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.

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

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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

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

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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

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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. 

10 Rad Gifts for Hikers

Greg Rosenke/Unsplash
Greg Rosenke/Unsplash

The popularity of bird-watching, camping, and hiking has skyrocketed this year. Whether your gift recipients are weekend warriors or seasoned dirtbags, they'll appreciate these tools and gear for getting most out of their hiking experience.

1. Stanley Nesting Two-Cup Cookset; $14

Amazon

Stanley’s compact and lightweight cookset includes a 20-ounce stainless steel pot with a locking handle, a vented lid, and two insulated 10-ounce tumblers. It’s the perfect size for brewing hot coffee, rehydrating soup, or boiling water while out on the trail with a buddy. And as some hardcore backpackers note in their Amazon reviews, your favorite hiker can take the tumblers out and stuff the pot with a camp stove, matches, and other necessities to make good use of space in their pack.

Buy it: Amazon

2. Osprey Sirrus and Stratos 24-Liter Hiking Packs; $140

Amazon

Osprey’s packs are designed with trail-tested details to maximize comfort and ease of use. The Sirrus pack (pictured) is sized for women, while the Stratos fits men’s proportions. Both include an internal sleeve for a hydration reservoir, exterior mesh and hipbelt pockets, an attachment for carrying trekking poles, and a built-in rain cover.

Buy them: Amazon, Amazon

3. Yeti Rambler 18-Ounce Bottle; $48

Amazon

Nothing beats ice-cold water after a summer hike or a sip of hot tea during a winter walk. The Yeti Rambler can serve up both: Beverages can stay hot or cold for hours thanks to its insulated construction, and its steel body (in a variety of colors) is basically indestructible. It will add weight to your hiker's pack, though—for a lighter-weight, non-insulated option, the tried-and-true Camelbak Chute water bottle is incredibly sturdy and leakproof.

Buy it: Amazon

4. Mappinners Greatest 100 Hikes of the National Parks Scratch-Off Poster; $30

Amazon

The perfect gift for park baggers in your life (or yourself), this 16-inch-by-20-inch poster features epic hikes like Angel’s Landing in Zion National Park and Half Dome in Yosemite National Park. Once the hike is complete, you can scratch off the gold foil to reveal an illustration of the park.

Buy it: Amazon

5. National Geographic Adventure Edition Road Atlas; $19

Amazon

Hikers can use this brand-new, updated road atlas to plan their next adventure. In addition to comprehensive maps of all 50 states, Puerto Rico, Canada, and Mexico, they'll get National Geographic’s top 100 outdoor destinations, useful details about the most popular national parks, and points on the maps noting off-the-beaten-path places to explore.  

Buy it: Amazon

6. Adventure Medical Kits Hiker First-Aid Kit; $25

Amazon

This handy 67-piece kit is stuffed with all the things you hope your hiker will never need in the wilderness. Not only does it contain supplies for pain, cuts and scrapes, burns, and blisters (every hiker’s nemesis!), the items are organized clearly in the bag to make it easy to find tweezers or an alcohol wipe in an emergency.

Buy it: Amazon

7. Hiker Hunger Ultralight Trekking Poles; $70

Amazon

Trekking poles will help increase your hiker's balance and stability and reduce strain on their lower body by distributing it to their arms and shoulders. This pair is made of carbon fiber, a super-strong and lightweight material. From the sweat-absorbing cork handles to the selection of pole tips for different terrain, these poles answer every need on the trail. 

Buy it: Amazon

8. Leatherman Signal Camping Multitool; $120

Amazon

What can’t this multitool do? This gadget contains 19 hiking-friendly tools in a 4.5-inch package, including pliers, screwdrivers, bottle opener, saw, knife, hammer, wire cutter, and even an emergency whistle.

Buy it: Amazon

9. RAVPower Power Bank; $24

Amazon

Don’t let your hiker get caught off the grid with a dead phone. They can charge RAVPower’s compact power bank before they head out on the trail, and then use it to quickly juice up a phone or tablet when the batteries get low. Its 3-inch-by-5-inch profile won’t take up much room in a pack or purse.

Buy it: Amazon

10. Pack of Four Indestructible Field Books; $14

Amazon

Neither rain, nor snow, nor hail will be a match for these waterproof, tearproof 3.5-inch-by-5.5-inch notebooks. Your hiker can stick one in their pocket along with a regular pen or pencil to record details of their hike or brainstorm their next viral Tweet.

Buy it: Amazon

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8-Year-Old Twinkies Intrigue Fungus Experts

Photog Bill, Flickr // CC BY-NC-ND 2.0
Photog Bill, Flickr // CC BY-NC-ND 2.0

An 8-year-old box of Twinkies has disproved the myth that the Hostess product never goes bad. Upon closer inspection, it also revealed that the snack may decompose differently than your average foodstuff. As NPR reports, the varying mold growth on the Twinkies was so unusual, it caught the attention of fungi scientists.

The story of the putrefied pastries began in 2012, when nature photographer Colin Purrington learned of Hostess Brands' impending bankruptcy. Fearing Twinkies would disappear from shelves for good, he bought a box for posterity. It sat in his basement until he was hit with a junk food craving earlier this year.

When Purrington ripped open the 8-year-old package, he was surprised to find that the snacks inside were inedible. He had assumed Twinkies were indestructible, but the contents of the box proved otherwise: The cake he had bitten into tasted like what he described as "old sock," another one had a dark spot on it, and a third had shriveled up and turned gray.

After Purrington shared photos of his discovery online, he sent the Twinkies to Brian Lovett and Matt Kasson, scientists at West Virginia University in Morgantown who specialize in fungi. In the past, the researchers have studied how well fungi grow on preservative-laden food, like Peeps. They suspected that fungal growth was responsible for the Twinkies' decay—they just didn't know what kind.

The shriveled-up Twinkie appeared to be shrink-wrapped in its wrapper, a result of the fungus inside consuming more gases than it produced. The fungus may have stopped growing once it ran out of air in the package. When they unwrapped it, the scientists encountered a rock-solid material they needed a bone marrow biopsy tool to penetrate. The Twinkie wasn't hard all the way through, however: The soft, creamy center survived, suggesting the fungi was more attracted to the cake than the filling.

Samples of the Twinkie marked with the dark circle pointed to the presence of Cladosporium, one of the most common airborne molds found indoors. The researchers haven't been able to obtain a living spore sample from the hardened pastry, but they're still trying. Further lab dish tests will hopefully reveal the species of fungi that mummified one of the world's most shelf-stable baked goods.

[h/t NPR]