Set of Numbers

A set of numbers is a collection of numbers that share specific properties and are grouped together under a common name. These sets form the basis for the structure and categorization of numbers in mathematics. The common sets of numbers are:

1. : The set of natural numbers. These include all positive integers from onward, e.g., , etc.

2. : The set of integers. This set includes all whole numbers, both positive and negative, as well as zero, e.g., , etc.

3. : The set of rational numbers. These are numbers that can be expressed as a fraction of two integers, e.g., , etc.

4. : The set of real numbers. This set includes all rational and irrational numbers, which are numbers that cannot be expressed as a simple fraction, e.g., , etc.

5. : The set of complex numbers. These include all numbers that can be expressed in the form , where and are real numbers and is the imaginary unit.

6. : The set of quaternion numbers. These are an extension of complex numbers that can be expressed as , where are real numbers and are the fundamental quaternion units which satisfy .

Here’s a representation of the hierarchical relationship between the sets:

Each set is a subset of the sets that follow it. For example, every integer is a rational number, and every rational number is a real number.

Set of Natural Numbers

The set of natural numbers, denoted as ℕ, includes all positive integers starting from 0. In some definitions, the set starts from 1 instead of 0.

Set of Integers

The set of integers, denoted as ℤ, includes all positive and negative whole numbers, along with zero. For example, the numbers -2, 0 and 3 are all integers, but numbers like 1/2 or the square root of 2 are not.

Set of Rational Numbers

The set of rational numbers, denoted as ℚ, can be defined by the quotient of two numbers belonging to the set of integers, where the divisor is non-zero.

Set of Real Numbers

The set of real numbers, denoted as ℝ, includes all rational and irrational numbers. It represents the entire continuum of possible number values from negative infinity to positive infinity.

Set of Complex Numbers

The set of complex numbers contains all possible complex numbers. Each complex number has a real part and an complex part.