Life is nonlinear. So handle it using Math.
A thousand examples can make a mathematical statement look convincing. One counterexample can show that it is false. This is among the first lessons we teach in mathematics, and it remains just as
Ten voices on beauty, imagination, uncertainty, and the patient work of understanding. Mathematics is usually introduced through symbols. We learn to calculate, solve equations, draw figures, and
Some equations become easier when we stop searching for their solutions and ask what any solution would have to look like. The problem, A1 from the 2024 William Lowell Putnam Mathematical Competition,
Choose four consecutive integers. Multiply them together, then add one. The result is always a perfect square. For example, 1 × 2 × 3 × 4 + 1 = 25 = 5² 2 × 3 × 4 × 5 + 1 = 121 = 11² 3 × 4 × 5 × 6 + 1
An AI-assisted construction has brought fresh attention to the Hopf problem. The mathematics turns on a subtle question: when do local complex coordinates fit together? A sphere is easy to describe
The number 1.001 is only slightly larger than 1. If we multiply a thousand copies of it together, will the result exceed 2? At first glance, it is difficult to judge. Each multiplication increases the
When I say that I teach mathematics, people have a fairly clear picture of my work. There are lectures, students, examinations, and answer sheets. But mathematical research is harder to explain. What
When I look at an algebra problem, I first ask whether the expressions have a relationship worth preserving. Expanding everything can make that relationship harder to see. Consider the equation
In the vast landscape of mathematics, some equations appear deceptively simple yet hold surprising depths. The equation 2ˣ = x³² is a prime example. It pits an exponential function against a
Isaac Newton is universally celebrated for his laws of motion, universal gravitation, and the development of calculus. In the world of mathematics, his name is also associated to a famous and slightly
For centuries, understanding how fluids move has been important in physics and engineering. It helps explain things such as air flowing over an airplane wing, weather systems, and blood flowing
Imagine Norman, a super salesperson, sells a bike originally costing him $79 to a customer for $99. The customer pays with a $150 check, and since the banks are closed, Norman cashes it with a
