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


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.


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."


woman enjoys listening to music in headphones

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.


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.


people playing pool

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.


tiled bathroom with shower stall

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. 

5 Signs Humans Are Still Evolving

Lealisa Westerhoff, AFP/Getty Images
Lealisa Westerhoff, AFP/Getty Images

When we think of human evolution, our minds wander back to the millions of years it took natural selection to produce modern-day man. Recent research suggests that, despite modern technology and industrialization, humans continue to evolve. "It is a common misunderstanding that evolution took place a long time ago, and that to understand ourselves we must look back to the hunter-gatherer days of humans," Dr. Virpi Lummaa, a professor at the University of Turku, told Gizmodo.

But not only are we still evolving, we're doing so even faster than before. In the last 10,000 years, the pace of our evolution has sped up, creating more mutations in our genes, and more natural selections from those mutations. Here are some clues that show humans are continuing to evolve.

1. Humans drink milk.

Historically, the gene that regulated humans' ability to digest lactose shut down as we were weaned off our mothers' breast milk. But when we began domesticating cows, sheep, and goats, being able to drink milk became a nutritionally advantageous quality, and people with the genetic mutation that allowed them to digest lactose were better able to propagate their genes.

The gene was first identified in 2002 in a population of northern Europeans that lived between 6000 and 5000 years ago. The genetic mutation for digesting milk is now carried by more than 95 percent of northern European descendants. In addition, a 2006 study suggests this tolerance for lactose developed again, independently of the European population, 3000 years ago in East Africa.

2. We're losing our wisdom teeth.

Our ancestors had much bigger jaws than we do, which helped them chew a tough diet of roots, nuts, and leaves. And what meat they ate they tore apart with their teeth, all of which led to worn-down chompers that needed replacing. Enter the wisdom teeth: A third set of molars is believed to be the evolutionary answer to accommodate our ancestors' eating habits.

Today, we have utensils to cut our food. Our meals are softer and easier to chew, and our jaws are much smaller, which is why wisdom teeth are often impacted when they come in — there just isn't room for them. Unlike the appendix, wisdom teeth have become vestigial organs. One estimate says 35 percent of the population is born without wisdom teeth, and some say they may disappear altogether.

3. We're resisting infectious diseases.

In 2007, a group of researchers looking for signs of recent evolution identified 1800 genes that have only become prevalent in humans in the last 40,000 years, many of which are devoted to fighting infectious diseases like malaria. More than a dozen new genetic variants for fighting malaria are spreading rapidly among Africans. Another study found that natural selection has favored city-dwellers. Living in cities has produced a genetic variant that allows us to be more resistant to diseases like tuberculosis and leprosy. "This seems to be an elegant example of evolution in action," says Dr. Ian Barnes, an evolutionary biologist at London's Natural History Museum, said in 2010 statement. "It flags up the importance of a very recent aspect of our evolution as a species, the development of cities as a selective force."

4. Our brains are shrinking.

While we may like to believe our big brains make us smarter than the rest of the animal world, our brains have actually been shrinking over the last 30,000 years. The average volume of the human brain has decreased from 1500 cubic centimeters to 1350 cubic centimeters, which is an amount equivalent to the size of a tennis ball.

There are several different conclusions as to why this is: One group of researchers suspects our shrinking brains mean we are in fact getting dumber. Historically, brain size decreased as societies became larger and more complex, suggesting that the safety net of modern society negated the correlation between intelligence and survival. But another, more encouraging theory says our brains are shrinking not because we're getting dumber, but because smaller brains are more efficient. This theory suggests that, as they shrink, our brains are being rewired to work faster but take up less room. There's also a theory that smaller brains are an evolutionary advantage because they make us less aggressive beings, allowing us to work together to solve problems, rather than tear each other to shreds.

5. Some of us have blue eyes.

Originally, we all had brown eyes. But about 10,000 years ago, someone who lived near the Black Sea developed a genetic mutation that turned brown eyes blue. While the reason blue eyes have persisted remains a bit of a mystery, one theory is that they act as a sort of paternity test. “There is strong evolutionary pressure for a man not to invest his paternal resources in another man’s child,” Bruno Laeng, lead author of a 2006 study on the development of blue eyes, told The New York Times. Because it is virtually impossible for two blue-eyed mates to create a brown-eyed baby, our blue-eyed male ancestors may have sought out blue-eyed mates as a way of ensuring fidelity. This would partially explain why, in a recent study, blue-eyed men rated blue-eyed women as more attractive compared to brown-eyed women, whereas females and brown-eyed men expressed no preference.

Now Ear This: A New App Can Detect a Child's Ear Infection


Generally speaking, using an internet connection to diagnose a medical condition is rarely recommended. But technology is getting better at outpacing skepticism over handheld devices guiding decisions and suggesting treatment relating to health care. The most recent example is an app that promises to identify one of the key symptoms of ear infections in kids.

The Associated Press reports that researchers at the University of Washington are close to finalizing an app that would allow a parent to assess whether or not their child has an ear infection using their phone, some paper, and some soft noises. A small piece of paper is folded into a funnel shape and inserted into the ear canal to focus the app's sounds (which resemble bird chirps) toward the child’s ear. The app measures sound waves bouncing off the eardrum. If pus or fluid is present, the sound waves will be altered, indicating a possible infection. The parent would then receive a text from the app notifying them of the presence of buildup in the middle ear.

The University of Washington tested the efficacy of the app by evaluating roughly 50 patients scheduled to undergo ear surgery at Seattle Children’s Hospital. The app was able to identify fluid in patients' ears about 85 percent of the time. That’s roughly as well as traditional exams, which involve visual identification as well as specialized acoustic devices.

While the system looks promising, not all cases of fluid in the ear are the result of infections or require medical attention. Parents would need to evaluate other symptoms, such as fever, if they intend to use the app to decide whether or not to seek medical attention. It may prove most beneficial in children with persistent fluid accumulation, a condition that needs to be monitored over the course of months when deciding whether a drain tube needs to be placed. Checking for fluid at home would save both time and money compared to repeated visits to a physician.

The app does not yet have Food and Drug Administration (FDA) approval and there is no timetable for when it might be commercially available. If it passes muster, it would join a number of FDA-approved “smart” medical diagnostic tools, including the AliveKor CardiaBand for the Apple Watch, which conducts EKG monitoring for heart irregularities.

[h/t WGRZ]