Art museum of math formulas for pi
Art museum of math formulas for pi
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- Beautiful Formulas for pi=3.14… The number $\pi$ is defined a s the ratio of a circle's circumference $C$ to its diameter $d$:
\[\pi=\frac{C}{d}.\]
$\pi$ in decimal starts with 3.14... and never end.
I will show you several beautiful formulas for $\pi$.
Art Museum of formulas for $\pi$ […]
- Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$
Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying
\[T\left(\, \begin{bmatrix}
1 \\
2
\end{bmatrix}\,\right)=\begin{bmatrix}
3 \\
4 \\
5
\end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} […]
- Can a Student Pass By Randomly Answering Multiple Choice Questions?
A final exam of the course Probability 101 consists of 10 multiple-choice questions. Each question has 4 possible answers and only one of them is a correct answer. To pass the course, 8 or more correct answers are necessary. Assume that a student has not studied probability at all and […]
- The Rotation Matrix is an Orthogonal Transformation
Let $\mathbb{R}^2$ be the vector space of size-2 column vectors. This vector space has an inner product defined by $ \langle \mathbf{v} , \mathbf{w} \rangle = \mathbf{v}^\trans \mathbf{w}$. A linear transformation $T : \R^2 \rightarrow \R^2$ is called an orthogonal transformation if […]
- Mathematics About the Number 2018 Happy New Year 2018!!
Here are several mathematical facts about the number 2018.
Is 2018 a Prime Number?
The number 2018 is an even number, so in particular 2018 is not a prime number.
The prime factorization of 2018 is
\[2018=2\cdot 1009.\]
Here $2$ and $1009$ are […]
- The Sum of Cosine Squared in an Inner Product Space
Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$.
Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$.
Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$.
Prove that
\[\cos […]
- The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements
Let $G$ be an abelian group.
Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.
Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.
Also determine whether the statement is true if $G$ is a […]
- The Cyclotomic Field of 8-th Roots of Unity is $\Q(\zeta_8)=\Q(i, \sqrt{2})$
Let $\zeta_8$ be a primitive $8$-th root of unity.
Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$.
Proof.
Recall that the extension degree of the cyclotomic field of $n$-th roots of unity is given by […]