Essential Math Formulas
Explore key formulas used in algebra and related topics.
= (reiθ)n = rneinθ = rncis nθ
Algebra & Numbers: More FAQs
Further questions about different types of numbers and concepts in algebra.
An algebraic number is any number (real or complex) that is a root of a non-zero polynomial equation with integer coefficients. For example, the number 2 is algebraic because it is a root of x - 2 = 0 (a polynomial with integer coefficients). The number √2 is also algebraic because it's a root of x² - 2 = 0.
Yes, the set of algebraic numbers is countable. This is a significant result in mathematics. It means that even though there are infinitely many algebraic numbers, they can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, ...). This is proven by showing that the set of polynomials with integer coefficients is countable, and each polynomial has a finite number of roots.
Algebraic number theory is a branch of mathematics that studies algebraic numbers. It extends concepts from number theory (which primarily deals with integers and rational numbers) to algebraic numbers and algebraic integers (roots of monic polynomials with integer coefficients). It involves studying algebraic structures like number fields, rings of integers, and ideals.
Number theory and algebra are distinct but closely related fields, and they heavily overlap. Many problems in number theory are solved using algebraic tools, and algebra provides the framework for areas like algebraic number theory. While you can study basic number theory without deep algebra and vice versa, more advanced work in either field invariably draws upon the other.
'x' (and other letters like 'a', 'b', 'm', 'y', 'h') in algebra typically represents a variable. A variable is a symbol (usually a letter) that stands for a number whose value may vary. In an equation or expression, 'x' represents an unknown quantity that you might be trying to solve for or a value that can take on different numbers within a specified set (like real numbers or integers).
Similar to 'x', letters like 'a', 'b', 'm', 'y', and 'h' are usually used as variables or parameters in algebra. Their value depends on the specific problem or context. They represent numbers whose value is either unknown and needs to be found (like in an equation) or represents a general quantity that can change (like in a formula or function).
The number written before an algebraic symbol (a variable or term) is called a coefficient. It is a numerical factor that multiplies the variable or term. For example, in the expression 5x, 5 is the coefficient. In the term -3y², -3 is the coefficient. If there is no number written (like in 'z'), the coefficient is understood to be 1 (or -1 if the term is negative, like -z).
Yes, the field of complex numbers is algebraically closed. This is a fundamental result known as the Fundamental Theorem of Algebra. It states that any non-constant polynomial with complex coefficients has at least one complex root. A direct consequence is that any non-constant polynomial with complex coefficients can be factored into linear factors over the complex numbers. This means you don't need to extend the complex number system further to find roots of such polynomials.
Yes, 0 is a real number. The set of real numbers includes all rational and irrational numbers, and 0 is an integer (which is a rational number). It sits at the origin on the number line and is a critical number in algebra, serving as the additive identity (a + 0 = a) and playing a key role in definitions, properties, and solving equations.
