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

iStock
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
iStock

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
iStock

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
iStock

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. 

When You Feel "Chemistry" With Someone, What's Actually Going On?

iStock
iStock

We know chemistry when we feel it with another person, but we don't always know why we're drawn to one person over another. Is it just a cascade of neurotransmitters and hormones conspiring to rush you toward reproduction? Is it attraction borne of a set of shared values? Or is it bonding over specific experiences that create intimacy?

It's probably a combination of all three, plus ineffable qualities that even matchmaking services can't perfectly nail down.

"Scientists now assume, with very few exceptions, that any behavior has features of both genetics and history. It's nature and nurture," Nicole Prause, a sexual psychophysiologist and neuroscientist, tells Mental Floss. She is the founder of Liberos, a Los Angeles-based independent research center that works in collaboration with the University of Georgia and the University of Pittsburgh to study human sexual behavior and develop sexuality-related biotechnology.

Scientists who study attraction take into consideration everything from genetics, psychology, and family history to traumas, which have been shown to impact a person's ability to bond or feel desire.

THE (BRAIN) CHEMISTRY OF LOVE

Helen Fisher, a biological anthropologist at Rutgers University, Match.com's science advisor, and the author of Anatomy of Love: A Natural History of Mating, Marriage, and Why We Stray, breaks down "love" into three distinct stages: lust, attraction, and attachment. In each stage, your body chemistry behaves differently. It turns out that "chemistry" is, at least in part, actual chemistry. Biochemistry, specifically.

In the lust and attraction phases, your body is directing the show, as people can feel desire without knowing anything personal about the object of that desire. Lust, Fisher asserts in a seminal 1997 paper [PDF], is nothing more than the existence of a sex drive, or "the craving for sexual gratification," she writes. It's a sensation driven by estrogens and androgens, the female and male sex hormones, based in the biological drive to reproduce.

Attraction may be influenced less than lust by physiological factors—the appeal of someone's features, or the way they make you laugh—but your body is still calling the shots at this stage, pumping you full of the hormones cortisol, adrenaline, and dopamine, effecting your brain in a way that's not unlike the way illicit substances do.

Fisher has collaborated multiple times on the science of attraction with social psychologist Arthur Aron, a research professor at Stony Brook University in New York. Aron and his wife Elaine, who is also a psychologist, are known for studying what makes relationships begin—and last.

In a 2016 study in Frontiers in Psychology, the researchers proposed that "romantic love is a natural (and often positive) addiction that evolved from mammalian antecedents by 4 million years ago as a survival mechanism to encourage hominin pair-bonding and reproduction, seen cross-culturally today."

In the attraction phase, your body produces increased amounts of dopamine, the feel-good chemical that is also responsible for pain relief. Using fMRI brain imaging, Aron's studies have shown that "if you're thinking about a person you're intensely in love with, your brain activates the dopamine reward system, which is the same system that responds to cocaine," he tells Mental Floss.

Earlier, Fisher's 1997 paper found that new couples often show "increased energy, less need for sleep or food, focused attention and exquisite delight in smallest details of this novel relationship."

The attachment phase is characterized by increases in oxytocin and vasopressin; these hormones are thought to promote bonding and positive social behaviors to sustain connections over time in order to fulfill parental duties.

There is no hard and fast timeline for how long each phase lasts, as it can vary widely due to gender, age, and other environmental factors, Fisher writes.

Additionally, while oxytocin has long gotten the credit for being the love hormone, Prause says that scientists are now "kind of over oxytocin," because it has broader functions than simply bonding. It also plays a role in the contraction of the uterus to stimulate birth, instigating lactation, and sexual arousal; low levels have been linked to autism spectrum disorders. 

Now they're focusing on a charmingly named hormone known as kisspeptin (no, really). Produced in the hypothalamus, kisspeptin plays a role in the onset of puberty, and may increase libido, regulate the gonadal steroids that fuel the sex drive, and help the body maintain pregnancy. But Prause says there is a lot more study about the role kisspeptin plays in attraction.

CHEMICAL AND PERSONAL BONDS

Biology may explain our initial attraction and the "honeymoon" phase of a relationship, but it doesn't necessarily explain why a person's love of obscure movies or joy of hiking tickles your fancy, or what makes you want to settle down.

The Arons' numerous studies on this subject have found connection boils down to something quite simple: "What makes people attracted to the point of falling in love—presuming the person is reasonably appropriate for them—is that they feel the other person likes them," he says. 

In the process of doing research for her book How To Fall in Love With Anyone, writer Mandy Len Catron of Vancouver became her own test subject when she came across the research the Arons are most well-known for: their 36 questions, which promote bonding.

The questions were originally designed to "generate intimacy, a sense of feeling similar, and the sense that the other person likes you," Aron explains. Romantic love wasn't the goal. "It was a way of creating closeness between strangers."

The Arons first tested their questions by pairing up students during a regular class section of a large psychology course, as they related in a paper in the journal Personality and Social Psychology Bulletin. Some students were paired with someone of the same sex, while others were matched with someone of the opposite sex. Each partner then answered a series of 36 increasingly personal questions, which took about 45 minutes each. (Question 2: "Would you like to be famous? In what way?" Question 35: "Of all the people in your family, whose death would you find most disturbing? Why?") Small talk during class hadn't made them bond, but the questions made the students feel closer.

