🧮 Welcome to CollabMath

A collaborative space where ideas meet rigor — write, discuss, and explore mathematics together.

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What is CollabMath?

CollabMath is a platform built for mathematicians, students, and curious minds. Whether you're working through a proof, sharing an elegant identity, or debugging a derivation — this is your space.

We support full Markdown and KaTeX rendering, so your math looks exactly as it should.

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✍️ Writing Math

You can write inline math like $e^{i\pi} + 1 = 0$ seamlessly within sentences, or display it as a full block:

$$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$$

Display equations are great for derivations. Here's the Cauchy integral formula:

$$f(z_0) = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{z - z_0} \, dz$$

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📐 A Sample Proof

Theorem (Pythagorean Theorem): For a right triangle with legs $a$, $b$ and hypotenuse $c$,
$a^2 + b^2 = c^2$

Proof sketch. Consider a square of side $(a + b)$. Arranged inside it are four congruent right triangles, each of area $\frac{1}{2}ab$, surrounding a central square of side $c$.

$$(a + b)^2 = 4 \cdot \frac{1}{2}ab + c^2 \implies a^2 + 2ab + b^2 = 2ab + c^2 \implies a^2 + b^2 = c^2 \qquad \blacksquare$$

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🔢 Formatting Features

Text Formatting

You can write bold, italic, strikethrough, and inline code for variable names like f(x).

Lists

Ordered and unordered lists work great for proof steps:

1. Assume $\epsilon > 0$
2. Choose $\delta = \epsilon / 2$
3. Show $|f(x) - L| < \epsilon$ whenever $0 < |x - a| < \delta$

Some common function families:

• Polynomials: $p(x) = a_n x^n + \cdots + a_1 x + a_0$
• Exponentials: $f(x) = e^x$
• Trigonometric: $\sin^2 x + \cos^2 x = 1$

Tables

Code Blocks

Algorithms and computations live naturally alongside math:

def newton_raphson(f, df, x0, tol=1e-9):
    """Solve f(x) = 0 near x0 using Newton's method."""
    x = x0
    while abs(f(x)) > tol:
        x -= f(x) / df(x)
    return x

The iteration step corresponds to the update rule:

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

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🧵 CollabMath Thread Conventions

Use blockquotes to reply to or highlight another user's statement:

"Is there a closed form for $\sum_{n=1}^{\infty} \frac{1}{n^2}$?"

Yes! This is the Basel problem, solved by Euler in 1734:

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

More generally, the Riemann zeta function is defined for $\text{Re}(s) > 1$ as:

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$

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🔗 Horizontal Rules & Section Dividers

Use --- to cleanly divide a long thread into sections, keeping discussions readable even as they grow.

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Happy proving. May your limits converge and your series be absolutely convergent. 🎓