## 백지영 씨가 자랑스럽다

백지영, 그가 구설수에 오르며 가수로서의 활동을 중단할 수밖에 없었던 그 때, 그가 정말 안쓰러웠다. 그리고 그를 그렇게 몰고 간 사람이 미웠다. 그러나 이제 백지영 씨는 다시 돌아왔고 가수로서 무대에 섰다. 당당한 그가 자랑스럽다. 비단 백지영 씨만의 이야기가 아니다. 우리 나라에는 백지영 씨의 일과 비슷한 사건으로 수모를 겪은 사람들이 많이 있다. 우리가 사는 세상은 그러한 일로 난처함을 겪는 사람들이… Read More »

## Which is better, e or π?

Top ln(e10) reasons why e is better than pi : e is easier to spell than pi. pi ≒ 3.14 while e ≒ 2.718281828459045. The character for e can be found on a keyboard, but pi sure can't. Everybody fights for their piece of the pie. ln(pi1) is a really nasty number, but ln(e1) = 1. e is… Read More »

## Euler Phi-Function is a Multiplicative Function

Let $$n$$ be a positive integer. Recall that the Euler phi-function $$\phi(n)$$ is defined as the number of positive integers less than or equal to $$n$$ and relatively prime to $$n.$$ Note that $$\phi(1)=1.$$ We have seen that Euler used this function to generalize the Fermat's Little Theorem. It is sometimes needed to calculate the value $$\phi(n)$$ of… Read More »

## Euler’s Generalization of Fermat’s Little Theorem

Fermat's Little Theorem says: Theorem 1. (Fermat) If $$p$$ is a prime number and $$(a,~p)=1,$$ that is, if $$a$$ and $$p$$ are relatively primes, then $$a^{p-1}\equiv 1$$ $$({\rm mod}~ p).$$ Euler gave a generalization of Fermat's theorem. His generalization will follow at once from next theorem, which is proceed by counting, using essentially the same argument as in… Read More »

## Fermat’s Little Theorem and Finite Prime Fields

The fact that "$$\mathbb{Z}_p$$ is a field if and only if $$p$$ is a prime" can be derived from the fact that "every finite integral domain is a field." Also, it is easily derived that every field include a subfield which is isomorphic to one of $$\mathbb{Q}$$ or $$\mathbb{Z}_p$$ for a prime $$p$$. That's why the fields $$\mathbb{Z}_p$$… Read More »

## “Associativity, Left Identity, Left Inverse Elements” Makes a Group

In most undergraduate algebra textbooks, you can find the definition of a group as following: A nonempty set $$G$$ with a binary operation satisfying the following conditions is called a group: Associative rule: $$a(bc)=(ab)c$$ for all $$a,~b,~c \in G$$. Existence of an identity element: There exists $$e\in G$$ s.t. $$ea = ae = a$$ for all \(a \in… Read More »

## Welcome to my Room :)

꿈과 현실의 중간 쯤, 아니면 그 어느 곳도 아닌 다른 곳. 보라도 아니고 파랑도 아닌 그 어디 쯤. 힘들 땐 그 짐을 잠시 내려 놓아요. 내가 당신의 곁에 있어 줄게요. There once was a child living everyday expecting tomorrow to be different from today.