Number and Algebra Formulae AI HL only

Real numbers encompass all numbers that can be placed on a continuous number line. This includes all rational numbers (like integers and fractions) and all irrational numbers (like √2 or π). In typical algebra courses, variables usually represent real numbers unless specified otherwise.

Yes, absolutely. Zero (0) is an integer, and all integers are also rational numbers, and all rational numbers are real numbers. So, 0 is definitively a real number in algebra and all of mathematics. It holds a central position on the number line.

A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. This category includes all integers (e.g., 7 = 7/1), all terminating decimals (e.g., 0.75 = 3/4), and all repeating decimals (e.g., 0.111... = 1/9).

Natural numbers are the counting numbers: {1, 2, 3, 4, ...}. These are positive integers. In some fields (like set theory) or depending on the context, 0 is sometimes included, but in basic algebra, they usually start from 1.

Whole numbers are the natural numbers combined with zero: {0, 1, 2, 3, ...}. They are the non-negative integers.

A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying the equation \(i^2 = -1\). The term \(a\) is the real part, and \(bi\) is the imaginary part. Complex numbers extend the real numbers and are used to solve equations (like \(x^2 + 1 = 0\)) that have no real solutions. They are typically introduced in Algebra 2 or college-level algebra courses.

An imaginary number is a complex number where the real part is zero, meaning it can be written in the form \(bi\), where \(b\) is a real number and \(i\) is the imaginary unit (\(\sqrt{-1}\)). Examples include \(3i\), \(-i\), or \(\sqrt{5}i\). The number \(i\) itself is the basic imaginary unit.

An algebraic number is any number that is a root of a non-zero polynomial equation with integer coefficients. All rational numbers are algebraic, and many irrational numbers (like \(\sqrt{2}\)) are also algebraic. Transcendental numbers (like π and e) are those that are *not* algebraic.

Yes, the set of all algebraic numbers is countable. This means they can be put into a one-to-one correspondence with the natural numbers, even though there are infinitely many of them. This is proven by showing that the set of polynomials with integer coefficients is countable, and each polynomial has a finite number of roots.

A signed number is simply a number with a positive (+) or negative (-) sign indicating its position relative to zero on the number line. Positive numbers are greater than zero, and negative numbers are less than zero. All real numbers (except zero, which has no sign) can be considered signed numbers.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. (e.g., 2, 3, 5, 7, 11). While the definition comes from number theory, prime numbers are used in algebra, particularly when factoring expressions or working with concepts like least common multiple (LCM) or greatest common divisor (GCD) of numbers involved in algebraic terms.

A radical number is a number expressed using a radical symbol (√). This typically involves square roots, cube roots, or roots of higher indices (e.g., √5, ³√7). Radical numbers can be rational (like √9 = 3) or irrational (like √2). Algebra often involves simplifying expressions containing radicals or solving equations with variables under radical signs.

In algebra, letters like \(x, a, b, h, m, p, t,\) and \(y\) are used as symbols to represent numbers. These symbols are called variables or parameters.

• Variables: Represent unknown quantities or values that can change. \(x\) and \(y\) are very commonly used for variables, especially in equations and when graphing.
• Parameters: Represent values that are fixed in a particular context but can change in different contexts (like \(m\) and \(b\) in \(y = mx + b\)).

The specific number they represent depends entirely on the equation, expression, or problem you are working with.

The number multiplying a variable or term (the number written directly before the algebraic symbol) is called the coefficient. For example, in the term \(8y\), 8 is the coefficient. In \(-z\), the coefficient is -1.

In algebra, if you let a variable (like \(n\) or \(x\)) represent "a number," then "twice a number" is represented as \(2n\) or \(2x\). This shows the number multiplied by 2.

A number line is a straight line where every point corresponds to a unique real number. Numbers are ordered on the line, with zero at the center, positive numbers to the right, and negative numbers to the left. In algebra, number lines are used to visualize numbers, represent intervals, understand inequalities, and graph solutions to equations or inequalities.

Number Theory and Algebra are distinct branches of mathematics, but they are closely related and often use techniques from each other. Number Theory focuses specifically on the properties of integers and related concepts (like prime numbers, divisibility). While basic number theory can be studied separately, advanced number theory, particularly Algebraic Number Theory, heavily relies on abstract algebraic structures and methods. So, while not synonymous, there's a significant overlap and interdependence, especially at higher levels.