An example of a mathematical proof is Euclid’s proof of the infinitude of primes, which shows that there are infinitely many prime numbers.
More detailed answer to your request
Euclid’s proof of the infinitude of primes is a classic example of a mathematical proof. The proof argues that there are infinitely many prime numbers by contradiction. It begins by assuming that there are only finitely many primes, and then goes on to construct a new number that is not in the original list of primes. This new number is shown to be either prime or divisible by a prime that is not on the original list, effectively disproving the assumption that there are only finitely many primes.
The full proof is quite long and involves several steps, but it is considered to be one of the most elegant proofs in mathematics. It has inspired countless other proofs over the centuries, and it remains one of the most important results in number theory.
One interesting fact about Euclid’s proof is that it is more than 2,000 years old, but it still holds up as a rigorous and powerful argument. Another interesting fact is that the concept of “proof by contradiction” that Euclid used is still a widely used technique in mathematics to this day.
As the famous mathematician Paul Erdős once said, “God may not play dice with the universe, but something strange is going on with the prime numbers.” Indeed, the study of prime numbers continues to be a fascinating area of research in mathematics.
| n | Primes <=n | n/ln(n) |
|---|---|---|
| 10 | 2, 3, 5, 7 | 4.3 |
| 10^2 | 25 | 21.7 |
| 10^3 | 168 | 144.8 |
| 10^4 | 1,229 | 1,083.2 |
| 10^5 | 9,592 | 8,632.4 |
| 10^6 | 78,498 | 72,382.4 |
| 10^7 | 664,579 | 620,420.3 |
Table sourced from: https://primes.utm.edu/howmany.shtml
Other approaches of answering your query
What is an example of proof in math? An example of a proof is for the theorem "Suppose that a, b, and n are whole numbers. If n does not divide a times b, then n does not divide a and b." For proof by contrapositive, suppose that n divides a or b. Then n certainly divides a times b, since it divides one of its factors.
For example, direct proof can be used to prove that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2 a and y = 2 b, respectively, for some integers a and b. Then the sum is x + y = 2 a + 2 b = 2 ( a + b ). Therefore x + y has 2 as a factor and, by definition, is even.
Further study
- Proof by contradiction – for example, proving the square roots of prime numbers are irrational numbers by assuming they are rational and showing it cannot happen.
Direct proof, proof by contraposition, and proof by contradiction. Well, there are many more proof techniques (e.g., induction, casework, incognito…)
Here is an example of each.
Direct proof that if [math]n[/math] is even, then [math]n^2[/math] is even: Since [math]n[/math] is even, [math]n=2k[/math] for some integer [math]k[/math]. Moreover, [math]n^2=(2k)^2=4k^2=2(2k^2)=2\ell[/math] where [math]\ell[/math] is the integer [math]2k^2[/math]. Therefore, [math]n^2[/math] is even.
Proof by contraposition that if [math]n^2[/math] is even, then [math]n[/math] is even: We first suppose [math]n[/math] is not even, then we want to prove that the hypothesis can’t hold. Well, since [math]n[/math] is not even, it is odd, say [math]n=2k+1[/math] for some integer [math]k[/math]. With that, [math]n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1=2\ell+1[/math] where [math]\ell=2k^2+2k[/math]. Consequently, [math]n^2[/math] is odd, and hence not even.
Proof by contradiction that [math]\sqrt 2[/math] is irratio…
Response via video
The speaker in the video highlights that mathematics is beyond just computation; rather, it involves discovering patterns and meaning in apparent chaos. Mathematical proofs serve as a creative art that helps in conscious understanding of abstract mathematical truth. By giving an example and questioning why adding odd numbers resulted in the same product as multiplying sequences of numbers by themselves, he emphasizes the importance of proof in understanding mathematical truth. The speaker concludes that intuition is essential when approaching problems from all sides and that it is vital to find ways to solve problems, not just to rely on calculation.
More intriguing questions on the topic
Additionally, What is a simple mathematical proof?
As an answer to this: A mathematical proof is a way to show that a mathematical theorem is true. To prove a theorem is to show that theorem holds in all cases (where it claims to hold). To prove a statement, one can either use axioms, or theorems which have already been shown to be true.
In this way, What is the most famous mathematical proof? The Pythagorean Theorem is arguably the most famous statement in mathematics, and the fourth most beautiful equation.
What are the 3 types of proofs?
As a response to this: There are many different ways to go about proving something, we’ll discuss 3 methods: direct proof, proof by contradiction, proof by induction. We’ll talk about what each of these proofs are, when and how they’re used.
People also ask, How do I write a proof in math?
Answer to this: A sentence must begin with a WORD, not with mathematical notation (such as a numeral, a variable or a logical symbol). This cannot be stressed enough – every sentence in a proof must begin with a word, not a symbol! A sentence must end with PUNCTUATION, even if the sentence ends with a string of mathematical notation.
Are math proofs worth studying?
The answer is: Yes, you should try to understand proofs. A proof of something often reveals an insight that is at least interesting, and often widely applicable. Nobody every has to prove things outside of formal mathematics, but proof is often very useful. Take the programming language Idris, for example.
Consequently, What type of reasoning does a mathematical proof use? Answer to this: Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory.
Consequently, What are the different types of mathematics?
The response is: Branches Of Mathematics. The main branches of mathematics are algebra, number theory, geometry and arithmetic. Based on these branches, other branches have been discovered. Before the advent of the modern age, the study of mathematics was very limited. But over a period of time, mathematics has been developed as a vast and diverse topic.