Algebraic number — Safekipedia Discoverer

In mathematics, an algebraic number is a special kind of number that solves a certain type of equation called a polynomial. These equations have integer or rational numbers as their coefficients. For example, the golden ratio, which appears in nature and art, is an algebraic number because it solves the equation x² – x – 1 = 0.

Algebraic numbers include everyday numbers we use often, such as integers (like 1, 2, or –3) and rational numbers (such as 1/2 or 4/5). They also include more complex numbers, like square roots and even some complex numbers. These numbers can be added, subtracted, multiplied, and divided to get other algebraic numbers, forming what mathematicians call a field.

Not all numbers are algebraic. Some important numbers, like π (the ratio of a circle’s circumference to its diameter) and the number e (used in growth and decay problems), are called transcendental because they do not solve any polynomial equation with integer coefficients. While there are infinitely many algebraic numbers, most numbers on the number line are actually transcendental.

Examples

All rational numbers are algebraic. This means numbers like fractions, such as 3 divided by 4, are algebraic because they can be found as solutions to certain equations.

Many famous numbers, like the square root of 2 or the golden ratio, are also algebraic. They can be solutions to equations with whole number coefficients, even if they cannot be written as simple fractions. For example, the golden ratio is a solution to the equation x² − x − 1 = 0.

Properties

Algebraic numbers on the complex plane colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4). The larger points come from polynomials with smaller integer coefficients.

An algebraic number is a number that solves a special kind of math problem called a polynomial equation. These equations use whole numbers or fractions as coefficients. For example, the golden ratio, which is (1 + √5)/2, solves the equation x² − x − 1 = 0, making it an algebraic number.

All regular numbers you know, like whole numbers, fractions, and square roots of whole numbers, are algebraic numbers. They have unique simple equations that show their properties clearly. These numbers are very common in math, but there are also many numbers that aren’t algebraic, which are called transcendental numbers.

Field

Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black.[further explanation needed]

When you add, subtract, multiply, or divide two algebraic numbers (as long as you don't divide by zero), the result is also an algebraic number. This is because these operations create new numbers that still follow the same rules as the original algebraic numbers.

All algebraic numbers together form a special group called a field. This means that any root of a polynomial with algebraic number coefficients will also be an algebraic number. In other words, the algebraic numbers are "closed" under taking roots of polynomials.

Related fields

Algebraic numbers are numbers that can be found using equations with integer coefficients. For example, you can get some algebraic numbers by using basic math operations like adding, subtracting, multiplying, and dividing, and also by taking roots (like square roots). However, not all algebraic numbers can be found this way. Some come from more complex equations, like one that needs a fifth-degree polynomial to solve.

A special kind of number called a "closed-form number" can be defined using polynomials and other math tools like exponentials and logarithms. Algebraic numbers fit into this group, but there are also other numbers, like some very special ones, that are not algebraic but still have neat definitions.

Algebraic integers

Main article: Algebraic integer

Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top.

An algebraic integer is a special type of algebraic number. It is a number that solves a polynomial equation where the coefficients are whole numbers and the first coefficient is 1. For example, numbers like (5 + 13\sqrt{2}), (2 - 6i), and (\frac{1}{2}(1 + i\sqrt{3})) are algebraic integers.

Algebraic integers behave nicely under addition, subtraction, and multiplication—they stay within the group of algebraic integers. They are like whole numbers but extend beyond them, similar to how rational numbers include fractions but also whole numbers. In more advanced math, algebraic integers help form structures called rings and domains.

Special classes

Algebraic numbers can be grouped into special classes based on their properties. Some of these classes include Algebraic solution, which are solutions to polynomial equations that can be expressed using only square roots and other basic operations. Other examples are Gaussian integer and Eisenstein integer, which extend the idea of whole numbers into complex number systems. These special classes help mathematicians study algebraic numbers in more detail.