All Types of Numbers in Mathematics: From Natural to Surreal and Beyond
Numbers are astonishing in and of themselves. Just think about it — all a person needs is a pen, a sheet of paper, and numbers themselves to spend an entire lifetime exploring this vast, infinite world full of mind-bending connections and patterns.
The Everyday Types of Numbers in Mathematics
Every day we encounter natural, rational, and sometimes even negative numbers. But even among the seemingly familiar natural numbers, there are many subtypes with their own nature, beauty, and endless unsolved problems: prime, semiprime, composite, twin primes, amicable numbers, happy numbers, narcissistic numbers, palindromic numbers, Catalan numbers, Mersenne numbers, Fibonacci numbers, Sophie Germain numbers, Carmichael numbers, Harshad numbers, Lychrel numbers, Bell numbers, Stirling numbers, Proth numbers, Wilson numbers, Pell numbers, Leonardo numbers, Lucky numbers, Goodstein numbers, Sylvester numbers, Ulam numbers, and figurate numbers such as square, cubic, triangular, pentagonal, and hexagonal.
And yet — that is only the very tip of an enormous iceberg.
Beyond the Familiar: Irrational, Complex, and Other Advanced Number Types
There are also irrational, algebraic, transcendental, complex, hypercomplex, and p-adic numbers, each opening up a deeper layer of mathematical reality. The further you go, the stranger and more beautiful the landscape becomes.
Numbers can also be represented in a variety of forms and numeral systems: the familiar decimal, binary, ternary, sexagesimal, and any other base you can imagine. They can be written as continued fractions, in Knuth’s up-arrow notation, or using Conway’s chained arrow notation.
The Fascinating Properties of Numbers at the Edges of Mathematics
The zoo of number types does not stop there. Mathematics also gives us computable numbers, definable numbers, constructible numbers, figurate numbers, ordinal numbers, cardinal numbers, infinitesimals, transfinite numbers, hyperreal numbers, surreal numbers, odd numbers, trigonometric numbers, and periods — and that is still far from a complete list.
Over the past twelve months, four episodes dedicated to different types of numbers have been published on this channel:
