We invited Geoffrey Shen to lead us through an exercise, starting from the seemingly simplest questions, triggering and expanding our in-depth thinking on mathematical problems, and cultivating core mathematical literacy.
In the last article, we discussed the story of Pi.
The article raised a question:
Why is the circumference-to-diameter ratio of two circles of different sizes the same?
This proof is not simple; it requires the concept of limits, which is a part of higher mathematics.
Apart from limits, the proof also requires a theorem:
The corresponding sides of similar triangles are proportional.
Most students are familiar with this theorem, but if asked to prove it, the difficulty is similar to the question of why the circumference-to-diameter ratio of two different-sized circles is the same.
The seemingly obvious conclusion that the corresponding sides of similar triangles are proportional requires highly sophisticated proof techniques and the use of a very important theorem, the Pythagorean Theorem.
For any right-angled triangle, the sum of the squares of the lengths of the two sides forming the right angle equals the square of the length of the hypotenuse – this is the Pythagorean Theorem.
To express the Pythagorean Theorem algebraically, it is:
In ancient China, the Pythagorean Theorem was known as Gou-Gu theorem. In the West, it is known as the Pythagorean Theorem.
Why does the same theorem have different names?
During China’s Western Han Dynasty, there was a mathematical book called “Zhou Bi Suan Jing” which contained the record “a Gou of three, a Gu of four, a hypotenuse of five.”
“Gou” and “Gu” referred to the two legs forming the right angle in a right-angled triangle in ancient Chinese terminology, while “hypotenuse” referred to the longest side of the right-angled triangle.
“A Gou of three, a Gu of four, a hypotenuse of five” means that for a right-angled triangle with the two legs forming the right angle being 3 and 4, the length of the hypotenuse would be 5.
Although “a Gou of three, a Gu of four, a hypotenuse of five,” is correct, it is regrettable that it cannot be considered a theorem, as it is at most a specific example that conforms to the theorem.
“A Gou of three, a Gu of four, a hypotenuse of five,” “a Gou of five, a Gu of twelve, a hypotenuse of thirteen,” and “a Gou of seven, a Gu of twenty-four, a hypotenuse of twenty-five” – all of these are correct but are merely special cases.
In the realm of mathematics, special cases are only special cases. No matter how many special cases there are, they may not represent a general rule, and even less can they constitute a theorem.
Pythagoras was an ancient Greek mathematician who discovered this theorem several hundred years after the “Zhou Bi Suan Jing.”
However, Pythagoras provided a rigorous proof that for any right-angled triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two sides forming the right angle.
The proof is very simple, so much so that even elementary school students can understand it.
Given a square ABCD with a side lengths of a + b, take four points EFGH on the four sides, dividing each side into parts a and b. There are two methods, as shown in the illustration below.
In the first method (see left part of the diagram), connect EFGH, which happens to form a square.
Thus, the area of square ABCD is equal to the area of square EFGH plus the area of the four right-angled triangles.
The four right-angled triangles can be combined to form two rectangles with lengths and widths of a and b, respectively. So:
In the second method (see right part of the diagram), the area of square ABCD is equal to the area of the two smaller squares plus the area of the two rectangles.
The two rectangles, each having a length and width of a and b, can also be split into four right-angled triangles. So:
The results calculated by both methods must be equal,
hence:
Pythagoras’s proof established this important theorem about right-angled triangles, which was ultimately named after him.
The Pythagorean Theorem can indeed be said to be aptly named.
The Pythagorean Theorem can be considered the most important theorem in geometry, without peer, because the proof of many other geometric theorems are inseparable from it.
For instance, as mentioned above, to prove that two circles of different sizes have the same ratio of circumference to diameter, you need to use the proportionality of corresponding sides in similar triangles. And to prove this proportionality, you must make use of the Pythagorean Theorem.
The Cosine Rule (Law of Cosines) in trigonometry looks a bit similar to the Pythagorean Theorem:
The Cosine Rule can be seen as an extension of the Pythagorean Theorem. The Pythagorean Theorem is only applicable to right-angled triangles, but the Cosine Rule applies to any triangle, including right-angled triangles. When angle C is 90 degrees, its cosine value is zero, and the last term in the law of cosines equation disappears, leaving exactly the Pythagorean Theorem.
Thus, the Pythagorean Theorem can be said to be a special case of the Cosine Rule.
However, the proof of the Cosine Rule must utilize the Pythagorean Theorem.
Without the Pythagorean Theorem, there would be no Cosine Rule.
There is another important formula in trigonometry called the compound angle formulas:
The compound angle formulas are the starting point for all angle-related formulas in trigonometry. Whether we use the reciprocal identities, the double angle formulas, or the auxiliary angle formulas, they can all be viewed as special cases of the compound angle formulas.
The proof of the compound angle formulas relies on the Cosine Rule, and the proof of the Cosine Rule cannot be done without the Pythagorean Theorem.
The Pythagorean Theorem is like the cornerstone at the very bottom of the edifice of geometry.
What’s even more surprising is that, through the Pythagorean Theorem, the existence of irrational numbers can be found.
Take a right-angled triangle with both legs having a length of 1. By using the Pythagorean Theorem, it can be calculated that the length of its hypotenuse is equal to the square root of 2.
The square root of 2 is an irrational number.
The term “irrational number” does not mean a number that behaves irrationally.
If a number can be expressed as the quotient of two integers, it is a rational number.
A number that cannot be expressed as the quotient of two integers is an irrational number.
Using a proof by contradiction, it is not difficult to prove that the square root of 2 cannot be expressed as the quotient of two integers; hence, the square root of 2 is an irrational number.
There are many irrational numbers, such as the square roots of 2, 3, 5, and so on, and p is also an irrational number.
In fact, there are infinitely many rational numbers, but there are even more irrational numbers.
Before the discovery of the Pythagorean Theorem, people were unaware of the existence of irrational numbers and believed that all numbers were rational.
As a geometric theorem, the Pythagorean Theorem also contributed an unexpected by-product: in addition to rational numbers, there are irrational numbers.
Indeed, it is the combination of rational and irrational numbers that make up the complete family of numbers.
As one of the greatest mathematicians in history, Pythagoras could have achieved even greater heights with the discovery of irrational numbers.
Regrettably, Pythagoras was steadfast in his belief that all numbers could be expressed as the quotient of two integers; only such numbers were perfect in his view.
A proponent of perfectionism, he could not accept any imperfect numbers.
When he encountered the existence of irrational numbers, which he considered a flaw and a subversion of the perfection of numbers, he chose to reject irrational numbers and to pretend they did not exist.
What’s even more shocking is that when Hippasus, a student of Pythagoras, discovered that the square root of 2 was an irrational number and discussed it with Pythagoras, Pythagoras, in an attempt to conceal the existence of irrational numbers, went so far as to throw Hippasus into the sea…
The pursuit of extreme perfection led to an extremely imperfect result.
The Pythagorean Theorem is a great discovery; it has immortalized Pythagoras’s name throughout history but has also stained his reputation.
A perfect theorem, in the hands of someone pursuing perfection, can end up being imperfect.
Perfect truth, interpreted by the imperfect, can be rendered imperfect.
Even great individuals have their flaws and blind spots.
As ordinary people, we should be more cautious and reflective.
Only with a humble heart can we continuously climb higher in the tower of truth.