> ## Documentation Index
> Fetch the complete documentation index at: https://resources.latex-cloud-studio.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Mathematical Equations

> Master complex mathematical equations in LaTeX. Learn multi-line equations, alignment, numbering, and advanced equation environments.

Building on the math basics, this guide covers advanced equation formatting, multi-line equations, alignment, and professional mathematical typesetting.

<Info>
  **Prerequisites**: Familiarity with basic math mode (`$...$` and `\[...\]`). See our [Mathematics Basics](/learn/latex/mathematics/basics) guide if needed.

  **Related topics**: [Mathematical symbols](/learn/latex/mathematics/symbols) | [Text formatting in math](/learn/latex/text-formatting) | [Cross-referencing equations](/learn/latex/cross-referencing)
</Info>

## Single Equations

### Basic Display Equations

<Tabs>
  <Tab title="Code">
    <CodeGroup>
      ```latex display-equations.tex theme={null}
      \documentclass{article}
      \usepackage{amsmath}
      \begin{document}

      % Simple display equation
      \[ E = mc^2 \]

      % With equation number
      \begin{equation}
        E = mc^2
      \end{equation}

      % Without equation number
      \begin{equation*}
        E = mc^2
      \end{equation*}

      % Reference an equation
      \begin{equation}\label{eq:energy}
        E = mc^2
      \end{equation}
      As shown in Equation \ref{eq:energy}, energy and mass are related.

      % Using \eqref for parentheses
      As shown in Equation \eqref{eq:energy}...

      \end{document}
      ```
    </CodeGroup>
  </Tab>

  <Tab title="Rendered output">
    <Card title="Rendered output" icon="eye">
      **Display equation:** $E = mc^2$

      **Numbered equation:** $E = mc^2 \tag{1}$
    </Card>
  </Tab>
</Tabs>

### Equation Numbering Control

<CodeGroup>
  ```latex equation-numbering.tex theme={null}
  % Suppress numbering for specific equation
  \begin{equation}
    a^2 + b^2 = c^2 \nonumber
  \end{equation}

  % Custom numbering
  \begin{equation}
    F = ma \tag{Newton's 2nd Law}
  \end{equation}

  % Numbered with custom tag
  \begin{equation}
    e^{i\pi} + 1 = 0 \tag{$\star$}
  \end{equation}

  % Subequations
  \begin{subequations}
  \begin{equation}
    x + y = 5
  \end{equation}
  \begin{equation}
    2x - y = 1
  \end{equation}
  \end{subequations}
  ```
</CodeGroup>

## Multi-line Equations

### Split Environment

For single equations that are too long:

<Tabs>
  <Tab title="Code">
    <CodeGroup>
      ```latex split-environment.tex theme={null}
      \documentclass{article}
      \usepackage{amsmath}
      \begin{document}

      \begin{equation}
      \begin{split}
        (a + b)^4 &= (a + b)^2(a + b)^2 \\
                  &= (a^2 + 2ab + b^2)(a^2 + 2ab + b^2) \\
                  &= a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
      \end{split}
      \end{equation}

      % Left-aligned split
      \begin{equation}
      \begin{split}
        \text{LHS} &= \text{some long expression} \\
        &\quad + \text{continuation} \\
        &\quad + \text{more terms} \\
        &= \text{RHS}
      \end{split}
      \end{equation}

      \end{document}
      ```
    </CodeGroup>
  </Tab>

  <Tab title="Rendered output">
    <Card title="Rendered output" icon="eye">
      $\begin{aligned}
            (a + b)^4 &= (a + b)^2(a + b)^2 \\
                      &= (a^2 + 2ab + b^2)(a^2 + 2ab + b^2) \\
                      &= a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
            \end{aligned} \tag{2}$
    </Card>
  </Tab>
</Tabs>

### Multline Environment

For very long single equations:

