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

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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. 

More Than 350 Franklin Expedition Artifacts Retrieved from Shipwreck of HMS Erebus

Drone image above the HMS Erebus shipwreck.
Drone image above the HMS Erebus shipwreck.
Parks Canada's Underwater Archaeology Team

From a shallow Arctic gulf, a treasure trove of objects from the HMS Erebus shipwreck has been brought to the surface for the first time in more than 170 years. The items could offer new clues about the doomed Franklin expedition, which left England in 1845 to search for the Northwest Passage. All 129 people perished from still-uncertain causes—a mystery that was fictionalized in the AMC series The Terror in 2018.

Marc-André Bernier, head of underwater archaeology at Parks Canada, said in a teleconference from Ottawa that this year’s research season was the most successful since the discovery of the HMS Erebus shipwreck in 2014. Parks Canada divers and Inuit located the HMS Terror, the second ship of the Franklin expedition, in 2016.

Parks Canada diver at HMS Erebus shipwreck
A Parks Canada diver retrieves a glass decanter at the HMS Erebus shipwreck.
Parks Canada's Underwater Archaeology Team

From mid-August to mid-September, 2019, the Parks Canada and Inuit research team began systematically excavating the large and complex shipwreck. “We focused on areas that had not been disturbed since the ship had sunk,” Bernier said. “Right now, our focus is the cabins of the officers, and we’re working our way toward the higher officers. That’s where we think we have a better chance of finding more clues to what happened to the expedition, which is one of the major objectives.”

Over a total of 93 dives this year, archaeologists concentrated on three crew members’ cabins on the port side amidships: one belonging to the third lieutenant, one for the steward, and one likely for the ice master. In drawers underneath the third lieutenant’s bed, they discovered a tin box with a pair of the officer’s epaulets in “pristine condition,” Bernier said. They may have belonged to James Walter Fairholme, one of the three lieutenants on the Erebus.

HMS Erebus shipwreck epaulets
A pair of epaulets, which may have belonged to third lieutenant James Walter Fairholme, was found at the HMS Erebus shipwreck.
Parks Canada's Underwater Archaeology Team

In the steward’s pantry, where items used to serve the captain were stored, divers carefully brushed away sediment to reveal dozens of plates, bowls, dish warmers, strainers, and more— about 50 serving pieces total. Bernier said some of the most exciting finds were personal objects that could be linked to individuals, such as a lead stamp with the inscription “Ed. Hoar,” for Edmund Hoar, the 23-year-old captain’s steward. They also found a piece of red sealing wax with a fingerprint of its last user.

Dishes at HMS Erebus shipwreck
Divers found dishes in the steward's pantry at the HMS Erebus shipwreck.
Parks Canada's Underwater Archaeology Team

Other intriguing items brought to the surface include a glass decanter, found in the officers’ mess area on the lower deck, which may have held brandy or port; a high-quality hairbrush with a few human hairs still in the bristles; and a cedar-wood pencil case. All of the artifacts are jointly owned by the Government of Canada and Inuit.

Hairbrush from HMS Erebus shipwreck
A hairbrush discovered at the HMS Erebus shipwreck still had a few human hairs in the bristles.
Parks Canada's Underwater Archaeology Team

The extensive recovery was made possible by a new research barge, which was moored over the shipwreck and provided hyperbaric chambers and hot-water suits. While wearing the suits, divers were able to stay in the frigid waters for about 90 minutes at a time; they spent over 100 hours examining the wreck this year.

The HMS Erebus’s size and excellent state of preservation mean there’s much more to discover, Bernier said. The Erebus is 108 feet long, and though the upper deck has collapsed, there are 20 cabins on the main deck. They’ve examined only three so far. “There are tens of thousands of artifacts still there,” Bernier tells Mental Floss. “We’re going to be very focused and save what needs to be saved, and go to places [in the wreck] where there are good chances of finding the most information that is valuable for the site.”

Parks Canada and Inuit archaeologists
Parks Canada and Inuit archaeologists set up instruments near the HMS Erebus shipwreck.
Parks Canada's Underwater Archaeology Team

As with the findings from previous research seasons, many questions about the shocking demise of the Franklin expedition remain unanswered. How and when did the HMS Erebus sink after both ships were abandoned in spring 1848, having been trapped in ice since September 1846? Which officers and crew were among the 24 men who had died by that time, and why so many?

Bernier tells Mental Floss there’s even a new mystery to solve. Near Edmund Hoar’s items, divers found another artifact that also bore the name of a crew member—mate Frederick Hornby. “Originally, when the ships set sail, he was not on Erebus, he was on Terror,” Bernier says. “So this object jumped ship at one point. How did that happen? Was Hornby transferred to Erebus; did they abandon one ship and put everybody on the other one? Was it something somebody recovered after he died? Was it given to somebody? With one object, we can start to see [new] questions. Hopefully, by piecing all of this together, we can actually start pushing the narrative of the story in some interesting direction.”

The Reason Our Teeth Are So Sensitive to Pain

This woman's tooth pain is actually helping her avoid further damage.
This woman's tooth pain is actually helping her avoid further damage.
champja/iStock via Getty Images

On a good day, your teeth can chew through tough steak and split hard candy into pieces without you feeling a thing. But sometimes, something as simple as slurping a frosty milkshake can send a shock through your tooth that feels even more painful than stubbing your toe.

According to Live Science, that sensitivity is a defense mechanism we’ve developed to protect damaged teeth from further injury.

“If you eat something too hot or chew something too cold, or if the tooth is worn down enough where the underlying tissue underneath is exposed, all of those things cause pain,” Julius Manz, American Dental Association spokesperson and director of the San Juan College dental hygiene program, told Live Science. “And then the pain causes the person not to use that tooth to try to protect it a little bit more.”

Teeth are made of three layers: enamel on the outside, pulp on the inside, and dentin between the two. Pulp, which contains blood vessels and nerves, is the layer that actually feels pain—but that doesn’t mean the other two layers aren’t involved. When your enamel (which isn’t alive and can’t feel anything at all) is worn down, it exposes the dentin, a tissue that will then allow especially hot or cold substances to stimulate the nerves in the pulp. Pulp can’t sense temperature, so it interprets just about every stimulus as pain.

If you do have a toothache, however, pulp might not be the (only) culprit. The periodontal ligament, which connects teeth to the jawbone, can also feel pain. As Manz explains, that sore feeling people sometimes get because of an orthodontic treatment like braces is usually coming from the periodontal ligament rather than the pulp.

To help you avoid tooth pain in the first place, here are seven tips for healthier teeth.

[h/t Live Science]

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