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

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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. 

A Low-Carb Diet Could Shorten Your Lifespan

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The Atkins, Paleo, and Keto diets may have different gimmicks, but they all share a common message: Carbs are bad and meat is good. Yet a new analysis reported by New Scientist suggests that anyone who buys into this belief may later come to regret it. According to the paper, published in The Lancet Public Health, people who eat a moderate amount of carbs actually live longer than those who avoid them.

For their study, researchers analyzed data previously collected from 15,400 participants in the U.S. They found that people who received about 50 to 55 percent of their calories from carbohydrates had the longest lifespans, roughly four years longer than those who got 30 percent or less of their energy from carbs.

This doesn't necessarily mean that the key to a healthy diet is to stock your pantry with pasta and croissants. The study also showed that people who got up to 70 percent or more of their energy from carbs died one year earlier on average than subjects in the 50 to 55 percent group. A closer examination at the eating of habits of people who ate fewer carbs revealed another layer to the phenomenon: When people avoided carbohydrates in favor of meat, their chances of early death rose, but the opposite was true for people who replaced carb-heavy foods with plant-based fats and proteins, such as nuts, beans, and vegetables.

These numbers point to something dietitians have long been aware of: Eating a diet that's based around animal products isn't ideal. Getting more of your protein from plant-based sources, on the other hand, can lower your blood pressure and reduce your risks of type 2 diabetes, heart disease, and certain cancers. Nonetheless, fad diets that forbid people from eating carbs while letting them eat as much steak as they want are still popular because they're an easy way to lose weight in a short amount of time. But as the research shows, the short-term results are rarely worth the long-term effects on your health.

[h/t New Scientist]

Why Is Pee Yellow?

Chloe Effron
Chloe Effron

WHY? is our attempt to answer all the questions every little kid asks. Do you have a question? Send it to why@mentalfloss.com.

Your body is kind of like a house. You bring things into your body by eating, drinking, and breathing. But just like the things we bring home to real houses, we don’t need every part of what we take in. So there are leftovers, or garbage. And if you let garbage sit around in your house or your body for too long, it gets gross and can make you sick. Your body takes out the garbage by peeing and pooping. These two things are part of your body’s excretory system (ECKS-krih-tore-eee SISS-tem), which is just a fancy way of saying “trash removal.” If your body is healthy, when you look in the toilet you should see brown poop and yellow pee.

Clear, light yellow pee is a sign that your excretory system and the rest of your body are working right. If your pee, or urine (YER-inn), is not see-through, that might mean you are sick. Dark yellow urine usually means that you aren’t drinking enough water. On the other hand, really pale or colorless pee can mean you might be drinking too much water! 

Your blood is filtered through two small organs called kidneys (KID-knees). Remember the garbage we talked about earlier? The chemicals called toxins (TOCK-sins) are like garbage in your blood. Your kidneys act like a net, catching the toxins and other leftovers and turning them into pee.

One part of your blood is called hemoglobin (HEE-moh-gloh-bin). This is what makes your blood red. Hemoglobin goes through a lot of changes as it passes through your body. When it reaches your kidneys, it turns yellow thanks to a chemical called urobilin (yer-ah-BY-lin). Urobilin is kind of like food coloring. The more water you add, the lighter it will be. That's why, if you see dark yellow pee in the toilet, it's time to ask your mom or dad for a cup of water. 

To learn more about pee, check out this article from Kids Health. 

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