Fundamental Algebra Concepts and Properties

Fundamental Algebra Concepts

• Product: Refers to the result of multiplication.
• Quotient: The outcome obtained from division.
• Sum: The result of adding two or more numbers.

Properties of Equality

• Reflexive Property: Any quantity is equal to itself (a = a).
• Symmetric Property: If a equals b, then b also equals a.
• Transitive Property: If a equals b and b equals c, then a equals c.

Inverses and Identities

• Additive Inverse: The opposite of a number; when added to the number, the result is zero (e.g., a + (-a) = 0).
• Multiplicative Inverse: The reciprocal of a number (e.g., for 5, the inverse is 1/5).
• Additive Identity: Adding zero to a number does not change its value (a + 0 = a).
• Multiplicative Identity: Any number multiplied by one remains unchanged (a × 1 = a).

Algebraic Properties

• Commutative Property: States that the order of addition or multiplication does not affect the result (e.g., a + b = b + a and ab = ba).
• Associative Property: Changing the grouping of numbers does not change their value (e.g., (a + b) + c = a + (b + c)).
• Distributive Property: This property relates multiplication to addition (e.g., a(b + c) = ab + ac).

Number Sets

• Smallest Whole Number: Defined as 0.
• Rational Numbers: Numbers expressible as a ratio of two integers, including fractions and whole numbers.
• Real Numbers: Comprise all rational and irrational numbers.
• Integers: Include all whole numbers, both positive and negative, plus zero.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction; their decimal representation is non-ending and non-repeating.
• Natural Numbers: The set of counting numbers starting from 1 onwards (1, 2, 3, ...).