Mathematical Formulas with ChatGPT

Mathematics encompasses a wide range of fields, each with its own set of important formulae. Here are some key formulae from various branches:

Algebra

1. Quadratic Formula: For the quadratic equation \( ax^2 + bx + c = 0 \),
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
2. Binomial Theorem: For any positive integer \( n \),
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

Geometry

1. Pythagorean Theorem: In a right triangle, the square of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides \( a \) and \( b \),
\[ c^2 = a^2 + b^2 \]
2. Area of a Circle: Given a circle with radius \( r \),
\[ \text{Area} = \pi r^2 \]
3. Circumference of a Circle:
\[ \text{Circumference} = 2\pi r \]

Trigonometry

1. Sine, Cosine, Tangent: For a right triangle with angle \( \theta \),
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \] \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
2. Pythagorean Identity:
\[ \sin^2\theta + \cos^2\theta = 1 \]

Calculus

1. Derivative of a Function: The derivative of a function \( f(x) \) with respect to \( x \) is defined as,
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]
2. Fundamental Theorem of Calculus:
\[ \int_a^b f(x) \, dx = F(b) - F(a) \] where \( F \) is an antiderivative of \( f \), i.e., \( F'(x) = f(x) \).

Probability and Statistics

1. Bayes' Theorem: For events \( A \) and \( B \),
\[ P(A|B) = \frac{P(B|A) P(A)}{P(B)} \]
2. Standard Deviation**: For a set of data \( x_1, x_2, \ldots, x_n \) with mean \( \mu \),
\[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2} \]

Linear Algebra

1. Matrix Multiplication: For matrices \( A \) and \( B \),
\[ (AB)_{ij} = \sum_{k} A_{ik} B_{kj} \]
2. Determinant of a 2x2 Matrix: For matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \),
\[ \text{det}(A) = ad - bc \]

Number Theory

1. Euclidean Algorithm: To find the greatest common divisor (gcd) of two integers \( a \) and \( b \),
\[ \text{gcd}(a, b) = \text{gcd}(b, a \mod b) \]
2. Fermat's Little Theorem: If \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then
\[ a^{p-1} \equiv 1 \pmod{p} \]

These formulae are foundational across many areas of mathematics and are frequently used in both academic studies and practical applications.

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