Pythagoras Theorem

Pythagoras Theorem is easily one of the most famous equations in science & mathematics which are familiar to anyone who has attended school. It states that in any right-angled triangle, the sum of the squares on the 2 smaller sides would be equal to the sum of the square of the hypotenuse.

If you really think about it, the simple phrase “The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides” is pure mathematical magic.

Discovered in Many Cultures

This theorem was known even to Sumerians, Babylonians, Chinese and the Hindus. Many cultures knew of different sets of 3 numbers (3, 4 & 5 or 5, 12 & 13), or Pythagorean Triples which could form right-angle triangles.

Egyptians used it for measuring out their fields in rectangular shapes. They used a looped rope 12 units long knotted so that when stretched, it formed a triangle with sides 3, 4 & 5. These persons were even called "rope stretchers"!

Hindus knew that a square constructed on the diagonal of another square is twice its area.

But these cultures possibly did not know "the reason why" such numbers formed a right-angle triangle!

Pythagoras is accepted as the first person to "prove" it mathematically!

Rhind or Ahmes Papyrus

The Rhind Mathematical Papyrus is sometimes called the “Ahmes Papyrus” in honor of the scribe who compiled it. The papyrus is from the Egyptian Middle Kingdom and dates to around 1650 BCE. It was purchased by Henry Rhind in Egypt in 1858 and placed in the British Museum in 1864 by the estate of Henry Rhind, thus it bears his name.

This papyrus was probably a mathematics textbook, used by scribes to learn to solve particular mathematical problems by writing down appropriate examples.

It contains problems which indicate an understanding of the theorem was later known as Pythagoras Theorem.

Plimpton 322

The earliest recorded arithmetical artifact is the fragmentary clay tablet known as Plimpton 322, dating back to around 1800 BC in Mesopotamia. This tablet contains a list of “Pythagorean triples” (a, b, c) satisfying, the relation stated by the theorem.

The tablet indicates advanced mathematical knowledge. The table’s layout suggests the use of an implicit identity related to Old Babylonian exercises. The purpose of the table remains unknown, but it might have served as a source of numerical examples for educational purposes.

The tablet also shows that the Babylonians had a sophisticated understanding of right-angled triangles and could solve problems related to their sides with exceptional accuracy.

Proofs of the Pythagoras Theorem

The credit for finding this goes to the Greeks. They "proved" the theorem. They took results about lines, angles etc which had been proved earlier and from these results, proved the theorem. This result then became common intellectual property.

We have no conclusive evidence as to who proved the theorem first. In fact there is no evidence as to how the theorem got associated with Pythagoras.

Euclid includes the proof of the Pythagoras Theorem and its converse as proposition 47 in his Book 1. But he did not call it the Pythagorean theorem. .

The logical simplicity and beauty of his proof has excited many to invent their own proofs. Some of the were geometric and some were algebraic. In 1968, the National Council of Math Teachers (NCTM) published a book listing 367 proofs of the theorem. We give one such proof at the end of this article.

Latest Proof of Pythagoras Theorem

In accepting valid proofs of the Pythagoras Theorem, proofs based on Trigonometry have not been accepted because, according to mathematicians, Trigonometry itself depends on the Pythagoras Theorem for its basic ideas.

Two US school students Ne'Kiya Jackson and Calcea Johnson, of St Mary's Academy, New Orleans, have developed a new trigonometric proof which is not dependent on the Pythagoras Theorem. For this they used a trigonometric formula proposed by Brahmagupta which did not depend of the Pythagoras Theorem.

Brahmagupta’s Law of Sines stated that which for a triangle ABC can be stated as given below.

SinA/a = SinB/b = SinC/c where A, B & C are the 3 angles and a,b & c are the lengths of the sides opposite to them.

An interesting video on this proof and on other proofs of Pythagoras Theorem which has been produced by Presh Talwalkar of Mind Your Decisions is available at https://www.youtube.com/watch?v=juFdo2bijic

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Importance of the Right Angled Triangle

The special relationship between the sides of a right angle convinced Greek mathematicians of the importance of the right angled triangle.

There is one more reason which convinced them of this.

Most shapes in this world could be approximated with polygons and any polygon can be subdivided into a number of triangles. Each triangle can be divided into two right angled triangles by drawing one of the altitudes. Any right angled triangle can be divided by drawing an altitude meeting the hypotenuse into two similar right angles.

Hence any shape can be divided into a series of right angled triangles! Hence the world is made of right angled triangles!

Numbers Vs Segments

Pythagoras theorem does not really talk in terms of lengths of the sides of the triangle, but in terms of areas of squares on the sides.

This is because in ancient times, number systems had not progressed beyond rational numbers. Lengths of line segments were denoted by the term "magnitude" and were considered a different kind of quantity other than numbers. Therefore, they worked only with ratios of segments.

Proving the Pythagoras Theorem using Areas

Also the Pythagoras theorem was easy to prove using the areas of squares on its sides.

If we drawn a right angled triangle with sides 3, 4 & 5on a square ruled paper, using simple geometry, we can see that the squares on each of these sides would have areas 9, 16 & 25 and 9 + 16 = 25.

Irrational Numbers & the Pythagoras Theorem

It was through Pythagoras Theorem that the existence of irrational numbers was discovered. In trying to find the hypotenuse of a right triangle with sides 1, Greeks arrived at a answer which today we would call,√2 or "square root of 2".

Using one of the earliest proofs in geometry, Greeks also realised that √2 was also a number which could not be expressed as a ratio of 2 integers. Hence √2 was not a rational number!

This was a big shock to the Pythagorean school, who had believed that the world could be completely understood using rational numbers. They tried to keep the discovery of non-rational numbers as a secret. Later these numbers were called Irrational Numbers.

Importance of the Pythagoras Theorem

Apart from Irrational Numbers, trigonometry, algebra, differential equations and even imaginary complex numbers were founded using the Pythagorean theorem.

Golden Triangle

φ is called the Golden Ratio.

A right angled triangle with the hypotenuse being ϕ^2 and the other 2 sides being ϕ & 1 can be called the Golden Right Angled Triangle, since ϕ^2 = ϕ + 1

Lengths of Line Segments

In coordinate geometry, the Pythagoras Theorem gave an easy way to find the distance of an line segment connecting any two points on the cartesian plane.

Extension of Pythagoras Theorem

Once the theorem can be thought of as a relation between areas, it can also be seen as a relation between areas of similar figures, not only squares.

Hence the area of a circle drawn with the hypotenuse of a right-angle triangle, will be equal to the sum of the areas of circles drawn on the other two sides! If 3 similar triangles are drawn on the 3 sides of a right-angle triangle, then the area of the triangle on the hypotenuse will be equal to the sum of the triangles on the other 2 sides.

Pythagorean triples

Any 3 numbers which satisfy the relation of the theorem are called “Pythagorean Triples”. (3, 4 & 5), (5, 12 & 13) and (8, 15 & 17) are some examples. In fact, using the following 3 algebraic expressions given below will give an infinite number of Pythagorean triples for different values of m & n.

(m2 - n2) ,mn, (m2 + n2)

Proof of Pythagoras Theorem

There has yet been no "formal proof" of the theorem starting from first principles. Most proofs are visual or use trigonometry, which itself is based on the Pythagoras Theorem!

We are providing one such "visual" proof which can be demonstrated with a sheet of paper which can be cut and rearranged to bring out the proof.

Chapter 23.8 gives a visual proof of the theorem.