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The properties of the Real Number System will prove useful when working with equations, functions and formulas in Algebra, as they allow for the creation of equivalent expressions which will often aid in solving problems. In addition, they can be used to help explain or justify solutions.
Understanding the properties of real numbers will help us simplify numerical and algebraic expressions, solve equations, and more as you progress in studying algebra. In other words, the properties or axioms of real numbers are just one of many basic foundations of mathematics