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. 

A Dracula Ant's Jaws Snap at 200 Mph—Making It the Fastest Animal Appendage on the Planet

Ant Lab, YouTube
Ant Lab, YouTube

As if Florida’s “skull-collecting” ants weren’t terrifying enough, we’re now going to be having nightmares about Dracula ants. A new study in the journal Royal Society Open Science reveals that a species of Dracula ant (Mystrium camillae), which is found in Australia and Southeast Asia, can snap its jaws shut at speeds of 90 meters per second—or the rough equivalent of 200 mph. This makes their jaws the fastest part of any animal on the planet, researchers said in a statement.

These findings come from a team of three researchers that includes Adrian Smith, who has also studied the gruesome ways that the skull-collecting ants (Formica archboldi) dismember trap-jaw ants, which were previously considered to be the fastest ants on record. But with jaw speeds of just over 100 miles per hour, they’re no match for this Dracula ant. (Fun fact: The Dracula ant subfamily is named after their habit of drinking the blood of their young through a process called "nondestructive cannibalism." Yikes.)

Senior author Andrew Suarez, of the University of Illinois, said the anatomy of this Dracula ant’s jaw is unusual. Instead of closing their jaws from an open position, which is what trap-jaw ants do, they use a spring-loading technique. The ants “press the tips of their mandibles together to build potential energy that is released when one mandible slides across the other, similar to a human finger snap,” researchers write.

They use this maneuver to smack other arthropods or push them away. Once they’re stunned, they can be dragged back to the Dracula ant’s nest, where the unlucky victims will be fed to Dracula ant larvae, Suarez said.

Researchers used X-ray imaging to observe the ants’ anatomy in three dimensions. High-speed cameras were also used to record their jaws snapping at remarkable speeds, which measure 5000 times faster than the blink of a human eye. Check out the ants in slow-motion in the video below.

14 Facts About Celiac Disease


Going gluten-free may be a modern diet trend, but people have been suffering from celiac disease—a chronic condition characterized by gluten intolerance—for centuries. Patients with celiac are ill-equipped to digest products made from certain grains containing gluten; wheat is the most common. In the short-term this can cause gastrointestinal distress, and in the long-term it can foster symptoms associated with early death.

Celiac diagnoses are more common than ever, which also means awareness of how to live with the condition is at an all-time high. Here are some things you might not know about celiac disease symptoms and treatments.

1. Celiac an autoimmune disease.

The bodies of people with celiac have a hostile reaction to gluten. When the protein moves through the digestive tract, the immune system responds by attacking the small intestine, causing inflammation that damages the lining of the organ. As this continues over time, the small intestine has trouble absorbing nutrients from other foods, which can lead to additional complications like anemia and osteoporosis.

2. You can get celiac disease from your parents.

Nearly all cases of celiac disease arise from certain variants of the genes HLA-DQA1 and HLA-DQB1. These genes help produce proteins in the body that allow the immune system to identify potentially dangerous foreign substances. Normally the immune system wouldn't label gliadin, a segment of the gluten protein, a threat, but due to mutations in these genes, the bodies of people with celiac treat gliadin as a hostile invader.

Because it's a genetic disorder, people with a first-degree relative (a sibling, parent, or child) with celiac have a 4 to 15 percent chance of having it themselves. And while almost all patients with celiac have these specific HLA-DQA1 and HLA-DQB1 variations, not everyone with the mutations will develop celiac. About 30 percent of the population has these gene variants, and only 3 percent of that group goes on to develop celiac disease.

3. Makeup might contribute to celiac disease symptoms.

People with celiac disease can’t properly process gluten, the protein naturally found in the grains like wheat, rye, and barley. Patients have to follow strict dietary guidelines and avoid most bread, pasta, and cereal, in order to manage their symptoms. But gluten isn’t limited to food products: It can also be found in some cosmetics. While makeup containing gluten causes no issues for many people with celiac, it can provoke rashes in others or lead to more problems if ingested. For those folks, gluten-free makeup is an option.

4. The name comes from 1st-century Greece.

A 1st-century Greek physician named Aretaeus of Cappadocia may have been the first person to describe celiac disease symptoms in writing [PDF]. He named it koiliakos after the Greek word koelia for abdomen, and he referred to people with the condition as coeliacs. In his description he wrote, “If the stomach be irretentive of the food and if it pass through undigested and crude, and nothing ascends into the body, we call such persons coeliacs.”

