Number And Algebra Formulae AA SL & AA HL

Negative numbers are numbers less than zero. In algebra, they are essential for representing values like debt, temperatures below freezing, or movement in the opposite direction on a number line (left of zero). Understanding how to perform operations (addition, subtraction, multiplication, division) with negative numbers, including rules for signs, is fundamental to solving equations and inequalities involving them.

Real numbers encompass all numbers that can be plotted on a number line. This includes all rational numbers (like fractions and integers) and all irrational numbers (like √2 and π). In most introductory algebra contexts, variables represent real numbers unless explicitly stated otherwise. They form the basis for graphing functions on a coordinate plane.

"All real numbers" refers to the entire set of numbers that can be found on the number line. When you solve an equation or inequality and the solution is "all real numbers," it means any value from the set of real numbers will satisfy the original statement. Similarly, if the domain of a function is "all real numbers," it means you can substitute any real number for the input variable.

Yes, absolutely. Negative numbers are a subset of the real numbers. Any number, positive, negative, or zero, that can be located on the number line is considered a real number. This includes negative integers (-3), negative fractions (-1/2), and negative irrational numbers (-√5).

Numbers in algebra are broadly classified into sets. The main ones include:

• Natural Numbers: Counting numbers {1, 2, 3, ...}. (Sometimes includes 0).
• Whole Numbers: Natural numbers plus zero {0, 1, 2, 3, ...}.
• Integers: All whole numbers and their opposites {..., -2, -1, 0, 1, 2, ...}.
• Rational Numbers: Numbers that can be written as a fraction p/q (p, q integers, q≠0). Includes integers, terminating, and repeating decimals.
• Irrational Numbers: Real numbers that cannot be written as a simple fraction (non-terminating, non-repeating decimals like π, √2).
• Real Numbers: The set of all rational and irrational numbers.
• Complex Numbers: Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Includes real numbers (when b=0) and imaginary numbers (when a=0).

Natural numbers are the basic counting numbers: {1, 2, 3, 4, ...}. They are positive integers excluding zero. In some contexts (like set theory or advanced algebra), the set may include 0, but in most elementary algebra, natural numbers start from 1.

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

Rational numbers are any numbers that can be expressed as a ratio or fraction p/q, where p and q are integers and q is not zero. This broad category includes all integers (e.g., 5 = 5/1), all terminating decimals (e.g., 0.25 = 1/4), and all repeating decimals (e.g., 0.333... = 1/3).

Irrational numbers are real numbers that cannot be expressed as a simple fraction p/q. When written as a decimal, they are non-terminating and non-repeating. Famous examples encountered in algebra include the square roots of non-perfect squares (like √2, √3, √7) and transcendental numbers like π (pi) and 'e' (Euler's number).

Complex numbers extend the real number system by including the imaginary unit 'i', where i² = -1. A complex number is written in the form a + bi, where 'a' and 'b' are real numbers. 'a' is called the real part, and 'bi' is called the imaginary part. Complex numbers allow us to find solutions to equations like x² + 1 = 0 that have no real solutions and are essential in higher-level math and many scientific fields.

Imaginary numbers are a subset of complex numbers where the real part is zero. They have the form bi, where 'b' is a real number and 'i' is the imaginary unit (√-1). For example, 3i, -0.5i, and √2 i are imaginary numbers. They are the imaginary component of a complex number.

Working with negative numbers requires applying specific rules to operations:

• Addition: Adding a negative is like subtracting (e.g., 5 + (-3) = 5 - 3 = 2). Subtracting a negative is like adding (e.g., 5 - (-3) = 5 + 3 = 8).
• Multiplication/Division: The product or quotient of two numbers with the same sign (both positive or both negative) is positive. The product or quotient of two numbers with different signs is negative.
• Equations/Inequalities: Remember these sign rules when adding, subtracting, multiplying, or dividing both sides of an equation or inequality. Multiplying or dividing an inequality by a negative number requires reversing the inequality sign.

In Algebra 2, you learn fundamental operations with complex numbers (a + bi):

• Adding/Subtracting: Combine the real parts and the imaginary parts separately: (a + bi) ± (c + di) = (a±c) + (b±d)i.
• Multiplying: Use the distributive property (FOIL method) and substitute i² with -1: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i.
• Dividing: Multiply the numerator and denominator by the complex conjugate of the denominator (conjugate of c + di is c - di). This eliminates 'i' from the denominator.
• Powers of i: Simplify powers of i (i¹, i², i³, i⁴, etc.) using the pattern i, -1, -i, 1, which repeats every four powers.
• Solving Quadratic Equations: Use the quadratic formula to find complex solutions when the discriminant (b² - 4ac) is negative.

Factoring in algebra means breaking down a number or an algebraic expression into a product of simpler expressions (its factors). For whole numbers, this often means finding its prime factors (e.g., 12 = 2 × 2 × 3). For algebraic expressions (like polynomials), it means writing the expression as a product of other polynomials (e.g., x² - 9 = (x - 3)(x + 3)). Factoring is a key skill used to simplify expressions, solve equations, and work with rational expressions.

To represent consecutive numbers algebraically, you typically use a variable and add integers. If 'n' represents the first number:

• Consecutive Integers: n, n + 1, n + 2, n + 3, ...
• Consecutive Even Integers: If 'n' is an even integer, then n, n + 2, n + 4, ... are consecutive even integers. Alternatively, start with 2n, and the consecutive evens are 2n, 2n + 2, 2n + 4, ...
• Consecutive Odd Integers: If 'n' is an odd integer, then n, n + 2, n + 4, ... are consecutive odd integers. Alternatively, start with 2n + 1, and the consecutive odds are 2n + 1, 2n + 3, 2n + 5, ...

Yes, absolutely. Complex numbers are a standard and important part of algebra, typically introduced in high school Algebra 2 or college algebra courses. The study of complex numbers, including their arithmetic operations, graphing, and use in solving polynomial equations, falls squarely within the domain of algebra.

Real numbers adhere to several fundamental properties that are essential for manipulating expressions and solving equations:

• Commutative Property: Order doesn't matter for addition (a + b = b + a) or multiplication (a × b = b × a).
• Associative Property: Grouping doesn't matter for addition (a + (b + c) = (a + b) + c) or multiplication (a × (b × c) = (a × b) × c).
• Distributive Property: Multiplication distributes over addition/subtraction (a × (b + c) = a × b + a × c).
• Identity Property: Adding 0 (a + 0 = a); Multiplying by 1 (a × 1 = a).
• Inverse Property: Adding the opposite results in 0 (a + (-a) = 0); Multiplying by the reciprocal results in 1 (a × (1/a) = 1, for a ≠ 0).

Classifying a number means identifying all the number sets it belongs to. You typically start with the most specific set it fits into and list the broader sets it also belongs to. The main hierarchy (from most specific to broadest) is often: Natural/Whole → Integers → Rational → Real → Complex. Irrational numbers are also Real but are separate from Rational numbers.

Example:

• 5 is Natural, Whole, Integer, Rational, Real, Complex.
• -2 is Integer, Rational, Real, Complex.
• 1/2 is Rational, Real, Complex.
• √2 is Irrational, Real, Complex.
• 3i is Imaginary, Complex.

No, you do not always round numbers in algebra. You should only round when the problem specifically instructs you to (e.g., "round your answer to two decimal places") or when dealing with real-world measurements or applications where rounding is practical or necessary. In pure algebraic problems, exact answers (like fractions or simplified radicals) are preferred unless rounding is specified.