Famous Theorems
What Is A Theorem?
The word “theorem” is related to “theory” and “theoretical,” all words that come from the Greek “theorein,” meaning “to look at.” The word later came to mean “study” and “speculate.” The word “theorem” now describes a statement that is considered true because it is justified through mathematical logic. In other words, it is not self-evident but rather established by a proof.
Theorems are at the heart of mathematics, and they are considered to describe facts that are absolutely true. Mathematics is fundamentally about discovering and understanding the laws that underpin our system of numbers, geometry, and algebra. Laws and principles are really just theorems that have broad applications.
The following are the ten most famous theorems students will likely encounter throughout their high school and college math classes.
- The Pythagorean Theorem
One of the best-known theorems is the Pythagorean Theorem. It was named for the Greek mathematician Pythagoras, who was the leader of a small group of mathematicians who worshipped math and devoted themselves to the study of numbers and philosophy. Although the theorem itself was known for 1000 years before Pythagoras’s time, he was the first person to come up with a proof.
The theorem is about relationships among side lengths in a right triangle and simply states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. If the hypotenuse (the side opposite the right angle, or the long side) is called c and the other two sides are a and b, we can also express this theorem with the formula a2 + b2 = c2.
- Euclid’s Proof Of The Infinitude Of Primes
Euclid was a Greek mathematician born in the mid-4th century BCE who was known as the father of geometry. Using just a small set of axioms, he derived all of the principles of what became known as Euclidean geometry, which dominated the study of mathematics until the 20th century. One of his most famous theorems is his proof that there is an infinite number of prime numbers.
His proof relies on the axiom that all non-prime, or composite, numbers can be broken down into prime factors. He then said that for any set of prime numbers, it is possible to find another prime number that’s not in the set by multiplying the numbers in the set and adding 1. For example, if a set of prime numbers is {3, 5, 11}, we can find a new prime with this equation: 1 + (3 x 5 x 11) = 166. The new number cannot have any factors in the original set. The factors of 166 are 1, 2, 86, and 166.
Therefore, if we take all the prime numbers that exist, multiply them, and add 1, we will get a new prime number. Therefore, the number of prime numbers has no limit.
- The Fundamental Theorem Of Arithmetic
Euclid is also responsible for a theorem known as the Fundamental Theorem of Arithmetic, which states that all numbers are composed of prime factors, the foundation of the previous theorem. This theorem states that every number higher than 13 is either a prime number or the product of prime factors. It also states that every number has a unique prime factorization that has only one possible representation. In other words, if the number 100 is factored into primes, we get 2 x 2 x 5 x 5. The theorem says that this is the only possible prime factorization of the number 100.
The theorem also explains why the number 1 is not considered a prime number. If it were, then another number could have multiple prime factorizations. In other words, 100 could be factored as 1 x 2 x 2 x 5 x 5 or 1 x 1 x 2 x 2 x 5 x 5, etc.
- The Four-Color Theorem
This theorem states that if a flat plane is divided into any number of contiguous regions, a minimum of four colors is required to color in each area so that no two colors touch.
The four-color theorem is a much more recent theorem than the ones previously discussed, and it’s famous because it was the first theorem proved with a computer. It was first proved in 1975 by Kenneth Appel and Wolfgang Haken using a computer program designed especially for the proof. Two more software-assisted proofs were made in 1997 and in 2005. However, mathematicians are still looking for a simpler and more elegant proof.
- The Square Root Of 2 Is Irrational
Although this proof has been attributed to the Pythagorean mathematicians, the real origins of it are unknown, and experts believe it came from ancient Egyptian sources. This theorem states that the square root of 2 is irrational, meaning that it can not be expressed as a ratio, or fraction, of two numbers.
“The square root of 2” can answer the question “what is the length of the diagonal of a 1-inch square?” According to the Pythagorean theorem above, a2 + b2 = c2, so 12 + 12 = 22. Thus, √12 + √12 = √22 and 1 + 1 = √2.
One way of proving that √2 is irrational is to start with the fact that when we square a rational number, each prime factor has an even exponent. Since √22 is 2 and the number 2 can be expressed as 21/11 (22/21 or 23/22, etc.), the square root of 2 must be irrational.
- Pi Is Irrational
Another famous irrational number is pi, also written with the Greek letter π. Pi is the ratio of the circumference (distance around) to the diameter (distance across) of any circle. While this ratio was observed by ancient cultures, Archimedes of Syracuse, who lived in the second century BCE, was the first to accurately calculate the value of pi. It wasn’t until the 19th century that a mathematician, John Heinrich Lambert, proved that pi is an irrational number, meaning that it cannot be expressed as a ratio of any two numbers. As an irrational number, pi is infinite and non-repeating.
- Fermat’s Little Theorem
Pierre de Fermat, a 17th-century French mathematician, is known as the father of modern number theory. His Little Theorem is a test to determine whether or not a number is prime. It states that if P is prime, then for any integer a, aP – a is divisible by P.
For example 23 – 2 = 6. 6 is divisible by 3 because 3 is prime. On the other hand, 34 – 3 = 78. 78 is not divisible by 4 because 4 is not prime. This theorem is useful for checking an integer whose primality is not known because it can quickly identify non-prime numbers.
- Fermat’s Last Theorem
This theorem is famous because it was considered unsolvable for more than 350 years. Fermat wrote it in the margin of a math text in 1638, and a mathematician named Andrew Wiles came up with the proof in 1995, after working on it for seven years.
The theorem states that when n is any number higher than 2, and x, y, z, and n are all natural numbers, there is no solution to the equation xn + yn = zn. The equation is solvable when n = 2, in which case the x, y, and z are known as “Pythagorean triplets.” However, if n is 3 or greater, x, y, and z can not be natural numbers.
- The Law Of Large Numbers
This theorem was first stated by Jakob Bernoulli, a Swiss mathematician who lived in the latter half of the 17th century. It has been called “Bernoulli’s Theorem,” but is more often referred to as the Law of Large Numbers. This law is related to the field of statistics, and it states that when we observe any process that has random outcomes, like a coin toss or a die roll, the average result will approach the expected result as the number of attempts increases.
In other words, if a coin is tossed, there is a 50% chance it will land on heads, so we can expect that when a coin is tossed repeatedly, it will land on heads 50% of the time. Of course, it’s possible to throw 5 or 10 heads in a row, meaning results may start off far from 50%. But as we keep tossing the coin, eventually the average results will stabilize close to 50%.
- The Prime Number Theorem
This theorem addresses the question of how prime numbers are distributed among the positive integers. Proved in 1896 by two independent mathematicians, Jacques Hadamard and Charles Jean de la Vallée-Poussin, the theorem states that prime numbers occur less frequently as they get larger and quantifies that frequency with the formula π(x) ~ x/log(x) where π(x) refers to the number of primes that are less than or equal to any real number x.
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