5. There are nearly 300 celiac disease symptoms.

Celiac disease may start in the gut, but it can be felt throughout the whole body. In children, the condition usually manifests as bloating, diarrhea, and abdominal discomfort, but as patients get older they start to experience more “non-classical” symptoms like anemia, arthritis, and fatigue. There are at least 281 symptoms associated with celiac disease, many of which overlap with other conditions and make celiac hard to diagnose. Other common symptoms of the disease include tooth discoloration, anxiety and depression, loss of fertility, and liver disorders. Celiac patients also have a greater chance of developing an additional autoimmune disorder, with the risk increasing the later in life the initial condition is diagnosed.

6. Some patients show no symptoms at all.

It’s not uncommon for celiac disease to be wrecking a patient’s digestive tract while showing no apparent symptoms. This form of the condition, sometimes called asymptomatic or “silent celiac disease,” likely contributes to part of the large number of people with celiac who are undiagnosed. People who are at high risk for the disease (the children of celiac sufferers, for example), or who have related conditions like type 1 diabetes and Down syndrome (both conditions that put patients at a greater risk for developing new autoimmune diseases) are encouraged to get tested for it even if they aren’t showing any signs.

7. It’s not the same as wheat sensitivity.

Celiac is often confused with wheat sensitivity, a separate condition that shares many symptoms with celiac, including gastrointestinal issues, depression, and fatigue. It’s often called gluten sensitivity or gluten intolerance, but because doctors still aren’t sure if gluten is the cause, many refer to it as non-celiac wheat sensitivity. There’s no test for it, but patients are often treated with the same gluten-free diet that’s prescribed to celiac patients.

8. It's not a wheat allergy either.

Celiac disease is often associated with wheat because it's one of the more common products containing gluten. While it's true that people with celiac can't eat wheat, the condition isn't a wheat allergy. Rather than reacting to the wheat, patients react to a specific protein that's found in the grain as well as others.

9. It can develop at any age.

Just because you don’t have celiac now doesn’t mean you’re in the clear for life: The disease can develop at any age, even in people who have tested negative for it previously. There are, however, two stages of life when symptoms are most likely to appear: early childhood (8 to 12 months) and middle adulthood (ages 40 to 60). People already genetically predisposed to celiac become more susceptible to it when the composition of their intestinal bacteria changes as they get older, either as a result of infection, surgery, antibiotics, or stress.

10. Not all grains are off-limits.

A gluten-free diet isn’t necessarily a grain-free diet. While it’s true that the popular grains wheat, barley, and rye contain gluten, there are plenty of grains and seeds that don’t and are safe for people with celiac to eat. These include quinoa, millet, amaranth, buckwheat, sorghum, and rice. Oats are also naturally gluten-free, but they're often contaminated with gluten during processing, so consumers with celiac should be cautious when buying them.

11. Celiac disease can be detected with a blood test.

Screenings for celiac disease used to be an involved process, with doctors monitoring patients’ reactions to their gluten-free diet over time. Today all it takes is a simple test to determine whether someone has celiac. People with the condition will have anti-tissue transglutaminase antibodies in their bloodstream. If a blood test confirms the presence of these proteins in a patient, doctors will then take a biopsy of their intestine to confirm the root cause.

12. The gluten-free diet doesn’t work for all patients.

Avoiding gluten is the most effective way to manage celiac disease, but the treatment doesn’t work 100 percent of the time. In up to a fifth of patients, the damaged intestinal lining does not recover even a year after switching to a gluten-free diet. Most cases of non-responsive celiac disease can be explained by people not following the diet closely enough, or by having other conditions like irritable bowel syndrome, lactose intolerance, or small intestine bacterial overgrowth that impede recovery. Just a small fraction of celiac disease sufferers don’t respond to a strict gluten-free diet and have no related conditions. These patients are usually prescribed steroids and immunosuppressants as alternative treatments.

13. If you don’t have celiac, gluten probably won’t hurt you.

The gluten-free diet trend has exploded in popularity in recent years, and most people who follow it have no medical reason to do so. Going gluten-free has been purported to do everything from help you lose weight to treat autism—but according to doctors, there’s no science behind these claims. Avoiding gluten may help some people feel better and more energetic because it forces them to cut heavily processed junk foods out of their diet. In such cases it’s the sugar and carbs that are making people feel sluggish—not the gluten protein. If you don’t have celiac or a gluten sensitivity, most experts recommend saving yourself the trouble by eating healthier in general rather than abstaining from gluten.

14. The numbers are growing.

A 2009 study found that four times as many people have celiac today than in the 1950s, and the spike can’t be explained by increased awareness alone. Researchers tested blood collected at the Warren Air Force Base between 1948 and 1954 and compared them to fresh samples from candidates living in one Minnesota county. The results supported the theory that celiac has become more prevalent in the last half-century. While experts aren’t exactly sure why the condition is more common today, it may have something to do with changes in how wheat is handled or the spread of gluten into medications and processed foods.