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1. Basic Arithmetic

Addition: 2 + 3 = 5

$2 + 3 = 5$

Multiplication: a × b

$a \times b$

Division: a ÷ b

$a \div b$

2. Fractions

Simple fraction: 1/2

$\frac{1}{2}$

Nested fraction: (a/b) / (c/d)

$\frac{\frac{a}{b}}{\frac{c}{d}}$

Continued fraction

$$1 + \cfrac{1}{2 + \cfrac{1}{3 + \cdots}}$$

3. Roots

Square root

$\sqrt{a^2 + b^2}$

Cube root

$\sqrt[3]{27}$

Nested radical

$\sqrt{1 + \sqrt{2 + \sqrt{3}}}$

4. Superscripts & Subscripts

Superscript: x²

$x^2$

Subscript: a_n

$a_n$

Combined: x_i^j

$x_i^j$

Nested exponent: e^(x²)

$e^{x^2}$

5. Greek Letters & Symbols

Greek alphabet sample

$\alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \pi, \sigma, \phi, \omega$

Mathematical symbols

$\int, \sum, \prod, \partial, \infty, \in, \forall, \exists, \nabla$

6. Summations

Sum of x_i from i=1 to n

$$\sum_{i=1}^{n} x_i$$

Double sum

$$\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}$$

Product notation

$$\prod_{k=1}^{n} a_k$$

7. Integrals

Indefinite integral

$$\int f(x)\,dx$$

Definite integral from a to b

$$\int_a^b f(x)\,dx$$

Double integral

$$\iint f(x,y)\,dx\,dy$$

Contour integral

$$\oint f(z)\,dz$$

8. Limits

Limit as x approaches infinity

$$\lim_{x \to \infty} f(x)$$

Limit as x approaches 0

$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$

9. Matrices

2×2 matrix

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

3×3 identity matrix

$$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Determinant

$$\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$

10. Systems of Equations

System of linear equations

$$\begin{cases} 2x + 3y = 7 \\ x - y = 1 \end{cases}$$

Piecewise function

$$f(x) = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

11. Calculus

Derivative

$$\frac{dy}{dx} = f'(x)$$

Partial derivative

$$\frac{\partial f}{\partial x}$$

Gradient

$$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$$

Divergence

$$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$

12. Set Notation

Union and intersection

$A \cup B, \quad A \cap B$

Set-builder notation

$$\{x \in \mathbb{R} \mid x > 0\}$$

Subset and membership

$A \subseteq B, \quad x \in A, \quad A \setminus B$

13. Logic

Logical operators

$p \land q, \quad p \lor q, \quad \lnot p$

Implication and biconditional

$p \Rightarrow q, \quad p \Leftrightarrow q$

Quantifiers

$$\forall x \in S, \; \exists y : P(x, y)$$

14. Famous Formulas

Quadratic formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Euler's identity

$$e^{i\pi} + 1 = 0$$

Cauchy's integral formula

$$f(a) = \frac{1}{2\pi i} \oint \frac{f(z)}{z - a}\,dz$$

Pythagorean theorem

$$a^2 + b^2 = c^2$$

Euler's formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

Fourier transform

$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi}\,dx$$

Maxwell's equation (Gauss's law)

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

Navier-Stokes (incompressible)

$$\rho\left(\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot \nabla \vec{v}\right) = -\nabla p + \mu \nabla^2 \vec{v} + \vec{f}$$

15. Chemistry-Style Expressions

Water formation

$$2H_2 + O_2 \rightarrow 2H_2O$$

Photosynthesis

$$6CO_2 + 6H_2O \xrightarrow{\text{light}} C_6H_{12}O_6 + 6O_2$$

16. Additional Expressions

Binomial theorem

$$\left(x + y\right)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$

Bayes' theorem

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Definition of e

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$

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Renderer: MathJax v4 (CDN) · Detection context: mjx-container[v4] · Input: LaTeX (TeX)

1. Basic Arithmetic

Addition: 2 + 3 = 5

$2 + 3 = 5$

Multiplication: a × b

$a \times b$

Division: a ÷ b

$a \div b$

2. Fractions

Simple fraction: 1/2

$\frac{1}{2}$

Nested fraction: (a/b) / (c/d)

