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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

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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

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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. 

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DNA Analysis of Loch Ness Could Reveal the Lake's Hidden Creatures
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Stakeouts, sonar studies, and a 24-hour video feed have all been set up in an effort to confirm the existence of the legendary Loch Ness Monster. Now, the Associated Press reports that an international team of scientists will use DNA analysis to learn what's really hiding in the depths of Scotland's most mysterious landmark.

The team, led by Neil Gemmell, who researches evolutionary genetics at the University of Otago in New Zealand, will collect 300 water samples from various locations and depths around the lake. The waters are filled with microscopic DNA fragments animals leave behind as they swim, mate, eat, poop, and die in the waters, and if Nessie is a resident, she's sure to leave bits of herself floating around as well.

After extracting the DNA from the organic material found in the water samples, the scientists plan to sequence it. The results will then be compared to the DNA profiles of known species. If there's evidence of an animal that's not normally found in the lake, or an entirely new species, the researchers will hopefully spot it.

Gemmell is a Nessie skeptic, and he says the point of the project isn't necessarily to discover new species. Rather, he wants to create a genetic profile of the lake while generating some buzz around the science behind it.

If the study goes according to plan, the database of Loch Ness's inhabitants should be complete by 2019. And though the results likely won't include a long-extinct plesiosaur, they may offer insights about other invasive species that now call the lake home.

[h/t AP]

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Essential Science
How Long to Steep Your Tea, According to Science
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The tea in your cabinet likely has vague instructions about how long to steep the leaves. Bigelow, for instance, suggests two to four minutes for black tea, and one to three minutes for green tea. According to Lipton, you should "try singing the National Anthem" while waiting for black tea leaves to infuse.

But while it's true that tea brewed for 30 seconds is technically just as drinkable as a forgotten mug of tea that's been steeping for 30 minutes, drinkable shouldn't be your goal. Taste and—depending on the tea you're drinking—antioxidant and caffeine levels all depend on the amount of time the leaves are in contact with the water. So how early is too early to pluck out a tea bag, and how long can you leave it in before passing the point of no return?

THE SCIENCE OF STEEPING

To achieve the perfect timing, you first need to understand the chemical process at work when you pour hot water over tea leaves. Black, green, white, and oolong tea all come from the leaves and buds of the same plant, Camellia sinensis. (Herbal teas aren't considered "true teas" because they don't come from C. sinensis.) The teas are processed differently: Green and white tea leaves are heated to dry them, limiting the amount of oxidation they get, while black and oolong tea leaves are exposed to oxygen before they're dried, creating the chemical reactions that give the tea its distinct color and flavor. Damaging the tea leaves—by macerating them, rolling them gently, or something in between—helps expose the chemicals inside their cells to varying levels of oxygen.

Both green and black teas contain a lot of the same chemical compounds that contribute to their flavor profiles and nutritional content. When the leaves are submerged in hot water, these compounds leach into the liquid through a process called osmotic diffusion, which occurs when there's fluid on both sides of a selectively permeable membrane—in this case, the tea leaf. Compounds on the surface of the leaf and in the interior cells damaged by processing will diffuse into the surrounding liquid until the compounds in both the leaf and the water reach equilibrium. In other words, if given enough time to steep, the liquid in your mug will become just as concentrated with tea compounds as the liquid in your tea leaves, and the ratio will stay that way.

Osmotic diffusion doesn't happen all at once—different compounds enter the water at different rates based on their molecular weight. The light, volatile chemicals that contribute to tea's aroma and flavor profile dissolve the fastest, which is why the smell from a bag of tea leaves becomes more potent the moment you dunk it in water. The next group of compounds to infuse with the water includes the micronutrients flavanols and polyphenols, which are antioxidants, and caffeine. They're followed by heavier flavanols and polyphenols such as tannins, which are the compounds responsible for tea's bitter flavor. (They're also what make your mouth feel dry after drinking a glass of wine.) Tea also has amino acids like theanine, which can offset the sharpness of tannins.

Water temperature is another factor to take into consideration when steeping your tea. High water temperature creates more kinetic energy, which encourages the compounds to dissolve. "The heat helps you to extract the compounds out of the tea leaves," Shengmin Sang, a North Carolina A&T State University researcher who studies the chemistry of tea, tells Mental Floss. "If you put it into cold water or low-temperature water, the efficiency to extract these compounds out of the leaves will be much lower." But not all water is equal: Bigelow Tea recommends using water at a rolling boil for black tea, and barely boiling water for green tea.

LOOSE LEAF VS. TEA BAGS

Osmotic diffusion takes place whether you use loose leaves or tea bags, but there are some notable differences between the two. When given room to expand, loose tea leaves swell to their full capacity, creating more room for water to flow in and extract all those desirable compounds. Tea that comes prepackaged in a bag, on the other hand, only has so much room to grow, and the quality suffers as a result. This is why some tea companies have started selling tea in roomier, pyramid-shaped bags, though the size matters more than the shape.

