KaTeX Math Examples and Tips

calendarDec 12, 2024 · 10 min read · katex math latex mathematics notation equations  ·
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KaTeX renders mathematical notation beautifully in web pages. This guide covers common patterns, syntax, and tips for writing mathematical expressions.

Use Case

Use KaTeX when you need to:

  • Display mathematical formulas and equations
  • Show scientific notation
  • Write technical documentation with math
  • Create educational content with equations
  • Document algorithms with mathematical foundations

Basic Syntax

Inline Math

Use single dollar signs for inline math: $x = y + z$ renders as $x = y + z$

Display Math

Use double dollar signs for display math (centered, on its own line):

1$$
2E = mc^2
3$$

Renders as:

$$ E = mc^2 $$

Common Mathematical Expressions

Fractions

1$$
2\frac{a}{b} \quad \frac{numerator}{denominator} \quad \frac{x^2 + 1}{x - 1}
3$$

$$ \frac{a}{b} \quad \frac{numerator}{denominator} \quad \frac{x^2 + 1}{x - 1} $$

Superscripts and Subscripts

1$$
2x^2 \quad x^{n+1} \quad x_i \quad x_{i+1} \quad x^{y^z}
3$$

$$ x^2 \quad x^{n+1} \quad x_i \quad x_{i+1} \quad x^{y^z} $$

Roots

1$$
2\sqrt{x} \quad \sqrt[n]{x} \quad \sqrt{x^2 + y^2}
3$$

$$ \sqrt{x} \quad \sqrt[n]{x} \quad \sqrt{x^2 + y^2} $$

Sums and Products

1$$
2\sum_{i=1}^{n} x_i \quad \prod_{i=1}^{n} x_i \quad \int_{a}^{b} f(x) dx
3$$

$$ \sum_{i=1}^{n} x_i \quad \prod_{i=1}^{n} x_i \quad \int_{a}^{b} f(x) dx $$

Greek Letters

1$$
2\alpha \quad \beta \quad \gamma \quad \delta \quad \epsilon \quad \theta \quad \lambda \quad \mu \quad \pi \quad \sigma \quad \phi \quad \omega
3$$

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

Capital Greek letters:

1$$
2\Alpha \quad \Beta \quad \Gamma \quad \Delta \quad \Theta \quad \Lambda \quad \Pi \quad \Sigma \quad \Phi \quad \Omega
3$$

$$ \Alpha \quad \Beta \quad \Gamma \quad \Delta \quad \Theta \quad \Lambda \quad \Pi \quad \Sigma \quad \Phi \quad \Omega $$

Operators and Relations

Comparison Operators

1$$
2< \quad > \quad \leq \quad \geq \quad \neq \quad \approx \quad \equiv
3$$

$$ < \quad > \quad \leq \quad \geq \quad \neq \quad \approx \quad \equiv $$

Set Operations

1$$
2\in \quad \notin \quad \subset \quad \subseteq \quad \cup \quad \cap \quad \setminus
3$$

$$ \in \quad \notin \quad \subset \quad \subseteq \quad \cup \quad \cap \quad \setminus $$

Logical Operators

1$$
2\land \quad \lor \quad \neg \quad \implies \quad \iff \quad \forall \quad \exists
3$$

$$ \land \quad \lor \quad \neg \quad \implies \quad \iff \quad \forall \quad \exists $$

Matrices and Vectors

Matrices

 1$$
 2\begin{pmatrix}
 3a & b \\\\
 4c & d
 5\end{pmatrix}
 6\quad
 7\begin{bmatrix}
 81 & 2 & 3 \\\\
 94 & 5 & 6 \\\\
107 & 8 & 9
11\end{bmatrix}
12$$

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \quad \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} $$

More Matrix Examples

Determinant

1$$
2\det(\mathbf{A}) = \begin{vmatrix}
3a & b \\\\
4c & d
5\end{vmatrix} = ad - bc
6$$

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

Matrix Types

1$$
2\begin{pmatrix} a & b \\\\ c & d \end{pmatrix} \quad
3\begin{bmatrix} a & b \\\\ c & d \end{bmatrix} \quad
4\begin{Bmatrix} a & b \\\\ c & d \end{Bmatrix} \quad
5\begin{vmatrix} a & b \\\\ c & d \end{vmatrix} \quad
6\begin{Vmatrix} a & b \\\\ c & d \end{Vmatrix}
7$$

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \quad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \quad \begin{Bmatrix} a & b \\ c & d \end{Bmatrix} \quad \begin{vmatrix} a & b \\ c & d \end{vmatrix} \quad \begin{Vmatrix} a & b \\ c & d \end{Vmatrix} $$

