Define real number with operation on real numbers.
A real number is a value that represents a quantity along a continuous line, often called the number line. Real numbers include both rational numbers (like integers, fractions) and irrational numbers (numbers that cannot be expressed as a simple fraction, like √2 or π). The set of real numbers is typically denoted by the symbol ℝ.
Operations on Real Numbers
1. Addition (+):
- The sum of two real numbers is also a real number.
- Example: 3 + 4.5 = 7.5
2. Subtraction (−):
- The difference between two real numbers is also a real number.
- Example: 7.5 - 2.3 = 5.2
3. Multiplication × or \:
- The product of two real numbers is also a real number.
- Example: 3 × 4 = 12
4. Division ÷ or /:
- The quotient of two real numbers is a real number, provided the divisor is not zero.
- Example: 10 ÷ 2 = 5
- Division by zero is undefined in real numbers.
5. Exponentiation (^):
- Raising a real number to the power of another real number results in a real number, depending on the values involved.
- Example: 23 = 8
Properties of Operations on Real Numbers
- Commutative Property:
- Addition: a + b = b + a
- Multiplication: a × b = b × a
- Associative Property:
- Addition: (a + b) + c = a + (b + c)
- Multiplication: (a × b) × c = a × (b × c)
- Distributive Property:
- Multiplication over Addition: ( a × (b + c) = a × b + a × c
- Identity Elements:
- Addition: ( a + 0 = a )
- Multiplication: a × 1 = a
- Inverse Elements:
- Additive Inverse: ( a + (-a) = 0)
- Multiplicative Inverse: ( a × 1/a = 1 (for a ≠ 0 )
Real numbers and their operations are fundamental in mathematics, forming the basis for algebra, calculus, and many other areas of study.