Some people may think that the theorem is true everywhere. However, the theorem is "not" true in more general number systems, like . This was first mentioned by Ernst Kummer in 1843, in his work on Fermat's last theorem. For more information about that: read .
The proof consists of two parts: "first" we show that every number can be written as a product of primes; "second" we show that if we write a number as a product of primes for a second time, then the two lists of prime numbers must be the same.
We show that if "not" every number greater than 1 can be written as a product of primes, we end up in some kind of impossibility. So after that we conclude that it "must" be true that every number can be written as a product of primes.
So, now see what happens when somebody says that he/she knows a positive integer, greater than 1, which can "not" be written as a product of primes. In that case we ask him/her to mention "all" the numbers, greater than 1, that can "not" be written as a product of primes. One of these numbers must be the "smallest": let's call it "n". Of course, this number "n" cannot be 1. Further, it cannot be a prime number, because a prime number is a 'product' of a single prime: itself. So it "must" be a product of numbers. Thus-
where both "a" and "b" are positive integers that are of course smaller than "n". But: "n" was the "smallest" number that can not be written as a product of primes. So it must be possible to write "a" and "b" as products of primes, because they are both smaller than "n". But then the product
can be written as a product of primes as well. This is an impossibility because we said that "n" can not be written as a product of primes.
We have now shown the impossibility which exists if the first part of the theorem would not be true. In this way we have now proven the first part of the theorem.
Now we have to prove that there is only one way to write a positive number greater than 1 as a product of prime numbers.
To do this, we use the following lemma: if a prime number "p" divides a product "ab", then it divides "a" or it divides "b" (Euclid's lemma).
First we now prove this lemma. Well, assume now that "p" does not divide "a". Then "p" and "a" are coprime and we have that says that there must be integers "x" and "y" such that
Multiplying everything with "b" gives
Remember that "ab" could be divided by "p". So now, on the left-hand side we have two terms that are divisible by "p". So the term on the right-hand side is also divisible by "p". We have now proven that if "p" does not divide "a", it must divide "b". That proves the lemma.
Now we will prove that we can write an integer greater than 1 in "only one" way as a product of prime numbers. Take two products of primes "A" and "B" which have the same outcome. So we know for the outcome of the products that "A" = "B". Take any prime "p" from the first product "A". It divides the "A", so it also divides "B". Using several times the lemma that we just proved, we can see that "p" must then divide at least one factor "b" of "B". But the factors are all primes themselves, so also "b" is prime. But we know that "p" is also prime, so "p" must be equal to "b". So now we divide "A" by "p" and also divide "B" by "p". And we get a result like "A*" = "B*". Again we can take a prime "p" from the first product "A*" and find out that is equal to some number in the product "B*". Continuing in this way, at the end we see that the prime factors of the two products must be exactly the same. This proves that we can write a positive integer as a product of primes in only one unique way.
A Portuguesa
"A Portuguesa" ( [ɐ puɾtuˈɣezɐ] ), meaning "Song of the Portuguese" ( 'The Portuguese [Song]') is the national anthem of Portugal, adopted in 1911. It was composed by Alfredo Keil, and its lyrics (words) were written by Henrique Lopes de Mendonça in response to the British Ultimatum for Portuguese troops to get rid of their positions in Africa between the (now former) colonies of Mozambique and Angola in the (also known as the Pink Map).
Welding
Welding is a way of heating pieces of metal using electricity or a flame so that they melt and stick together. There are many kinds of welding, including arc welding, resistance welding, and gas welding. The most common type is arc welding. Anyone who is near arc welding needs to wear a special helmet or goggles because the arc is so bright. Looking at the arc without visual protection may cause permanent eye damage. It is also important to cover all your skin because it can give you something like a sunburn. Hot sparks from the weld can burn any skin that is showing. One kind of welding that does not use an arc is Oxy-fuel welding (OFW), sometimes called gas welding. OFW uses a flame to heat up the metal. There are other kinds of welding that do not use an arc.
Any welding process that utilizes an electric arc is known as arc welding. The common forms of arc welding include:
Arc welding heats metals by making a high-current electric arc between pieces of metal to be joined and an electrode.
Use of the electrode varies based on the type of welding process. In SMAW, GMAW, and related welding processes, the electrode is consumed and becomes part of the weld. The electrode is usually made of the same kind of metal to be welded. Because the electrode is consumed by the welding process, the electrode must constantly be fed into the weld. The SMAW welding process features a "stick" electrode impregnated with a weld promoter known as flux, clamped to the end.
The GMAW welding process uses a thin wire on a rotating spool, as a continuous electrode. The size of this electrode varies from around 0.635 millimeters to about 4 millimeters. The welding machine has a motor-driven spool inside that feeds the wire electrode into the weld.
The TIG welding (GTAW) process features an electrode that is not consumed by the welding process as the metal that makes up the weld does not have any electricity flowing through it. The electrode is made of Tungsten, so used as it will not melt while immersed in the electrical arc. A filler metal, in the form of a rod, can be used to add metal to the weld area.
Almost all welding uses filler metal to fill in the small gap between the metal pieces. The extra metal helps to make the weld strong. Sometimes welds need to be made without any filler metal. Welding with no filler metal is called autogenous welding.
All types of welding require that the hot metal have protection. Dirt, rust, grease, and even the oxidation of the metal under the weld process can prevent a proper weld joint. As such all weld processes use one of two protection methods: flux, and shielding gas.
Welding flux may be used in a solid, liquid, or paste form. During welding, the flux will melt and some of it will evaporate. This creates a small pocket of gas around the weld. This pocket of gas prevents oxidation of the metal under weld. Melted flux, through a corrosive reaction, cleans contaminants that prevent a proper weld. After welding, the flux solidifies. This layer of solid flux is called slag, and must be removed from the weld. The SMAW weld process most commonly uses flux, and is most commonly used on steel.