Pythagorean Theorem

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Draw a triangle with one right angle and put a square on each of its three sides. The theorem says that the big square on the longest side has exactly the same area as the two smaller squares together.

You can see why by a rearrangement. Take a large square and place four copies of the triangle inside it, one in each corner, so that their longest sides enclose a tilted square in the middle. Now slide the same four triangles around inside the large square so that they form two rectangles. The leftover space is now two squares, one on each short side. The triangles cover the same area in both pictures, so the empty space must be the same too: the tilted square equals the two small squares.

In practice the theorem is a rule for distances. If you walk 3 km east and then 4 km north, you are 5 km from where you started, because 9 plus 16 is 25, and 25 is 5 times 5. The same rule works in three dimensions, where the diagonal of a box follows from applying it twice. Every distance computed from Cartesian coordinates uses it.

The Pythagorean theorem states a relationship between the three sides of a right-angled triangle where $a$ and $b$ represent the sides that form the right angle and $c$ is the longest side.

Concrete

Let a right triangle have legs $a, b$ and hypotenuse $c$ (the side opposite the right angle). Then

$$a^2 + b^2 = c^2 .$$

Proof by rearrangement

Place four copies of the triangle inside a square of side $a+b$ so that their hypotenuses form an inner square of side $c$. Counting the area of the outer square in two ways gives

$$(a+b)^2 = c^2 + 4\cdot\frac{ab}{2} \quad \Longrightarrow \quad a^2 + 2ab + b^2 = c^2 + 2ab \quad \Longrightarrow \quad a^2 + b^2 = c^2 .$$

Law of cosines and the converse

For an arbitrary triangle with angle $\gamma$ between the sides $a$ and $b$,

$$c^2 = a^2 + b^2 - 2ab\cos\gamma .$$

For $\gamma = 90^\circ$ this reduces to the Pythagorean theorem. Since $\cos\gamma = 0$ only for $\gamma = 90^\circ$ (with $0 < \gamma < 180^\circ$), the relation $a^2 + b^2 = c^2$ also implies that the triangle has a right angle, which is the converse of the theorem.

Distances and vectors

In Cartesian coordinates the distance between two points in $\mathbb{R}^n$ follows from applying the theorem repeatedly:

$$d = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \dots + (x_n - y_n)^2} .$$

In vector language, $|\vec u + \vec v|^2 = |\vec u|^2 + |\vec v|^2 + 2\,\vec u\cdot\vec v$, and the theorem is the statement that the last term vanishes when $\vec u \perp \vec v$. The magnitude of a velocity, $|\vec v| = \sqrt{v_x^2 + v_y^2 + v_z^2}$, is an example.

In special relativity the corresponding expression for the spacetime interval between two events has a relative minus sign, $\Delta s^2 = c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2$, so spacetime geometry is not Euclidean. Written for an infinitesimal displacement and with general coefficients, the distance formula becomes the line element of Riemannian geometry, as described in the Abstract layer.

$a^2+b^2=c^2$ if and only if the triangle is a right triangle

Abstract

Pythagoras’s Theorem, $$(ds)^2 = (dx)^2 + (dy)^2$$

Riemann realized some fundamental things:

1. Pythagoras’s Theorem not only told of the properties of right-angled triangles, but the function $(ds)$ (the positive square root of $(ds)^2$), was the ‘distance’ between two points;
2. more than that, it was the shortest ‘distance’ between two points;
3. most amazing of all, Pythagoras’s $(ds)^2$ was but one possibility - more generally, it could be written as:

$$(ds)^2 = g_{xx}(dx)(dx) +g_{xy}(dx)(dy) + g_{yx}(dy)(dx) + g_{yy}(dy)(dy) ,$$ where the $g_{ij}$ are coefficients to be determined. In Pythagoras’s Theorem, it just so happens that we have the values $g_{xx}=1$, $g_{xy}=0$, $g_{yx}=0$, and $g_{yy}=1$.

In other cases (other spaces),the $g_{ij}$ need not have these particular values, and need not be constants - they could even be functions of x and y. This discussion has, so far, been in a space of only two dimensions, but Riemann generalized his ‘distance function’ still further to n dimensions. […] Most remarkable of all, Riemann showed that all the geometric properties of an n-dimensional space can be completely determined by just this ‘distance’, $ds$ (the square root of the above equation. For example, if any of the coefficients, $g_{ij}$ , are not constant but are functions of the coordinates, then the corresponding space is not ‘flat’ (Euclidean) but ‘curved’. This, finally, is what ‘curved’ means: it is a measure of the departure from Euclidean space, and is determined by certain functions (Riemann’s ‘curvature functions’) of the $g_{ij}$ -coefficients. While the ‘distance’ is specific to the given space, its value is invariant as regards which coordinate representation has been adopted.

The Lazy Universe by Coopersmith

Why is it interesting?

The Pythagorean theorem enables us to draw accurate maps. This is achieved by covering the area that should be mapped with a virtual network of triangles. These triangles, via the Pythagorean theorem, allow us to measure distances and angles between stretches of land. This method is known as triangulation.

It is still used today, for example, to deduce the locations from given satellite data.