<CodeGroup>
  ```latex multline-environment.tex theme={null}
  \begin{multline}
    \int_0^1 \biggl\{ \sum_{i=1}^n x_i^2 + \sum_{j=1}^m y_j^2 
    + \sum_{k=1}^p z_k^2 \biggr\} \, dx \\
    + \int_1^2 \biggl\{ \sum_{i=1}^n x_i^3 + \sum_{j=1}^m y_j^3 
    + \sum_{k=1}^p z_k^3 \biggr\} \, dx \\
    = \text{some complicated result}
  \end{multline}

  % Control positioning
  \begin{multline}
    \text{First line flush left} \\
    \text{Middle lines centered} \\
    \shoveleft{\text{This line shoved left}} \\
    \shoveright{\text{This line shoved right}} \\
    \text{Last line flush right}
  \end{multline}
  ```
</CodeGroup>

## Aligned Equations

### Align Environment

The most versatile environment for multiple equations:

<Tabs>
  <Tab title="Code">
    <CodeGroup>
      ```latex align-environment.tex theme={null}
      \documentclass{article}
      \usepackage{amsmath}
      \begin{document}

      % Basic alignment
      \begin{align}
        2x + 3y &= 7 \\
        5x - 2y &= 4
      \end{align}

      % Multiple alignment points
      \begin{align}
        x &= a + b     &\qquad y &= c + d \\
        2x &= 2(a + b) &\qquad 3y &= 3(c + d)
      \end{align}

      % Without numbering
      \begin{align*}
        \sin^2\theta + \cos^2\theta &= 1 \\
        1 + \tan^2\theta &= \sec^2\theta \\
        1 + \cot^2\theta &= \csc^2\theta
      \end{align*}

      % Selective numbering
      \begin{align}
        a &= b + c \\
        d &= e + f \nonumber \\
        g &= h + i
      \end{align}

      \end{document}
      ```
    </CodeGroup>
  </Tab>

  <Tab title="Rendered output">
    <Card title="Rendered output" icon="eye">
      **Aligned equations (numbered):**
      $\begin{aligned}
            2x + 3y &= 7 \tag{3}\\
            5x - 2y &= 4 \tag{4}
            \end{aligned}$

      **Without numbering:**
      $\begin{aligned}
            \sin^2\theta + \cos^2\theta &= 1 \\
            1 + \tan^2\theta &= \sec^2\theta \\
            1 + \cot^2\theta &= \csc^2\theta
            \end{aligned}$
    </Card>
  </Tab>
</Tabs>

### Aligned Within Equation

<CodeGroup>
  ```latex aligned-within.tex theme={null}
  % For alignment within a single equation number
  \begin{equation}
  \begin{aligned}
    f(x) &= (x+a)(x+b) \\
         &= x^2 + (a+b)x + ab
  \end{aligned}
  \end{equation}

  % Multiple aligned blocks
  \begin{equation}
  \left\{
  \begin{aligned}
    x + y + z &= 1 \\
    2x - y + 3z &= 0 \\
    x - 2y - z &= 4
  \end{aligned}
  \right.
  \end{equation}
  ```
</CodeGroup>

## Equation Arrays

### Eqnarray (Deprecated)

<Warning>
  **Note**: `eqnarray` is deprecated. Use `align` instead. Shown here for reference only.
</Warning>

### Array Environment

For complex layouts:

<CodeGroup>
  ```latex array-equations.tex theme={null}
  \begin{equation}
  \begin{array}{lcl}
    f(x) &=& (x+1)^2 \\
         &=& x^2 + 2x + 1
  \end{array}
  \end{equation}

  % Multiple columns
  \begin{equation}
  \begin{array}{ccc}
    a_{11} & a_{12} & a_{13} \\
    a_{21} & a_{22} & a_{23} \\
    a_{31} & a_{32} & a_{33}
  \end{array}
  \end{equation}
  ```
</CodeGroup>

## Cases and Piecewise Functions

### Cases Environment

<Tabs>
  <Tab title="Code">
    <CodeGroup>
      ```latex cases-environment.tex theme={null}
      \documentclass{article}
      \usepackage{amsmath}
      \begin{document}

