GPS — Algebra Finds You on Earth
GPS — How Math Finds Your Location
The Global Positioning System uses algebra and the speed of light to pinpoint your location anywhere on Earth to within a few meters.
The Core Idea: Distance = Speed × Time
Each GPS satellite broadcasts a timed signal. Your phone measures how long the signal took to arrive:
$$d = c \times \Delta t$$
where $c = 299{,}792{,}458$ m/s (speed of light) and $\Delta t$ is the time delay.
Trilateration — Solving a System of Equations
One satellite gives you a sphere of possible locations (radius = distance). Two satellites narrow it to a circle. Three satellites give two points. Four satellites give one unique point.
Mathematically, each satellite gives an equation:
$$(x - x_i)^2 + (y - y_i)^2 + (z - z_i)^2 = d_i^2$$
where $(x_i, y_i, z_i)$ is the satellite's position and $d_i$ is the measured distance.
This is a system of quadratic equations — circles and spheres, the same conic sections from algebra II.
Two towers are located at $A = (0, 0)$ and $B = (10, 0)$ km. Your distances from each are $d_A = 5$ km and $d_B = 7$ km. Find your position.
From tower A: $x^2 + y^2 = 25$
From tower B: $(x-10)^2 + y^2 = 49$
Expand B: $x^2 - 20x + 100 + y^2 = 49$
Subtract A from B: $-20x + 100 = 24 \Rightarrow x = 3.8$ km
Substitute: $y^2 = 25 - 3.8^2 = 25 - 14.44 = 10.56 \Rightarrow y = \pm 3.25$ km
Two solutions — a third measurement would narrow it to one. This is exactly how GPS works, just in 3D.
The Numbers
| GPS Fact | Value |
|---|---|
| Satellites in orbit | 31 active |
| Orbital altitude | 20,200 km |
| Signal speed | 299,792,458 m/s |
| Signal travel time | ~67 milliseconds |
| Accuracy | ~3–5 meters |
| Clock precision needed | ~10 nanoseconds |
GPS solves a system of quadratic equations using the Pythagorean theorem in 3D. The algebra of circles and spheres — studied in geometry class — is what lets your phone know exactly where you are on the planet.