Real Numbers

1 chapter

Real numbers are a set of numbers that include rational and irrational numbers. They are used to represent all points on the number line. The set of real numbers is denoted by ℝ.

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Square and Cube Root

Rules for Calculating with Roots

Simplifying Square Roots (Partial Rooting)

SUBSETS OF REAL NUMBERS

Real numbers can be divided into two main subsets:

RATIONAL NUMBERS (ℚ): Numbers that can be written as a fraction p/q, where p and q are integers and q ≠ 0 (e.g., 1/2, -3/4, 5 which is 5/1).
IRRATIONAL NUMBERS: Numbers that cannot be written as a simple fraction and have an infinite, non-repeating decimal representation (e.g., √2, π, e).

PROPERTIES OF REAL NUMBERS

DENSITY: Between any two distinct real numbers, there is always another real number.
INFINITY (UNBOUNDEDNESS): The set ℝ is unbounded in both the positive and negative directions.
CLOSURE: Real numbers are closed under basic arithmetic operations (addition, subtraction, multiplication, and division, except for division by 0).

BASIC OPERATIONS

ADDITION: 2.5 + 1.3 = 3.8
SUBTRACTION: 7 – 4.2 = 2.8
MULTIPLICATION: 1.5 × 3 = 4.5
DIVISION: 5 ÷ 2 = 2.5

CONCLUSION

Real numbers represent the entire number line and allow for the notation of all possible quantities. Their structure is the foundation of many mathematical fields, including analysis and algebra.