Where Technical Writing Meets Scientific Writing – Euclid’s Proof

Technical writing shares a number of characteristics and overlaps with other types of writing, including business, creative, copy, and scientific writing.
Technical writers, for example, also make excellent science writers since they do not always have to write user manuals and other standard products of the trade. They can use their procedural writing skills, analytical thinking, and interest in scientific topics to write articles explaining scientific facts to a general population.
Here is an example, explaining the Greek mathematician Euclid’s (325-270 BC) proof that there are an infinite number of prime numbers; i.e., those natural numbers that can be divided only by themselves and 1.
– – – – – – – – – – – – – – – – – – – – – – – – – –
TITLE: Euclid’s Proof that There are an Infinite Number of Prime Numbers
Euclid, the Greek mathematician who lived 325-270 BC, has proven that there cannot be a finite number of prime numbers and that they are actually infinite in number.
Do you wonder how he did it?
Here are the steps that Euclid followed (with thanks to Douglas Hofstadter and his must-read book “I Am A Strange Loop”):
1. Let’s assume that there are a FINITE number of prime numbers in the universe. If we can prove that this is IMPOSSIBLE, then we will know that the other alternative MUST be true – that, there has to be an INFINITE number of prime natural numbers in the universe.
2. If there are finite number of prime numbers, there must be one last final prime number P that is bigger than all the others. There has to be such a number that defines the “upper border” of this group of all known prime numbers.
3. Let’s multiply EVERY prime number in this group with every other prime number to get a ridiculously large number that we’ll call Q. By definition, we can divide Q with ANY and ALL prime numbers in this set since Q is the multiplication of all of them.
4. Now let’s think of the very next natural number: Q+1. We know for sure that this is NOT a prime number since P is the last and biggest prime number and it is safely kept inside the FINITE balloon of all known prime numbers.
5. Since Q+1 is not prime, we know it has to be a COMPOSITE number. That is, it has to be divisible by something other than 1 or itself. Why? Because all non-prime numbers are composite numbers that can be divided by at least one prime number.
6. So what can be the prime number that divides Q+1? Can it be 2? No, because 2 is a prime number that divides Q. Since Q+1 is adjacent to Q, and since no two even numbers can be right next to one another along the continuum of natural numbers, we know that Q+1 cannot be an even number and thus cannot be divided by 2.
7. Can Q+1 be divided by the prime number 3? No, because Q is divisible by 3 and no two numbers divisible by 3 can be neighbors.
8. No matter which prime number “p” you select, it cannot divide both Q and Q+1. Since they always divide Q (by definition), they can never divide Q+1.
NOTE: Here is another way of saying the same thing: “Multiples of prime number “p” can never be next door neighbors along the continuum of positive natural numbers.”
9. This contradicts our original assumption that Q+1 must have a prime number divisor. Since that is not possible, we have proven that a fictitious number such as Q+1 cannot exist.
10. Since Q+1 cannot exist, this is a proof that there cannot be a finite number of prime numbers in the universe.
11. If there cannot be a finite number of prime numbers in the universe, this is a proof that there are INFINITE number of prime numbers in the universe.

No Comments Found

Leave a Reply