A number is a fundamental concept in mathematics, representing quantities or values. Typically, numbers are represented using the decimal number system and are organized into different sets representing different types of numbers:
- Natural Numbers: Also known as counting numbers, these include all positive integers starting from 1 (1, 2, 3, 4, 5, …).
- Whole Numbers: These include all natural numbers along with zero (0, 1, 2, 3, 4, 5, …).
- Integers: These include all whole numbers and their negatives (… -3, -2, -1, 0, 1, 2, 3, …).
- Rational Numbers: These are any numbers that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
- Real Numbers: These include all numbers that can be depicted on a number line, including rational numbers and irrational numbers (numbers that can’t be expressed as simple fractions, such as the square root of 2 or pi).
- Complex Numbers: These include all numbers that can be expressed as a + bi, where a and b are real numbers and i is the square root of -1. The real part is a, and b is the imaginary part.
Each type of number has different uses depending on the context. Over time, as our understanding of numbers evolved, different types of numbers have been defined and organized into these types.
The set of natural numbers, denoted as ℕ, includes all positive integers starting from 0. In some definitions, it also starts at 1 instead of zero.
The set of rational numbers can be defined by the quotient of two numbers belonging to the set of integers, where the divisor is non-zero.
The set of real numbers includes all rational and irrational numbers. It represents the entire continuum of possible number values from negative infinity to positive infinity.
The set of complex numbers contains all possible complex numbers. Each complex number has a real part and an complex part.