The Pythagorean Theorem in n Dimensions

In n dimensions, the length of a vector v = (v₁, v₂, …, v_n) is ‖v‖ = √v₁² + v₂² + ⋯ + v_n², called the Euclidean norm. It is the Pythagorean theorem applied one dimension at a time. In vector form: whenever u and v are perpendicular, ‖u + v‖² = ‖u‖² + ‖v‖².

This page builds the n-dimensional version step by step, proves the vector version with the dot product, and follows the idea into function spaces, Fourier series and machine learning. It assumes you know the ordinary theorem and the distance formula.

From two dimensions to n dimensions

In two dimensions, a vector (x, y) is the hypotenuse of a right triangle with legs x and y, so its length is √x² + y². In three dimensions you use the theorem twice: once across the floor of a box, and once up to the opposite corner.

For the box above, the floor diagonal satisfies d₁² = 6² + 3² = 45, and the space diagonal satisfies d² = d₁² + 2² = 49, so d = 7. Notice that you never need d₁ itself — the square root cancels — so d² = 6² + 3² + 2². Whole-number solutions like 2-3-6-7 are called Pythagorean quadruples; the three-dimensional case is covered in detail on the Pythagorean theorem in 3D.

Nothing stops the argument at three. Suppose the formula is true in n − 1 dimensions. Split a vector in ℝⁿ (the space of lists of n real numbers) into two pieces:

The first piece lies in the first n − 1 coordinate directions and the second points along the last one, so they are perpendicular. The theorem for those two legs, plus the formula already known for n − 1 dimensions, gives

By induction the formula holds in every dimension. The distance between points p and q is the length of p − q: d(p, q) = √∑(p_i − q_i)².

Vectors: orthogonality and the dot product

The coordinate formula hides a cleaner statement. The dot product of two vectors is u · v = u₁v₁ + u₂v₂ + ⋯ + u_nvn, and the squared length of a vector is its dot product with itself: ‖v‖² = v · v. Two vectors are called orthogonal (perpendicular), written u ⊥ v, when u · v = 0.

Now expand, using the fact that the dot product distributes like ordinary multiplication:

The middle term 2 u · v is the correction for vectors that are not perpendicular. Writing u · v = ‖u‖‖v‖cosθ turns the same expansion into the law of cosines, and the sign of the dot product tells you whether the angle between the vectors is acute or obtuse — the vector form of classifying triangles by their sides.

The theorem extends to any number of mutually orthogonal vectors. If v₁, …, v_k are pairwise perpendicular, every cross term vanishes and ‖v₁ + ⋯ + v_k‖² = ‖v₁‖² + ⋯ + ‖v_k‖². The coordinate formula ∑v_i² is the special case where the v_i point along the coordinate axes.

Inner-product spaces: the theorem without coordinates

Mathematicians noticed that the proof above used only three properties of the dot product, so they turned them into a definition. An inner product on a vector space is a rule ⟨u, v⟩ that is symmetric, linear in each slot, and positive (⟨v, v⟩ > 0 unless v = 0). Define length by ‖v‖ = √⟨v, v⟩ and call vectors orthogonal when ⟨u, v⟩ = 0. The same expansion then proves the Pythagorean theorem in every inner-product space.

The "vectors" need not be arrows. Functions on an interval form a vector space, and

is an inner product. With it, sinx and cosx are orthogonal, because ∫_−π^πsinxcosx dx = 0. Each has squared length π. The theorem predicts ‖sin + cos‖² = 2π, and a direct calculation agrees: ∫_−π^π(sinx + cosx)²dx = ∫_−π^π(1 + sin2x) dx = 2π. (The first step uses sin²x + cos²x = 1, the Pythagorean identity.)

Not every length obeys Pythagoras

Some useful ways of measuring length do not come from an inner product. The "taxicab" length ‖(x, y)‖₁ = |x| + |y| measures distance along a street grid. For the perpendicular vectors (1, 0) and (0, 1), it gives ‖(1, 1)‖₁² = 4, but 1² + 1² = 2. The Pythagorean theorem fails.

