

“That was testing the drug in anger, if you like,” Adams told The Telegraph in 2007. “But I hoped it really could work magic.”

As anyone who has ever been in that situation can probably predict, the ibuprofen did work magic on Adams’s hangover. After that, according to The Washington Post, the pharmaceutical company Adams worked for began promoting the drug as a general painkiller, and people started to stumble upon its use as a miracle hangover cure.

“It's funny now,” Adams told The Telegraph. “But over the years so many people have told me that ibuprofen really works for them, and did I know it was so good for hangovers? Of course, I had to admit I did.”

[h/t Smithsonian.com]