$\frac{\frac{a}{b}}{\frac{c}{d}}$

Continued fraction

$$1 + \cfrac{1}{2 + \cfrac{1}{3 + \cdots}}$$

3. Roots

Square root

$\sqrt{a^2 + b^2}$

Cube root

$\sqrt[3]{27}$

Nested radical

$\sqrt{1 + \sqrt{2 + \sqrt{3}}}$

4. Superscripts & Subscripts

Superscript: x²

$x^2$

Subscript: a_n

$a_n$

Combined: x_i^j

$x_i^j$

Nested exponent: e^(x²)

$e^{x^2}$

5. Greek Letters & Symbols

Greek alphabet sample

$\alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \pi, \sigma, \phi, \omega$

Mathematical symbols

$\int, \sum, \prod, \partial, \infty, \in, \forall, \exists, \nabla$

6. Summations

Sum of x_i from i=1 to n

$$\sum_{i=1}^{n} x_i$$

Double sum

$$\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}$$

Product notation

$$\prod_{k=1}^{n} a_k$$

7. Integrals

Indefinite integral

$$\int f(x)\,dx$$

Definite integral from a to b

$$\int_a^b f(x)\,dx$$

Double integral

$$\iint f(x,y)\,dx\,dy$$

Contour integral

$$\oint f(z)\,dz$$

8. Limits

Limit as x approaches infinity

$$\lim_{x \to \infty} f(x)$$

Limit as x approaches 0

$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$

9. Matrices

2×2 matrix

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

3×3 identity matrix

$$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Determinant

$$\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$

10. Systems of Equations

System of linear equations

$$\begin{cases} 2x + 3y = 7 \\ x - y = 1 \end{cases}$$

Piecewise function

$$f(x) = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

11. Calculus

Derivative

$$\frac{dy}{dx} = f'(x)$$

Partial derivative

$$\frac{\partial f}{\partial x}$$

Gradient

$$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$$

Divergence

$$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$

12. Set Notation

Union and intersection

$A \cup B, \quad A \cap B$

Set-builder notation

$$\{x \in \mathbb{R} \mid x > 0\}$$

Subset and membership

$A \subseteq B, \quad x \in A, \quad A \setminus B$

13. Logic

Logical operators

$p \land q, \quad p \lor q, \quad \lnot p$

Implication and biconditional

$p \Rightarrow q, \quad p \Leftrightarrow q$

Quantifiers

$$\forall x \in S, \; \exists y : P(x, y)$$

14. Famous Formulas

Quadratic formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Euler's identity

$$e^{i\pi} + 1 = 0$$

Cauchy's integral formula

$$f(a) = \frac{1}{2\pi i} \oint \frac{f(z)}{z - a}\,dz$$

Pythagorean theorem

$$a^2 + b^2 = c^2$$

Euler's formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

Fourier transform

$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi}\,dx$$

Maxwell's equation (Gauss's law)

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

Navier-Stokes (incompressible)

$$\rho\left(\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot \nabla \vec{v}\right) = -\nabla p + \mu \nabla^2 \vec{v} + \vec{f}$$

15. Chemistry-Style Expressions

Water formation

$$2H_2 + O_2 \rightarrow 2H_2O$$

Photosynthesis

$$6CO_2 + 6H_2O \xrightarrow{\text{light}} C_6H_{12}O_6 + 6O_2$$

16. Additional Expressions

Binomial theorem

$$\left(x + y\right)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$

Bayes' theorem

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Definition of e

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$

MathJax v4 Samples

1. Basic Arithmetic

2. Fractions

3. Roots

4. Superscripts & Subscripts

5. Greek Letters & Symbols

6. Summations

7. Integrals

8. Limits

9. Matrices

10. Systems of Equations

11. Calculus

12. Set Notation

13. Logic

14. Famous Formulas

15. Chemistry-Style Expressions

16. Additional Expressions

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MathJax v4 Samples

1. Basic Arithmetic

2. Fractions

3. Roots

4. Superscripts & Subscripts

5. Greek Letters & Symbols

6. Summations

7. Integrals

8. Limits

9. Matrices

10. Systems of Equations

11. Calculus

12. Set Notation

13. Logic

14. Famous Formulas

15. Chemistry-Style Expressions

16. Additional Expressions