Properties of Real Numbers

Study Notes

A real number is a value that can be represented on the number line, including all rational and irrational numbers.

Commutative property of addition and multiplication: The order of numbers being added or multiplied does not change the result.
Associative property of addition and multiplication: The order in which numbers are added or multiplied does not change the result when more than two numbers are involved.
Distributive property of multiplication over addition: The product of a number and the sum of two or more numbers is equal to the sum of the products of the number with each of the addends.

Rational numbers: Can be expressed as the quotient or fraction of two integers, e.g., 3/4 or 22/7.
Irrational numbers: Cannot be expressed as a finite decimal or fraction, e.g., π or e.
Integers: Whole numbers, either positive, negative, or zero, e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...
Non-negative real numbers: Real numbers greater than or equal to zero.
Positive real numbers: Real numbers greater than zero.

Addition: The result of adding two or more real numbers is always a real number.
Subtraction: The result of subtracting one real number from another is always a real number.
Multiplication: The result of multiplying two or more real numbers is always a real number.
Division: The result of dividing one real number by another is a real number, except for division by zero, which is undefined.

Decimal notation: Real numbers can be expressed in decimal notation, e.g., 3.14 or -0.5.
Scientific notation: Real numbers can be expressed in scientific notation, e.g., 1.23 × 10^4 or -4.56 × 10^-3.

A real number is a value that can be represented on the number line, including all rational and irrational numbers.

The order of numbers being added or multiplied does not change the result.
The order in which numbers are added or multiplied does not change the result when more than two numbers are involved.
The product of a number and the sum of two or more numbers is equal to the sum of the products of the number with each of the addends.

Rational numbers can be expressed as the quotient or fraction of two integers, e.g., 3/4 or 22/7.
Irrational numbers cannot be expressed as a finite decimal or fraction, e.g., π or e.
Integers are whole numbers, either positive, negative, or zero, e.g., ..., -3, -2, -1, 0, 1, 2, 3,....
Non-negative real numbers are real numbers greater than or equal to zero.
Positive real numbers are real numbers greater than zero.

The result of adding two or more real numbers is always a real number.
The result of subtracting one real number from another is always a real number.
The result of multiplying two or more real numbers is always a real number.
The result of dividing one real number by another is a real number, except for division by zero, which is undefined.

Real numbers can be expressed in decimal notation, e.g., 3.14 or -0.5.
Real numbers can be expressed in scientific notation, e.g., 1.23 × 10^4 or -4.56 × 10^-3.