Counting numbers are 1,2,3,4,5,... Integers are the numbers 0,1,-1,2,-2,3,-3,... "whole numbers" isn't well defined. It can mean counting numbers or it can mean integers. Natural numbers are positive integers: 1,2,3,4,5,... or sometimes non-negative integers: 0,1,2,3,4,5,... Rational numbers are any numbers that can be expressed as a ratio of any two integers: 1/2, 3/4, and note in particular: 5 = 5/1 = 10/2 is rational so integers are rational. Real numbers are what we think of as the number line. Any point on the number line is a real number. a point sqrt(2) away from the origin on the number line is a real number. Irrational numbers are those *real numbers* that can't be expressed as a ratio of two integers. and it goes on: Imaginary numbers are the square root of a negative number. Complex numbers are a sum of a real and an imaginary number. Algebraic numbers are real numbers which are the root of some polynomial: they are the ones you can compute using only plus,minus,times,divide,and raising to a power or fractional power. Transcendental numbers are real numbers which cannot be the root of a polynomial. e and pi are the most famous.