Fraktal

Enter two numbers to check if the first number is divisible by the second number: Check

 Show random mathematical shape.

It is \( \frac{2 + \sqrt{2}}{15} + \frac{\ln \left ( 1 + \sqrt{2} \right )}{3} \), approximately 0.52140543316472067833. Watch here to see why.

 Generate random polynomial with integer or rational coefficients. 

  Let \( K \) be a finite field. \( 1 + 1 = 0 \) in K if and only if for any \( f \in K \left [ x \right ] \) such that the degree of \( f \) is more or equal to 1, the polynomial \( f(X^2) \) is a reducible polynomial.

Clearly the set of continuous function \( C \left [ a, b \right ] \) is a vector space. Now, we have to prove that the function \( \left | \left | \cdot \right | \right | _{\infty} :  C \left [ a, b \right ] \to \mathbb{R}  \) such that  \( \left | \left | f \right | \right | _{\infty} = \displaystyle \max_{a\leq x\leq b} \left| f(x) \right| \) for every function \( f \in C \left [ a, b \right ] \) is really a norm.

 Generate a random species.

  It is easy to see that \( \mathrm{arg} \left ( \frac{a}{b} \right ) = \mathrm{arg} \left ( a \right )  - \mathrm{arg} \left ( b \right )  \) for any complex number \( a \) and \( b \). But, this is not the case for the principle value that is \( \mathrm{Arg} \) does not always equal to \( \mathrm{Arg} \left ( a \right ) - \mathrm{Arg} \left ( b \right )  \).

 Generate random image from Imgur. Warning! NSFW images may appear. This generator may do not work well in mobile or some browsers.

 Generate a random math expression.

  Let \(n\) be a natural number greater than \(1\). Suppose \( \left ( \mathbb{Z}_n, +,  \cdot \right ) \) be a ring of integers mod \(n\). Then, every element \(a \in \mathbb{Z}_n \) has multiplicative inverse if and only if \(n\) is a prime number. In other words,  \( \left ( \mathbb{Z}_n, +,  \cdot \right ) \) is a field if and only if \(n\) is a prime number.

 Any natural number coprimes with 10 has a mulitple that only contains digit 1 in its decimal representation.

A normed space or a normed vector space is a vector space over real or complex numbers on which a norm is defined. A norm is a real-valued function ||•|| defined on a vector space V with scalars in a field 𝔽 (the real numbers or the complex numbers) such that satisfies following properties: 1. || x || = 0 only if x = 0. 2. For every x ∈ V and ɑ ∈ 𝔽, || ɑ x || = | ɑ | || x ||. 3. For every x , y ∈ V, || x + y || ≤ || x || + || y || (triangle inequality). Maybe someone would be thinking the property "|| x || ≥ 0 for all  x  ∈ V (nonnegative)" must be included in the definition. This property actually can be proved by using the three other properties. Example

  Let f : ℝ n → ℝ and g : ℝ n → ℝ be scalar-valued differentiable function of several ( n ) variables. Then f and g satisfy the product rule (the Leibniz rule):

Watch random YouTube Video 

 Draw a graph based on adjacency matrix. 

 Generate a computable number.

 Generate a random food.

  Problem: Let be a function defined on the real line such that for every . Prove for all , i.e. is constant.   Show Solution Solution: Take any . Next, we obtain for all . So, for , . Then, take the limit as approaches , we get   Hence, the derivative is zero 0 everywhere. Therefore, must be a constant function.

  Problem: Let and be and , respectively, such that Determine with proof the matrix .