Algebra helps us navigate hard realities by making them abstract

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Six Math Essentials

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The origins of algebra date back at least 3, 500 years to the ancient Babylonians, but the name—and a systematic approach to solving linear and quadratic equations—came from the Islamic world. Originally called al-jabr by al-Khwārizmī (whose name inspired the word “algorithm”) in 830 CE, the Arabic word means “restoring broken parts.”

To solve a problem with algebra, we identify the key unknowns and give them names. To keep these names short, we use symbols such as x, y, and z. We tend to choose rarely used letters with a neutral connotation to help us generalize and make new connections.¹

Euclid and the ancient Greeks didn’t use variables like x, y, and z. Instead, they made comparisons between lengths, areas, and shapes—for example, x would be the length of a line, x2 would be the area of a square, and x3 would be the volume of a cube. In effect, they converted algebra problems into geometry problems. For the most part it worked, but it was a clunky approach. Algebra is elegant precisely because it frees us from needing a concrete interpretation for every expression.

When we first learn arithmetic, we work with concrete problems where we know the values of the numbers. Algebra offers an extra layer of abstraction that allows us to manipulate numbers whose values we don’t yet know. It’s a surprisingly powerful idea that we can say a lot about x without knowing exactly what x is, that we can say precise things about objects that we don’t precisely know.

Before the popularization of algebraic methods, every problem had its own rule that people had to learn to solve it. If you knew that 6 trees produced 20 bushels of fruit, how many bushels do 13 trees produce? The corresponding rule stated that if you multiply 13 by 20 and divide by 6, you get the answer. But a slight variation in the problem—for instance, asking how many trees produce 30 bushels of fruit—meant you had to learn a different rule. The rules often had complicated names and required a lot of rote memorization. Medieval math textbooks focused on techniques for solving specific practical problems. For example, they contained whole chapters on computus paschalis, the method for computing the convoluted calendrical concept of Easter, which involves the weekly calendar, the lunar calendar, and the solar calendar.² The computations employed what we now call algorithms, but at its core it’s an algebra problem (which Gauss later helped solve). If you know the fundamental principles of algebra, there’s no need to memorize so many bespoke laws.

In high school, my least favorite subject was organic chemistry. I had to memorize a raft of names: This molecule is an alkane, while this other one is an alkene. I preferred physics, because I could derive the answers to problems just by knowing a few basic laws, like Newton’s F = ma. For one hundred different problems, you don’t need one hundred separate techniques. All you need are three or four organizing principles of nature, plus algebra. That gives you the rest.

In principle, algebra lets us abstract anything and everything. That’s the key to its power but also explains why it’s such a roadblock for many middle and high school students. Because it’s so focused on abstracting away from reality, algebra is where a lot of people become traumatized out of mathematics. It’s natural to wonder why you need to solve the quadratic equation, for instance. Indeed, you’re unlikely to use it in your daily life; the benefits are indirect.

Algebraic thinking gives us the agility to manipulate problems in productive ways. A couple years ago, I had to install some heavy curtains at home, and I didn’t quite understand how all the parts fit together. Trying to unscrew and screw everything while standing on a stool was too much for me. But one thing you learn from solving algebra problems is to start with stripped down, special cases, and to work the details back in later. If you have a problem with the variables x, y, and z, you can set x equal to zero, so that everything involving x disappears. Then you can try to solve it when y equals zero. Once you have a better handle on these special cases, you put them together to solve the general case.

To install the curtains, I sat on the floor and first practiced putting them on the curtain rod. Then I removed the rod and tried to install it on the wall without the heavy curtains on it. Only after I could complete both steps separately did I attempt the full installation. My wife teased me for not doing it all at once, but I felt more confident breaking the task into parts and taking the methodical approach. (Assembling furniture is clearly not my strength.) Algebra teaches us to extract essential features from simplified versions of a problem before tackling it in its entirety.

Algebraic thinking counters our tendency to want to retain an excessive number of problem-specific details. If we include too many variables, they distract us from what’s relevant. More to the point, they obscure our view. Even if we’re able to solve that one problem, the extra variables can prevent us from seeing general relationships or underlying principles.

Ironically, one challenge of learning algebra is that it’s all too easy to apply it narrowly—many of us are taught to plug numbers into a formula without understanding why it works. That approach might get you through a math quiz, but it can also lead you astray. Take the 2008 global financial crisis. In the years leading up to it, investment bankers observed that mortgage default risk could be approximated by a Gaussian distribution, better known as a bell curve. Using that information, they managed risk in mortgage-backed securities by identifying the riskiest mortgages and bundling the rest in tranches to sell as a more reliable financial product. The math looked good on the surface, but not everyone who was structuring these products understood why the method seemed to work—and under what conditions it might fail. It was like consulting a medieval textbook for a formula to calculate risk.

People were using these formulas without examining the assumptions under the hood. In particular, the bell-shaped Gaussian distribution best describes independent, uncorrelated data, like the heights of a random group of people (or their math test scores). The financial models assumed that mortgage defaults were uncorrelated, or at least that any correlations were predictable. This made mortgage-backed securities seem safe—allaying concerns that a systemic shock could make a lot of people default at once. But mortgage defaults turned out to be more correlated, less Gaussian, and less predictable than the models assumed. The cascading defaults led to disaster, because the models and formulas that worked in normal situations suddenly stopped working.

Contrary to common misperceptions about algebra, it’s not about mindlessly following recipes. It’s about problem solving. And used correctly it can reveal the underlying reason why a calculation works or why a mathematical relationship is true. A famous story about the German astronomer Johannes Kepler illustrates this well. While purchasing a barrel of wine for his second wedding in Austria in 1613, Kepler questioned the wine merchant’s method for measuring the volume of wine in the barrel. With the barrel on its side, the merchant inserted a notched stick diagonally through the bunghole in the bulging middle of the barrel to the farthest corner, and just like that proclaimed the price Kepler would have to pay. How could these Austrian winemakers know the volume based solely on that stick? They must have had a table showing that for a given diagonal length, the barrel holds a certain volume of wine. But how could that account for different barrel shapes? Kepler knew that a long, skinny barrel could have the same diagonal measurement as a short, fat barrel while having a vastly different volume.

So Kepler did the algebra. He calculated that to maximize the amount of liquid for a particular diagonal length, the barrel had to have proportions as close as possible to a sphere. He found that, unlike the German wine barrels he was accustomed to, Austrian barrels closely matched his calculations for the optimal shape. His algebra further showed that small variations in the shape of these handmade barrels would not invalidate the stick measurement method, since the volume as a function of the height for a given diagonal looks like a parabola and doesn’t change much close to the peak volume. In the end, he proved that the wine merchant’s measurement was accurate, and he understood why it worked. What’s more, in applying rigorous algebraic thinking to the question of wine barrel shapes, Kepler invented a precursor to calculus.

¹ This differs from physics and other branches of science, where names often offer some intuition about the physical setting. Newton wisely wrote his second law of motion as F = ma, where F is force, m is mass, and a is acceleration.
² It’s the first Sunday after the paschal full moon—the first full moon on or after the spring equinox, when the sun sits directly above the equator.

Excerpted from Six Math Essentials by Terence Tao. Published by Farrar, Straus and Giroux, November 10, 2026. Copyright © 2026 by Terence Tao. All rights reserved.

Meet the Writer

Terence Tao

About Terence Tao

Dr. Terence Tao is a professor of mathematics at the University of California, Los Angeles and the author of “Six Math Essentials.”