Number theory is a branch of mathematics that deals with the properties and relationships of numbers. Some key theorems of number theory include:
The Fundamental Theorem of Arithmetic: This theorem states that every integer greater than 1 can be expressed as a unique product of prime numbers, up to the order of the factors.
Fermat's Little Theorem: This theorem states that if p is a prime number and a is any integer not divisible by p, then a^(p-1) is congruent to 1 (mod p).
Euler's Totient Theorem: This theorem states that if a and n are coprime positive integers, then a^φ(n) is congruent to 1 (mod n), where φ(n) is Euler's totient function.
The Chinese Remainder Theorem: This theorem states that if m1, m2, ..., mk are pairwise coprime positive integers, and a1, a2, ..., ak are any integers, then there exists an integer x that solves the system of congruences: x ≡ a1 (mod m1), x ≡ a2 (mod m2), ..., x ≡ ak (mod mk).
Wilson's Theorem: This theorem states that for any prime number p, (p-1)! is congruent to -1 (mod p).
Quadratic Reciprocity: This theorem is concerned with determining when a prime p is a quadratic residue (mod q), where q is another prime. It states that if p and q are distinct odd primes, then p is a quadratic residue (mod q) if and only if q is a quadratic residue (mod p).
Dirichlet's Theorem on Arithmetic Progressions: This theorem states that for any two coprime positive integers a and d, there are infinitely many primes of the form a + nd, where n is a non-negative integer.
Geometry
Euclid's Geometry is a mathematical system developed by the ancient Greek mathematician Euclid. Some key theorems of Euclid's Geometry include:
The Parallel Postulate: This postulate states that if a straight line intersects two other straight lines and the interior angles on the same side of the transversal sum to less than 180 degrees, then the two lines, when extended indefinitely, will intersect on that same side.
The Pythagorean Theorem: This theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
The Law of Cosines: This theorem relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, it states that the square of the length of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
The Law of Sines: This theorem relates the lengths of the sides of a triangle to the sine of one of its angles. Specifically, it states that the ratio of the length of one side to the sine of the opposite angle is equal to the same ratio for any other side and its opposite angle.
The Angle Bisector Theorem: This theorem states that in a triangle, an angle bisector divides the opposite side into two segments that are proportional to the lengths of the other two sides.
Combinatorics
Combinatorics is a branch of mathematics that deals with the study of counting and arranging objects. Some key theorems in combinatorics include:
The Multiplication Principle: This theorem states that if there are m ways to perform one task and n ways to perform another task, then there are mn ways to perform both tasks in order.
The Addition Principle: This theorem states that if there are m ways to perform one task and n ways to perform another task, then there are m + n ways to perform one of the tasks.
The Principle of Inclusion-Exclusion: This theorem provides a way to count the number of elements in the union of two or more sets. It involves adding the sizes of the individual sets, subtracting the sizes of their intersections, adding the sizes of the intersections of three sets, and so on.
Permutation and Combination Formulas: These formulas provide a way to count the number of ways to arrange or choose objects from a set. The permutation formula is used when order matters, and the combination formula is used when order does not matter.
Pigeonhole Principle: This theorem states that if n objects are placed into k containers and k < n, then at least one container must contain more than one object.
Catalan Numbers: These numbers appear in many different counting problems in combinatorics, including the number of ways to arrange parentheses in an expression and the number of paths that do not cross a given line in a triangular lattice.
Probability
Probability is a branch of mathematics that deals with the study of random events and their outcomes. Some key theorems in probability include:
Law of Large Numbers: This theorem states that as the number of trials in a probability experiment increases, the experimental probability of an event will approach the theoretical probability of that event.
Central Limit Theorem: This theorem states that the sum (or average) of a large number of independent and identically distributed random variables, each with finite mean and variance, will be approximately normally distributed.
Bayes' Theorem: This theorem provides a way to update probabilities based on new information. It states that the probability of an event given some evidence is proportional to the probability of the evidence given the event multiplied by the prior probability of the event.
Conditional Probability: This theorem provides a way to calculate the probability of an event given that another event has occurred. It states that the conditional probability of an event A given an event B is equal to the probability of the intersection of A and B divided by the probability of B.
Combinatorial Probability: This theorem provides a way to count the number of possible outcomes in a probability experiment. It includes the multiplication rule, permutation, and combination formulas.
Law of Total Probability: This theorem states that the probability of an event can be calculated as a weighted sum of the conditional probabilities of the event given each of several mutually exclusive and exhaustive events.
Statistics
Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. Some key theorems in statistics include:
Central Limit Theorem: This theorem states that the sum (or average) of a large number of independent and identically distributed random variables, each with finite mean and variance, will be approximately normally distributed.
Law of Large Numbers: This theorem states that as the sample size of a probability experiment increases, the sample mean will approach the population mean.
The Sampling Distribution of the Mean: This theorem states that the distribution of the sample means from a large number of samples will be approximately normal, regardless of the shape of the population distribution, and the mean and standard deviation of the sampling distribution can be calculated based on the mean and standard deviation of the population.
Hypothesis Testing: This theorem provides a framework for testing statistical hypotheses based on sample data. It involves formulating null and alternative hypotheses, calculating test statistics, and determining the p-value to make a decision about rejecting or failing to reject the null hypothesis.
Confidence Intervals: This theorem provides a range of values around a sample statistic that is likely to contain the true population parameter with a specified level of confidence. It involves calculating a margin of error based on the sample size, standard deviation, and confidence level.
Regression Analysis: This theorem provides a way to estimate the relationship between a dependent variable and one or more independent variables. It involves fitting a regression model, interpreting the coefficients, and assessing the goodness of fit.