Number and Algebra Formulae AI SL & AI HL

An algebraic number is any number (real or complex) that is a root (a solution) of a non-zero polynomial equation with integer coefficients. For instance, integers (like 5, which is a root of \(x - 5 = 0\)), rational numbers (like 1/2, a root of \(2x - 1 = 0\)), and numbers involving roots of rational numbers (like \(\sqrt{2}\), a root of \(x^2 - 2 = 0\), or the imaginary unit \(i\), a root of \(x^2 + 1 = 0\)) are all algebraic numbers.

Numbers that are *not* algebraic (like \(\pi\) or \(e\)) are called transcendental numbers.

Yes, the set of algebraic numbers is countable. This is a significant result in mathematics. Although there are infinitely many algebraic numbers, they can be put into a one-to-one correspondence with the set of natural numbers (\(\{1, 2, 3, ...\}\)). The proof relies on the fact that the set of all polynomials with integer coefficients is countable, and each polynomial has a finite number of roots.

Algebraic number theory is a major branch of number theory that uses the tools of abstract algebra (like rings, fields, and ideals) to study algebraic numbers and algebraic integers. It extends concepts from elementary number theory (which deals with integers and rational numbers) to more general algebraic structures arising from the roots of polynomials. It helps in understanding properties of numbers in number fields.

Number theory and algebra are distinct but deeply interconnected fields of mathematics. While you can study foundational topics in each independently, advanced work in number theory relies heavily on algebraic structures and techniques (leading to fields like Algebraic Number Theory). Conversely, problems in algebra are often motivated by questions arising from number theory. So, while not strictly a *part* of elementary algebra, advanced number theory is inseparable from advanced algebra.

In algebra, letters like \(x, a, b, m, y,\) and \(h\) typically represent variables or parameters. A variable is a symbol that stands for a numerical value that is unknown or can change. In an equation or expression, \(x\) often represents the primary unknown you are solving for, while other letters like \(a, b, m,\) etc., might represent other variables, known constants, or parameters that define a general case (like \(m\) and \(b\) in the linear function \(y = mx + b\)). Their exact meaning depends on the specific problem context.

The number written directly before a variable or group of variables (an algebraic symbol or term) is called a coefficient. It indicates how many times the variable term is being multiplied. For example:

• In \(7x\), the coefficient is 7.
• In \(-3y^2\), the coefficient is -3.
• In \(ab\), the coefficient is 1 (since \(1 \times ab = ab\)).
• In \(-z\), the coefficient is -1.

Factoring in algebra usually refers to factoring *algebraic expressions* (like polynomials), not just prime factoring numbers. It means rewriting an expression as a product of simpler expressions (its factors). Common techniques include:

• Greatest Common Factor (GCF): Pulling out the largest common factor from all terms (e.g., \(4x + 8y = 4(x + 2y)\)).
• Factoring Trinomials: Expressing a three-term polynomial (like \(x^2 + bx + c\)) as a product of two binomials (e.g., \(x^2 + 5x + 6 = (x + 2)(x + 3)\)).
• Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\).
• Grouping: Used for polynomials with four or more terms.

The goal is often to simplify expressions or solve equations (since if a product of factors is zero, at least one factor must be zero).

Yes, the field of complex numbers (\(\mathbb{C}\)) is algebraically closed. This is the statement of the Fundamental Theorem of Algebra. It means that any non-constant polynomial equation with complex coefficients has at least one root within the complex numbers. A consequence is that any such polynomial can be factored into a product of linear factors over the complex numbers. This is one reason the complex number system is so powerful.

By convention in algebra, the numerical coefficient usually comes first, followed by the variable(s). If there are multiple variables in a term, they are typically written in alphabetical order. For example, we write \(5x\), not \(x5\); \(-2ab\), not \(-2ba\). If a term is just a number without variables, it's usually written last in an expression (the constant term).