Real-World Problems Solved by Mathematics: Encryption, GPS, Routing, and More

We continue exploring applied mathematics problems that have shaped the tools we use every day.

Applied Mathematics Problems in Data Security: Encryption Over Open Channels

The problem: How can a client send confidential data — a card number, a token — to a server knowing only the server’s public key, while an interceptor cannot decrypt the message or forge the response? Everything rests on the computational difficulty of tasks like factoring enormous numbers into primes.

The solution: A public-key and private-key scheme: anyone with the public key can encrypt, but only the holder of the secret key can decrypt. For an attacker, brute-forcing the possibilities takes an astronomical amount of time — at least until quantum computers arrive.

Transmitting Messages Over a Noisy Channel

The problem: How do you encode bits so that, when part of the signal is corrupted — Wi-Fi interference, radio noise, memory errors — the receiver can still reconstruct the original message?

The solution: A calculated amount of redundancy is added in the form of check bits. An error alters them in a characteristic way, and by following that pattern, the system can locate and correct the fault. The receiver sees a “corrupted word” and, by a set of rules, shifts it toward the nearest valid codeword — recovering the original.

Real-Life Applied Mathematics Problems in Navigation: Finding the Fastest Route

The problem: Given a road map — with distances, speeds, and traffic as edge weights — how do you find the path between two points that minimizes travel time or distance?

The solution: Roads and intersections are represented as a weighted graph, and an algorithm is launched that expands step by step from the starting point, discarding provably suboptimal options. Under natural conditions, it is proven to find the shortest path — not merely a reasonable one.

Determining Position from Satellite Signals: GPS

The problem: By measuring the arrival times of signals from several satellites at known orbits, compute the receiver’s position on Earth.

The solution: Each measurement defines a sphere of possible locations for the receiver. The intersection of several such spheres yields a small region where the receiver actually is. Algorithms solve the resulting system of equations accounting for errors and corrections, iteratively refining the coordinates until the result is accurate enough.

Computing Areas, Volumes, and Other Quantities for Complex Shapes

The problem: Given curved boundaries — a roof, a tank, a machine part — find areas, volumes, centers of mass, and the work done by forces, either exactly or with a guaranteed margin of error.

The solution: The shape is mentally sliced into very small, simple pieces; the contribution of each piece is calculated and then summed. Mathematical analysis makes this rigorous through integrals. In practical calculations, numerical methods are used, which provide a controllable and predictable error.

This is part 2 of a series on solved mathematical problems from real life. More to follow.