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

      % Multiple conditions
      \begin{equation}
      \text{sgn}(x) = \begin{cases}
        1 & \text{if } x > 0 \\
        0 & \text{if } x = 0 \\
        -1 & \text{if } x < 0
      \end{cases}
      \end{equation}

      % Nested cases
      \begin{equation}
      f(x,y) = \begin{cases}
        \begin{cases}
          1 & \text{if } y > x \\
          0 & \text{if } y = x
        \end{cases} & \text{if } x \geq 0 \\
        -1 & \text{if } x < 0
      \end{cases}
      \end{equation}

      % Left cases
      \begin{equation}
      \begin{rcases}
        x^2 + y^2 = 1 \\
        x + y = 0
      \end{rcases} \text{defines a curve}
      \end{equation}

      \end{document}
      ```
    </CodeGroup>
  </Tab>

  <Tab title="Rendered output">
    <Card title="Rendered output" icon="eye">
      **Piecewise function:**
      $f(x) = \begin{cases}
            x^2 & \text{if } x \geq 0 \\
            -x^2 & \text{if } x < 0
            \end{cases} \tag{5}$

      **Sign function:**
      $\text{sgn}(x) = \begin{cases}
            1 & \text{if } x > 0 \\
            0 & \text{if } x = 0 \\
            -1 & \text{if } x < 0
            \end{cases} \tag{6}$
    </Card>
  </Tab>
</Tabs>

## Gathered and Centered Equations

<CodeGroup>
  ```latex gathered-equations.tex theme={null}
  % Multiple centered lines with one number
  \begin{equation}
  \begin{gathered}
    (a + b)^2 = a^2 + 2ab + b^2 \\
    (a - b)^2 = a^2 - 2ab + b^2 \\
    (a + b)(a - b) = a^2 - b^2
  \end{gathered}
  \end{equation}

  % Within text
  The identities $\begin{gathered}
    \sin^2\theta + \cos^2\theta = 1 \\
    \tan\theta = \frac{\sin\theta}{\cos\theta}
  \end{gathered}$ are fundamental.
  ```
</CodeGroup>

## Advanced Alignment

### Complex Alignment Patterns

<CodeGroup>
  ```latex complex-alignment.tex theme={null}
  % Aligning equals signs and operators
  \begin{align}
    f(x) &= x^2 + 2x + 1 \\
         &= (x + 1)^2 \\
         &> 0 \quad \text{for all } x \neq -1
  \end{align}

  % Multiple columns
  \begin{align}
    a_1 &= b_1 + c_1 & a_2 &= b_2 + c_2 \\
    d_1 &= e_1 + f_1 & d_2 &= e_2 + f_2
  \end{align}

  % Alignment with text
  \begin{align}
    2x + 3 &= 7 & &\text{(given)} \\
    2x &= 4 & &\text{(subtract 3)} \\
    x &= 2 & &\text{(divide by 2)}
  \end{align}
  ```
</CodeGroup>

### Intertext and Shortintertext

<CodeGroup>
  ```latex intertext.tex theme={null}
  \begin{align}
    a &= b + c \\
    &= d + e + f
  \intertext{Now we substitute the known values:}
    &= 1 + 2 + 3 \\
    &= 6
  \end{align}

  % Shorter spacing
  \begin{align}
    x^2 + y^2 &= r^2
  \shortintertext{and}
    x &= r\cos\theta \\
    y &= r\sin\theta
  \end{align}
  ```
</CodeGroup>

## Boxed and Highlighted Equations

<CodeGroup>
  ```latex boxed-equations.tex theme={null}
  % Simple box
  \begin{equation}
  \boxed{E = mc^2}
  \end{equation}

  % Colored box
  \usepackage{xcolor}
  \begin{equation}
  \colorbox{yellow}{$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$}
  \end{equation}

  % Framed important result
  \begin{equation}
  \boxed{
    \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
  }
  \end{equation}

  % Highlight part of equation
  \begin{equation}
  f(x) = \underbrace{x^2 + 2x}_{\text{quadratic}} + 
         \overbrace{3x + 4}^{\text{linear}}
  \end{equation}
  ```
</CodeGroup>