Large Matrices

1$$
2\mathbf{A} = \begin{bmatrix}
3a_{11} & a_{12} & \cdots & a_{1n} \\\\
4a_{21} & a_{22} & \cdots & a_{2n} \\\\
5\vdots & \vdots & \ddots & \vdots \\\\
6a_{m1} & a_{m2} & \cdots & a_{mn}
7\end{bmatrix}
8$$

$$ \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

Block Matrices

 1$$
 2\begin{bmatrix}
 3\mathbf{A} & \mathbf{B} \\\\
 4\mathbf{C} & \mathbf{D}
 5\end{bmatrix} = \begin{bmatrix}
 6a_{11} & a_{12} & b_{11} & b_{12} \\\\
 7a_{21} & a_{22} & b_{21} & b_{22} \\\\
 8c_{11} & c_{12} & d_{11} & d_{12} \\\\
 9c_{21} & c_{22} & d_{21} & d_{22}
10\end{bmatrix}
11$$

$$ \begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \\ a_{21} & a_{22} & b_{21} & b_{22} \\ c_{11} & c_{12} & d_{11} & d_{12} \\ c_{21} & c_{22} & d_{21} & d_{22} \end{bmatrix} $$

Vectors

1$$
2\vec{v} = \begin{pmatrix} x \\\\ y \\\\ z \end{pmatrix} \quad \mathbf{v} = \begin{bmatrix} v_1 \\\\ v_2 \\\\ v_3 \end{bmatrix}
3$$

$$ \vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \quad \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} $$

Functions and Special Notation

Common Functions

1$$
2\sin(x) \quad \cos(x) \quad \tan(x) \quad \log(x) \quad \ln(x) \quad \exp(x)
3$$

$$ \sin(x) \quad \cos(x) \quad \tan(x) \quad \log(x) \quad \ln(x) \quad \exp(x) $$

Limits

1$$
2\lim_{x \to \infty} f(x) \quad \lim_{n \to 0} \frac{\sin(n)}{n} = 1
3$$

$$ \lim_{x \to \infty} f(x) \quad \lim_{n \to 0} \frac{\sin(n)}{n} = 1 $$

Derivatives

1$$
2\frac{d}{dx}f(x) \quad f'(x) \quad \frac{\partial f}{\partial x} \quad \nabla f
3$$

$$ \frac{d}{dx}f(x) \quad f'(x) \quad \frac{\partial f}{\partial x} \quad \nabla f $$

Aligned Equations

Single Alignment

1$$
2\begin{aligned}
3x &= a + b \\\\
4y &= c + d \\\\
5z &= x + y
6\end{aligned}
7$$

$$ \begin{aligned} x &= a + b \\ y &= c + d \\ z &= x + y \end{aligned} $$

Multi-line Equations

1$$
2\begin{align}
3f(x) &= x^2 + 2x + 1 \\\\
4     &= (x + 1)^2 \\\\
5     &= x^2 + 2x + 1
6\end{align}
7$$

$$ \begin{align} f(x) &= x^2 + 2x + 1 \\ &= (x + 1)^2 \\ &= x^2 + 2x + 1 \end{align} $$

Cases and Piecewise Functions

Basic Cases

1$$
2f(x) = \begin{cases}
3x^2 & \text{if } x \geq 0 \\\\
4-x^2 & \text{if } x < 0
5\end{cases}
6$$

$$ f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases} $$

Conditional Expressions

1$$
2a = \begin{cases}
32 & \text{if } b = 2 \\\\
44 & \text{if } b = 4 \\\\
56 & \text{if } b = 6 \\\\
60 & \text{otherwise}
7\end{cases}
8$$

$$ a = \begin{cases} 2 & \text{if } b = 2 \\ 4 & \text{if } b = 4 \\ 6 & \text{if } b = 6 \\ 0 & \text{otherwise} \end{cases} $$

Multiple Conditions

1$$
2f(x) = \begin{cases}
3x + 1 & \text{if } x < 0 \\\\
4x^2 & \text{if } 0 \leq x < 1 \\\\
52x - 1 & \text{if } x \geq 1
6\end{cases}
7$$

$$ f(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ x^2 & \text{if } 0 \leq x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases} $$

Conditional Matrix

1$$
2\mathbf{A} = \begin{cases}
3\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \end{bmatrix} & \text{if } \det(\mathbf{B}) \neq 0 \\\\
4\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \end{bmatrix} & \text{otherwise}
5\end{cases}
6$$

$$ \mathbf{A} = \begin{cases} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} & \text{if } \det(\mathbf{B}) \neq 0 \\ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} & \text{otherwise} \end{cases} $$

Conditional with Logical Operators

1$$
2y = \begin{cases}
3x^2 & \text{if } x > 0 \text{ and } x \neq 1 \\\\
4\log(x) & \text{if } x > 0 \text{ and } x = 1 \\\\
50 & \text{if } x \leq 0
6\end{cases}
7$$

$$ y = \begin{cases} x^2 & \text{if } x > 0 \text{ and } x \neq 1 \\ \log(x) & \text{if } x > 0 \text{ and } x = 1 \\ 0 & \text{if } x \leq 0 \end{cases} $$

Spacing and Formatting

Manual Spacing

1$$
2a\,b \quad a\;b \quad a\:b \quad a\!b \quad a\ b \quad a\quad b \quad a\qquad b
3$$

$$ a,b \quad a;b \quad a:b \quad a!b \quad a\ b \quad a\quad b \quad a\qquad b $$

