Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Powers, Roots, etc.: Calculation Rules

Description

This lecture covers the rules of calculation for powers, roots, and more, focusing on positive numbers and integers. It explains conventions, properties, and notations related to these mathematical concepts, including reciprocals and logarithms.

In MOOCs (3)

Warm-up for EPFL

Analysis I (part 1): Prelude, basic concepts, real numbers

Analysis I

Official source

Quiz

Related concepts (32)

Quadratic integer

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0 with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of rational integers, such as , and the complex number i = , which generates the Gaussian integers.

Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic.

Eisenstein integer

In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form where a and b are integers and is a primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set.

Coprime integers

In number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also a is prime to b or a is coprime with b. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor.

Algebraic integer

In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted by OK, is the intersection of K and A: it can also be characterised as the maximal order of the field K.

Related lectures (29)

Cartesian Product and Induction

Introduces Cartesian product and induction for proofs using integers and sets.

Elementary Algebra: Numeric Sets

Explores elementary algebra concepts related to numeric sets and prime numbers, including unique factorization and properties.

Basic Properties MOOC: Analysis I (part 1): Prelude, basic concepts, real numbers MOOC: Analysis I

Covers basic properties of natural numbers, including order relations and inverses.

Integers: Elementary Concepts

Covers fundamental concepts related to integers, including properties of well-ordered sets and the principle of induction.

Binary Integer Representation MOOC: Information, Computation, Communication: Introduction to computational thinking