Today we’ll try to realign our intuition with the standard textbook proof of “ is a Euclidean ring”. denotes the set of all complex numbers of the form where and are integers. . Let us take two complex numbers . Let and , where and . Then if there exists a complex number such thatContinue reading “Proving J[i] is a Euclidean ring.”

# Category Archives: Uncategorized

## Euclidean rings and prime factorization

Now we will talk about the factorization of elements in Euclidean rings. On pg.146 of “Topics in Algebra” by Herstein, it says: “Let be a Euclidean ring. Then every element in is either a unit in or can be written as the product of a finite number of prime elements in .” This seems elementary.Continue reading “Euclidean rings and prime factorization”

## Euclidean rings and generators of ideals

This is to address a point that has just been glazed over in “Topics in Algebra” by Herstein. In a Euclidean ring, for any two elements such that . Also, there exists a function such that . We also know that the element with the lowest d-value generates the whole ring . The proof ofContinue reading “Euclidean rings and generators of ideals”

## Integral domains and characteristics

Today we shall talk about the characteristic of an integral domain, concentrating mainly on misconceptions and important points. An integral domain is a commutative ring with the property that if and , then . Hence, if , then or (or both). The characteristic of an integral domain is the lowest positive integer such that .Continue reading “Integral domains and characteristics”

## Ordinals- just what exactly are they?!

If ordinals have not confused you, you haven’t really made a serious attempt to understand them. Let me illustrate this. If I have 5 fruits (all different) and 5 plates (all different), then I can bijectively map the fruits to plates. However, I arrange the fruits or plates, I can still bijectively map them. Let’sContinue reading “Ordinals- just what exactly are they?!”

## Why substitution works in indefinite integration

Let’s integrate . We know the trick: substitute for . We get . Substituting into the original equation, we get . Let us assume remains positive throughout the interval under consideration. Then we get the integral as or . I have performed similar operations for close to five years of my life now. But IContinue reading “Why substitution works in indefinite integration”

## Fermat’s Last Theorem

When in high school, spurred by Mr. Scheelbeek’s end-of-term inspirational lecture on Fermat’s Last Theorem, I tried proving the same for…about one and a half long years! For documentation purposes, I’m attaching my proof. Feel free to outline the flaws in the comments section. Let us assume FLT is true. i.e. . We know (Continue reading “Fermat’s Last Theorem”

## Binomial probability distribution

What exactly is binomial distribution? Q. A manufacturing process is estimated to produce nonconforming items. If a random sample of the five items is chosen, find the probability of getting two nonconforming items. Now one could say let there be items. Then the required probability woud be . In what order the items are chosenContinue reading “Binomial probability distribution”

## Continuous linear operators are bounded.: decoding the proof, and how the mathematician chances upon it

Here we try to prove that a linear operator, if continuous, is bounded. Continuity implies: for any for We want the following result: , where is a constant, and is any vector in . What constants can be construed from and , knowing that they are prone to change? As is a linear operator, isContinue reading “Continuous linear operators are bounded.: decoding the proof, and how the mathematician chances upon it”

## Linear operators mapping finite dimensional vector spaces are bounded,

Theorem: Every linear operator , where is finite dimensional, is bounded. Proof where . What we learn from here is where . Similarly, where Another proof of the assertion is which is a constant. Note: why does this not work in infinite dimensional spaces? Because the max and min of and might not exist.