But even before the tea touches the water, there's a difference in quality. Loose leaf tea usually consists of whole leaves, while most teabags are filled with broken pieces of tea leaves called dust or fannings, which have less-nuanced flavors and infuse fewer antioxidants than whole leaves, no matter how long you let them steep.

So if you have a choice, go with loose leaf. But if tea bags are all you have on hand, don't bother adjusting your brewing method: The difference in taste and antioxidants isn't something that can be fixed with a few extra minutes, and according to Sang, you should follow the same steeping times for both tea bags and loose leaf.

To calculate the perfect brew times for what's in your mug, first consider what you want most out of your drink.

IF YOU DRINK TEA TO BE HEALTHY

Suggested steeping time: 2 minutes, 30 seconds to 5 minutes

Tea leaves are packed with beneficial compounds. Research indicates that flavanols such as catechins and epicatechins, found in both green and black teas, help suppress inflammation and curb plaque build-up in arteries. Drinking tea may improve vascular reactivity, which dictates how well blood vessels adjust to stress. According an analysis of multiple tea-related studies published in the European Journal of Epidemiology in 2015, drinking three cups of tea a day reduces your risk of coronary heart disease by 27 percent, cardiac death by 26 percent, and total mortality by 24 percent. Polyphenolic antioxidants in tea may also protect against diabetes, depression, and liver disease.

Past research has shown that it takes 100 to 150 seconds to extract half the polyphenol content from green and black tea leaves. According to a study published in 2016 in the journal Beverages, you can get more polyphenols into your drink if you allow the leaves more time to steep. However, the returns may not be worth the extra effort: Most of the compounds the researchers measured after 10 minutes of steeping were extracted in the first 5 minutes.

Sang makes another argument for not waiting too long to drink your tea. Antioxidants are slightly unstable, which means they will eventually break down and lose their healthy properties after infusing with water. “After you extract the compounds from the tea bag, you can not keep the solution for too long,” he says. “Because these compounds are not stable, they will be oxidized. So if you brew it in the morning, then you drink it in the afternoon, that's not good.” This oxidation can occur even after the tea leaves are removed from the cup, so if your tea has been sitting out for a few hours, it's better to brew a new batch than to pop it in the microwave.

IF YOU DRINK TEA FOR THE CAFFEINE BOOST

Suggested steeping time: 3 to 5 minutes

Though less potent than its rival coffee, a properly brewed cup of tea packs a caffeine punch. According to a 2008 study published in the Journal of Analytical Toxicology [PDF], letting your tea brew for at least a few minutes has a big impact on the caffeine content. The study found that after brewing for one minute, a cup of regular Lipton black tea had 17 milligrams of caffeine per 6 ounces of water, 38 milligrams per 6 ounces after three minutes, and 47 milligrams per 6 ounces after five. (The nutritional information for Lipton black tea says a serving contains 55 milligrams of caffeine per 8 ounces, so it's pretty accurate.)

Some people may use those numbers as an excuse to steep their tea past the five-minute mark in an attempt to reach 100 percent dissolution. But a longer brewing time doesn't necessarily equal a stronger caffeine kick. Yes, more caffeine molecules will enter the tea, but so will other compounds like thearubigins. Caffeine works because it's perfectly shaped to bind to certain neuroreceptors in your brain, thus blocking the chemicals that tell you to feel tired. But caffeine is the right shape to bind to thearubigins as well, and if that happens first, less caffeine will get to those neuroreceptors. So if you're looking for a highly caffeinated cup of tea, you should remove the leaves after most of the caffeine has been extracted—after about three to five minutes—rather than waiting for every last milligram of caffeine to dissolve.

IF YOU DRINK TEA BECAUSE IT TASTES GOOD

Suggested steeping time: 1 to 3 minutes

There's nothing wrong with enjoying a cup of tea for taste alone. Flavor is the most subjective factor influenced by steeping times, but for the sake of simplicity, let's assume you prefer a pronounced tea taste that's not overshadowed by bitterness. To extract those more delicate flavors, you don't need to steep your tea leaves for very long at all. Some of the first volatile organic compounds to break down in tea are geraniol and phenylacetaldehyde, tied to a tea's floral aroma, and linalool and linalool oxide, which give tea its sweetness.

The other compounds we associate with tea's distinctive taste are tannins. They're the difference between an aromatic, fruity cup of tea and a bitter cup that needs to be diluted with milk before it's palatable. But tannins aren't all bad: Some people prefer their tea to have a bracing astringency. Because tannins are some of the last molecules to dissolve into tea, if you want to add some bitter complexity to your drink, steep your tea for a minute or two longer than you normally would. A good way to keep track of the strength of your tea is to look at the color: Like tannins, pigments are heavy compounds, so if you see your tea getting darker, that means it's getting stronger as well.

And what about herbal teas? Feel free to leave the leaves in as long as you like. Because herbal teas are high in aromatic compounds and low in tannins, drinkers can be more liberal with their steep times without worrying about getting that astringent taste. Some teas, like rooibos and chamomile, also contain antioxidants, which is another reason to take your time.

And if you're new to the world of tea and aren't sure what your preferences are, put a kettle on the stove and start experimenting.

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