Foundations of Algebra Chapter 1 Flashcards

Key Algebra Concepts

• Variable: Symbol (typically a letter) representing one or more numbers. Essential for forming equations and expressions.

• Integers: Collection of whole numbers including both positive and negative values, along with zero.

• Order of Operations: Established process to evaluate mathematical expressions (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Real Numbers: Complete set of numbers encompassing both rational (fractions, integers) and irrational numbers (non-repeating, non-terminating decimals).

• Additive Inverse: For any number 'a', its additive inverse is '-a', such that their sum equals zero.

• Evaluate: Substituting numbers for given variables and simplifying the expression.

• Simplify: Process of reducing a mathematical expression to its most basic form.

• Equivalent Expression: Different expressions that yield the same value for their respective variables.

• Algebraic Expressions: Composed of numerical values, variables, and operations, these phrases do not have an equality sign.

• Coefficient: Number in front of a variable in a term, indicating how many times the variable is multiplied.

• Term: Units in an expression that can be a single number, variable, or a combination of numbers and variables.

• Like Terms: Terms that share exactly the same variable(s) raised to the same power, allowing for them to be combined.

Types of Numbers

• Quantity: A measurable or countable amount.

• Numerical Expression: Consists solely of numbers and operational symbols without any variables.

• Power: Represents a number multiplied by itself a defined number of times, characterized by a base and an exponent.

• Exponent: Indicates how many times to multiply the base by itself.

• Base: The number that is repeatedly multiplied in a power expression.

• Radicand: The number or expression located under a radical sign, indicating the value to extract the root from.

• Radical: Symbol used to denote the square root or other roots of a number.

• Rational Number: Any number expressible as a fraction where both numerator and denominator are integers.

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3,...).

• Whole Numbers: Combination of natural numbers and zero (0, 1, 2, 3,...).

• Irrational Number: Cannot be expressed as a fraction of two integers; represented by non-terminating, non-repeating decimals.

• Absolute Value: The distance of a number from zero on a number line, always expressed as a positive value.

• Opposite: Two numbers that are symmetrically located relative to zero on the number line.

• Reciprocal (Multiplicative Inverse): For any non-zero number, the reciprocal is a number which when multiplied with the original equals one (e.g., the reciprocal of 2/3 is 3/2).

Key Algebra Properties

• Distributive Property: A critical algebraic principle that allows for distribution of multiplication over addition (a(b + c) = ab + ac).

• Constant: A value that remains fixed; it does not change or contain any variables.

• Open Sentence: An equation featuring one or more variables, indicating that it may hold different truths based on the variable's value.

• Equation: A mathematical statement asserting that two expressions are equal (represented with an equals sign).

• Solution of an Equation (one variable): Specific value assigned to a variable that makes the equation valid.

• Solution of an Equation (two variables): Any ordered pair (x,y) that satisfies the equality of the equation.

• Square Root: A value which, when multiplied by itself, retrieves the original number (e.g., √4 = 2).

• Perfect Square: The result of squaring a whole number (e.g., 1, 4, 9 are perfect squares).

• Inequality: Mathematical expressions that show the comparative relationships between two values or expressions using inequality symbols (>, <, ≥, ≤).