There is a sharp test. A norm comes from an inner product exactly when it satisfies the parallelogram law, ‖u + v‖² + ‖u − v‖² = 2‖u‖² + 2‖v‖² (a theorem of Pascual Jordan and John von Neumann, 1935). The Euclidean norm passes; the taxicab norm does not. In this sense the Pythagorean theorem is what makes Euclidean length special.

Projections and the closest point

One of the theorem's most important jobs is finding the closest point. Let W be a line or plane (more generally, a subspace) through the origin, and let v be any vector. Drop a perpendicular from v to W; its foot p is the orthogonal projection of v onto W.

For any other point w in W, the vector v − p is perpendicular to p − w (which lies in W). By the theorem,

So the projection is the closest point of W to v. This is the principle behind least-squares fitting: the best-fit line through data is the projection of the data vector onto the space of possible lines, and "least squares" means least squared Pythagorean distance.

If e₁, …, e_k are perpendicular unit vectors in W, the projection is p = ∑⟨v, e_i⟩e_i, and the theorem gives ‖v‖² = ‖p‖² + ‖v − p‖². Dropping the last term yields Bessel's inequality, ∑⟨v, e_i⟩² ≤ ‖v‖² — a fancy way of saying a leg is never longer than the hypotenuse.

Parseval's identity: Pythagoras in infinitely many dimensions

When the perpendicular unit vectors are numerous enough to build every vector (a complete orthonormal basis), nothing is left over, and Bessel's inequality becomes an equation:

This is Parseval's identity, named after Marc-Antoine Parseval, who stated a version for series in a memoir presented in 1799. The sum can have infinitely many terms: it is literally the Pythagorean theorem with infinitely many perpendicular legs. Complete inner-product spaces in which this works are called Hilbert spaces, after David Hilbert.

The classic example is Fourier series. On [−π, π] the functions cosnx and sinnx are mutually orthogonal, and a function can be written as f(x) = a₀ over 2 + ∑_{n ≥ 1}(a_ncosnx + b_nsinnx). Parseval's identity then reads

The legs add up only gradually: the first Fourier term alone captures about 61% of the squared length, the first 10 terms about 94%, and the first 100 about 99.4%. In signal processing the same identity says that the energy of a signal equals the total energy of its frequency components. It also tells engineers exactly how much error they introduce when they drop the smallest components — the squared error is the sum of the squares of what was dropped — a key idea behind data compression.

Euclidean distance in data science

In machine learning each item — a customer, an image, a song — is often described by a list of numbers called a feature vector, so a dataset is a cloud of points in a high-dimensional space. Euclidean distance, computed with the n-dimensional theorem, measures how similar two items are.

• k-nearest neighbors classifies a new point by looking at the closest labeled points.
• k-means clustering groups points so that the total of the squared distances to each group's center is as small as possible — a sum of Pythagorean squares.
• Feature scaling matters: if one feature is measured in dollars and another in kilometers, the large numbers dominate the sum of squares, so data are usually standardized first.

High dimensions also hold surprises, and the theorem explains them. For two random points in the unit cube in n dimensions, each coordinate difference contributes 1 over 6 on average to the squared distance, so the expected squared distance is exactly n over 6. In 1,000 dimensions a typical distance is about 12.9, and the distances between random pairs typically vary by only about 2% around that value. When every point is roughly equally far from every other, "nearest neighbor" loses meaning — part of what data scientists call the curse of dimensionality. Read more on the Pythagorean theorem in computer science.

Other higher-dimensional versions

The norm formula generalizes lengths. There is also a version for areas: in a box-corner tetrahedron, the squared area of the slanted face equals the sum of the squared areas of the three right-angled faces — de Gua's theorem — and it too extends to n dimensions. And if space itself is curved, the formula holds only for very small triangles, as explained on the Pythagorean theorem in non-Euclidean geometry.