here is some about euclidean division. The article states that although euclid doesn't know about the theorem but still it named after him (see) . The reason for it is also mentioned (written as "The term Euclidean division was introduced.........") but i can't understood it (probably because i'm not familiar with abstract algebra). Can someone help me understanding that.
maths query: article states that Theorem was named after Euclid, tho Euclid didn't know it
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HallsofIvy
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I would have used the phrase "Euclidean Algorithm" or "Euclidean Division Algorithm". But, what, exactly, are you asking? Are you just asking about the name? Or are you asking about the algorithm itself? The latter is just the basic division you learned in second or third grade! 4 divides into 15 three times with remainder 3: 4(3)+ 3= 15. 7 divides into 37 5 times with remainder 2: 7(5)+ 2= 37. 9 divides into 4 0 times with remainder 4: 9(0)+ 4= 4.
Given any two whole numbers, a and b, with \(\displaystyle b\ne 0\), a is the "dividend" and b the "divisor" there exist two whole numbers, q and r, such that "b divides into a q times with remainder b": b= aq+ r. "q" is the "quotient" and r is the "remainder.
Given any two whole numbers, a and b, with \(\displaystyle b\ne 0\), a is the "dividend" and b the "divisor" there exist two whole numbers, q and r, such that "b divides into a q times with remainder b": b= aq+ r. "q" is the "quotient" and r is the "remainder.
I'm asking to understand the lines in " " in my thread above
Dr.Peterson
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I think you're saying that you don't understand the sentence
in the Wikipedia page, because you don't understand what Euclidean rings are. I think all you need to understand is that Euclidean rings are an abstract entity that is a generalization of integers in some sense. When mathematicians invent a generalization, that is often the first time they need a special name for it. (For example, the term "commutative" was first used when group theory started studying things other than numbers that have such a property, so they needed a name for the property distinct from the entities it applies to.)
In general, it is not necessary to fully understand the context in which a term arose, in order to use the term itself. I wouldn't be concerned about it, unless you have some reason to learn about Euclidean rings.
It may also be important to realize that the term "Euclidean division" as used there is commonly just called "integer division", and is not the same thing as the Euclidean Algorithm (which has its own Wikipedia page). The theorem on the page you referred to is just called the Division Algorithm, and basically just says that you can divide integers. (Oddly, as the page mentions, this is a theorem, not an algorithm, despite that name.) These terms do tend to confuse people.
The term "Euclidean division" was introduced during the 20th century as a shorthand for "division of Euclidean rings".
in the Wikipedia page, because you don't understand what Euclidean rings are. I think all you need to understand is that Euclidean rings are an abstract entity that is a generalization of integers in some sense. When mathematicians invent a generalization, that is often the first time they need a special name for it. (For example, the term "commutative" was first used when group theory started studying things other than numbers that have such a property, so they needed a name for the property distinct from the entities it applies to.)
In general, it is not necessary to fully understand the context in which a term arose, in order to use the term itself. I wouldn't be concerned about it, unless you have some reason to learn about Euclidean rings.
It may also be important to realize that the term "Euclidean division" as used there is commonly just called "integer division", and is not the same thing as the Euclidean Algorithm (which has its own Wikipedia page). The theorem on the page you referred to is just called the Division Algorithm, and basically just says that you can divide integers. (Oddly, as the page mentions, this is a theorem, not an algorithm, despite that name.) These terms do tend to confuse people.