In another version of the study, heterosexual, opposite-sex pairs follow the 36-question session with four minutes of staring deeply into each other's eyes.

Catron decided to test these methods out with a casual acquaintance, Mark, over beers at a local bar one night. They were both dating other people at the time, and no one exclusively. As she answered the questions and listened to Mark's answers, "I felt totally absorbed by the conversation in a way that was unlike any of the other first dates I was having at the time with people I met online," Catron tells Mental Floss.

She was ready to skip the four minutes of soulful eye gazing, but Mark thought they should try it. "It was deeply uncomfortable, but it was also an important part of the experience," she recalls. "It's so intimate, it requires you to let your guard down."

The process instilled in Catron a deep feeling of trust in Mark and a desire to know him better. Within three months, they began dating in earnest. Now, more than three years later, they live together in a condo they bought.

The Arons' questions offer "accelerated intimacy," she says, in a time of increasingly online-driven dating experiences.

A LITTLE MYSTERY, A LOT OF SHARED VALUES

Despite all that we’ve learned, scientists may only ever be able to brush up against the edge of a true understanding of "chemistry." “We understand a fair amount about what happens when [attraction has] already occurred, but we're really bad at predicting when it will happen," Prause says. "People who try to claim magical matchmaking, or that they're going to somehow chemically manipulate an aphrodisiac or something—well good luck! Because we can't figure it out.”

And anyway, what's romance without a little mystery?

If you must have a definitive answer to the puzzle of interpersonal chemistry, Prause says to keep this in mind: "The best predictor of long-term outcomes is shared values."

This piece originally ran in 2018.

11 Squeaky-Clean Facts About Spit

iStock/fotolinchen
iStock/fotolinchen

Though most people find the thought of saliva rather disgusting, spit plays a vital role in our lives. It allows us to comfortably chew, swallow, and digest. It fights off bacteria in our mouths and elsewhere, and leads the mouth’s bold fight against cavities. Here are 11 facts that might have you reconsidering that unsung hero of bodily fluids: spit.

1. Spit is mostly water.

Saliva consists of about 99 percent water. The other 1 percent is made up of electrolytes and organic substances, including digestive enzymes and small quantities of uric acid, cholesterol, and mucins (the proteins that form mucus).

2. There's a medical standard for how much spit you should have.

Healthy individuals accumulate between 2 and 6 cups of spit a day. That’s without stimulation from activities like eating or chewing gum, which open the spit floodgates [PDF].

3. Saliva production has a circadian rhythm.

Your body typically produces the most saliva in the late afternoon, and the least at night. Salivation is controlled by the autonomic nervous system (much like your heartbeat), meaning it’s an unconscious process.

4. There are five different kinds of spit.

Salivation has five distinct phases, most triggered by the passage of food through the body. Not all of them are a good thing. The first type of salivation is cephalic, the kind that occurs when you see or smell something delicious. The buccal phase is the body’s reflexive response to the actual presence of food in the mouth (which aids in swallowing). The esophageal involves the stimulation of the salivary glands as food moves through the esophagus. The gastric phase happens when something irritates your stomach—like when you’re just about to puke. The intestinal phase is triggered by a food that doesn’t agree with you passing through the upper intestine.

5. Spit can battle bacteria.

There’s a reason the phrase “lick your wounds” came about. Spit is full of infection-battling white blood cells. And, according to a 2015 study in the journal Blood, neutrophils—a type of white blood cell—are more effective at killing bacteria if they come from saliva than from anywhere else in the body. So adding saliva to a wound gives the body a powerful backup as it fights off infection.

6. Spit keeps you from getting cavities.

The calcium, fluoride, and phosphate in saliva strengthen your teeth. Spit also fights cavity-causing bacteria, washes away bits of food, and neutralizes plaque acids, reducing tooth decay and cavities. That’s why chewing gum gets dentists’ stamp of approval—chewing increases the flow of saliva, thus protecting your oral health.

7. You need spit if you want to taste anything.

Saliva acts like a solvent for tastes, ferrying dissolved deliciousness to the sites of taste receptors. It also keeps those receptors healthy by preventing them from drying out and protecting them from bacterial infection. Many people who have dry mouth (or xerostomia) find their sense of taste affected by their oral cavity’s parched conditions. Because many medications have dry mouth as a side effect, scientists have developed artificial saliva sprays that mimic the lubrication of real spit.

8. Swapping spit exchanges millions of bacteria.

A 10-second kiss involves the transfer of some 80 million bacteria, one study found.

9. People aren’t born drooling.

Babies don’t start drooling until they’re 2 to 4 months old. Unfortunately, they also don’t really know what to do with their spit. They don’t have full control of the muscles of their mouth until they’re around 2 years old, so they can’t really swallow it effectively. Which is why we invented bibs.

10. Stress can leave you spit-less.

The body’s fight-or-flight response is designed to give you the energy and strength needed to overcome a near-death experience, like, say, running into a bear or giving a big presentation at work. Your blood pressure goes up, the heart beats faster, and the lungs take in more oxygen. This is not the time to sit around and digest a meal, so the digestion system slows down production, including that of saliva.

11. A lack of spit was once used as an admission of guilt.

In some ancient societies, saliva was used as a basic lie detector. In ancient India, accused liars had to chew grains of rice. If they were telling the truth, they would have enough saliva to spit them back out again. If someone was lying, their mouth would go dry and the rice would stick in their throat.

SECTIONS

arrow
LIVE SMARTER