Text in Math

1$$
2\text{for all } x \in \mathbb{R} \quad \text{where } n > 0
3$$

$$ \text{for all } x \in \mathbb{R} \quad \text{where } n > 0 $$

Number Sets

1$$
2\mathbb{N} \quad \mathbb{Z} \quad \mathbb{Q} \quad \mathbb{R} \quad \mathbb{C}
3$$

$$ \mathbb{N} \quad \mathbb{Z} \quad \mathbb{Q} \quad \mathbb{R} \quad \mathbb{C} $$

Practical Examples

Example 1: Quadratic Formula

1$$
2x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
3$$

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

Example 2: Euler's Identity

1$$
2e^{i\pi} + 1 = 0
3$$

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

Example 3: Bayes' Theorem

1$$
2P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}
3$$

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

Example 4: Matrix Multiplication

 1$$
 2\mathbf{C} = \mathbf{A} \mathbf{B} = \begin{bmatrix}
 3a_{11} & a_{12} \\\\
 4a_{21} & a_{22}
 5\end{bmatrix}
 6\begin{bmatrix}
 7b_{11} & b_{12} \\\\
 8b_{21} & b_{22}
 9\end{bmatrix}
10$$

$$ \mathbf{C} = \mathbf{A} \mathbf{B} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} $$

Example 5: Gradient Descent Update

1$$
2\theta_{t+1} = \theta_t - \alpha \nabla_\theta J(\theta_t)
3$$

$$ \theta_{t+1} = \theta_t - \alpha \nabla_\theta J(\theta_t) $$

Example 6: Neural Network Forward Pass

 1$$
 2\mathbf{h} = \sigma(\mathbf{W}\mathbf{x} + \mathbf{b})
 3$$
 4$$
 5
 6where $\sigma$ is the activation function, $\mathbf{W}$ is the weight matrix, $\mathbf{x}$ is the input vector, and $\mathbf{b}$ is the bias vector.
 7
 8### Example 7: Loss Function
 9
10```markdown
11$$
12\mathcal{L} = -\frac{1}{N}\sum_{i=1}^{N} \left[ y_i \log(\hat{y}_i) + (1-y_i)\log(1-\hat{y}_i) \right]
13$$

$$ \mathcal{L} = -\frac{1}{N}\sum_{i=1}^{N} \left[ y_i \log(\hat{y}_i) + (1-y_i)\log(1-\hat{y}_i) \right] $$

Tips and Best Practices

1. Use Display Math for Important Equations

Display math (double $$) is better for:

  • Standalone equations
  • Multi-line expressions
  • Complex formulas

Inline math (single $) is better for:

  • Short expressions in text
  • Variables mentioned in sentences
  • Simple formulas

2. Escape Special Characters

In markdown, you may need to escape:

  • $ → \$ (if not used for math)
  • _ → \_ (in non-math contexts)
  • * → \* (in non-math contexts)

3. Use Aligned Environments for Multi-line

1$$
2\begin{aligned}
3f(x) &= x^2 + 2x + 1 \\\\
4     &= (x+1)^2
5\end{aligned}
6$$

Better than:

1$$
2f(x) = x^2 + 2x + 1 = (x+1)^2
3$$

4. Label Important Equations

While KaTeX doesn't support \label like LaTeX, you can add text labels:

1$$
2\text{(1)} \quad E = mc^2
3$$

5. Use Text Mode for Words

Always use \text{} for words in math mode:

1$$
2\text{for } x \in \mathbb{R} \text{ and } y > 0
3$$

Not:

1$$
2for x \in \mathbb{R} and y > 0  \quad \text{(incorrect)}
3$$

Common Mistakes to Avoid

Mistake 1: Mixing Math and Text

❌ Wrong:

1The value is $x$ where $x > 0$ and positive.

✅ Correct:

1The value is $x$ where $x > 0$ and is positive.

Mistake 2: Forgetting Curly Braces

❌ Wrong:

1$x^10$  (renders as $x^10$)

✅ Correct:

1$x^{10}$  (renders as $x^{10}$)

Mistake 3: Incorrect Spacing

❌ Wrong:

1$f(x)=x^2+1$  (no spacing)

✅ Correct:

1$f(x) = x^2 + 1$  (proper spacing)

Advanced Examples

Example: Fourier Transform

1$$
2F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt
3$$

$$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt $$

Example: Schrödinger Equation

1$$
2i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)
3$$

$$ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t) $$

Example: Einstein Field Equations

1$$
2R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}
3$$

$$ R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} $$

Resources

Gotchas/Warnings

  • ⚠️ Dollar Signs: In markdown, $ starts math mode - escape with \$ if needed
  • ⚠️ Curly Braces: Always use {} for multi-character subscripts/superscripts
  • ⚠️ Text Mode: Use \text{} for words, not raw text in math mode
  • ⚠️ Spacing: Math mode removes spaces - use \,, \;, \quad, etc. for spacing
  • ⚠️ Alignment: Use aligned or align environments for multi-line equations
  • ⚠️ Backslashes: In markdown, you may need \\ for line breaks in matrices

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