## Equation Spacing

### Manual Spacing in Equations

<CodeGroup>
  ```latex equation-spacing.tex theme={null}
  % Spacing commands
  \begin{align}
    f(x) &= x^2+2x+1 \\              % No space
    f(x) &= x^2 + 2x + 1 \\          % Normal space
    f(x) &= x^2\,+\,2x\,+\,1 \\      % Thin space
    f(x) &= x^2\:+\:2x\:+\:1 \\      % Medium space
    f(x) &= x^2\;+\;2x\;+\;1 \\      % Thick space
    f(x) &= x^2\quad+\quad2x\quad+\quad1  % Quad space
  \end{align}

  % Negative space
  \begin{equation}
    \int\!\!\!\int f(x,y) \, dx \, dy  % Bring integral signs closer
  \end{equation}
  ```
</CodeGroup>

## Best Practices

<Tip>
  **Equation guidelines:**

  1. **Consistency**: Use the same alignment style throughout
  2. **Readability**: Don't over-align; prioritize clarity
  3. **References**: Label important equations for cross-referencing
  4. **Spacing**: Use `\,` before differentials in integrals
  5. **Breaking**: Break long equations at operators (+, -, =)
  6. **Grouping**: Use `subequations` for related equations
</Tip>

## Common Patterns

### System of Equations

<Tabs>
  <Tab title="Code">
    <CodeGroup>
      ```latex system-equations.tex theme={null}
      % Using array
      \begin{equation}
      \left\{
      \begin{array}{rcrcrcr}
        2x & + & 3y & - &  z & = & 4 \\
         x & - &  y & + & 2z & = & 1 \\
        3x & + &  y & - &  z & = & 0
      \end{array}
      \right.
      \end{equation}

      % Using aligned
      \begin{equation}
      \left\{
      \begin{aligned}
        2x + 3y - z &= 4 \\
        x - y + 2z &= 1 \\
        3x + y - z &= 0
      \end{aligned}
      \right.
      \end{equation}
      ```
    </CodeGroup>
  </Tab>

  <Tab title="Rendered output">
    <Card title="Rendered output" icon="eye">
      $\left\{
            \begin{aligned}
            2x + 3y - z &= 4 \\
            x - y + 2z &= 1 \\
            3x + y - z &= 0
            \end{aligned}
            \right. \tag{7}$
    </Card>
  </Tab>
</Tabs>

### Derivations

<Tabs>
  <Tab title="Code">
    <CodeGroup>
      ```latex derivations.tex theme={null}
      \begin{align}
        (x + y)^2 &= (x + y)(x + y) \\
                  &= x(x + y) + y(x + y) \\
                  &= x^2 + xy + yx + y^2 \\
                  &= x^2 + 2xy + y^2
      \end{align}
      ```
    </CodeGroup>
  </Tab>

  <Tab title="Rendered output">
    <Card title="Rendered output" icon="eye">
      $\begin{aligned}
            (x + y)^2 &= (x + y)(x + y) \\
                      &= x(x + y) + y(x + y) \\
                      &= x^2 + xy + yx + y^2 \\
                      &= x^2 + 2xy + y^2
            \end{aligned}$
    </Card>
  </Tab>
</Tabs>

## Quick Reference

| Environment | Purpose                    | Numbering       |
| ----------- | -------------------------- | --------------- |
| `equation`  | Single equation            | Yes             |
| `equation*` | Single equation            | No              |
| `align`     | Multiple aligned equations | Yes (each)      |
| `align*`    | Multiple aligned equations | No              |
| `split`     | Split long equation        | One number      |
| `multline`  | Very long equation         | One number      |
| `gather`    | Centered equations         | Yes (each)      |
| `cases`     | Piecewise functions        | Within equation |

***

<Info>
  **Next**: Explore [Mathematical symbols](/learn/latex/mathematics/symbols) for a comprehensive guide to mathematical notation in LaTeX.
</Info>