Mathematics

Comprehensive math reference with formulas and explanations

Sum of the first $n$ natural numbers

\(1 + 2 + \cdots + n = \frac{n(n+1)}{2}\)

Variables:

  • \(n is a positive integer.\)

Description/Usage:

Gives the sum of all natural numbers from \(1\) up to \(n\). Often illustrated by pairing terms (e.g. \(1+n\), \(2+(n-1)\), etc.) to simplify the addition.

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Fermat's Little Theorem

\(a^{p-1} \equiv 1 \pmod p\)

Variables:

  • \(p is a prime number, a is an integer not divisible by p.\)

Description/Usage:

In modular arithmetic, this theorem states that if \(p\) is prime, then \(a^{p-1}\) leaves a remainder of \(1\) upon division by \(p\). It's fundamental for simplifying exponents mod \(p\).

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Quadratic Formula

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

Variables:

Description/Usage:

Gives the solutions for a quadratic equation of any form. It is typically illustrated by the parabola of \(y = ax^2+bx+c\) intersecting the \(x\)-axis at \(x\) values given by this formula (the roots).

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Pythagorean Theorem

\(c^2 = a^2 + b^2\)

Variables:

  • \(a and b are the lengths of the legs (perpendicular sides) of a right triangle, and c is the length of the hypotenuse (side opposite the right angle).\)

Description/Usage:

Relates the side lengths of a right-angled triangle. It is used to find the third side given the other two.

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Trigonometric Ratios (right triangle definitions)

\(\displaystyle \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}\)

Variables:

Description/Usage:

Defines sine, cosine, and tangent as ratios of sides in a right-angled triangle.

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Distance Between Two Points (2D)

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Variables:

  • \((x_1, y_1) and (x_2, y_2) are the coordinates of the two points.\)

Description/Usage:

Calculates the straight-line distance between two points in the plane, derived from the Pythagorean theorem.

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