Numbers are the "things" we create to count and measure other "things". I know that sounds a little unhelpful, but that's just how simple the idea of a number is. 1 - or one - is a number we use to represent a single, individual "thing". That thing can be a person, place, idea, action, concept, relationship... It can be anything. And when we decide to compare these things to other things, we start to define the relationships between numbers. When we use more numbers to define these relationships, we're creating numerical relationships. The number two is larger than one. How much larger? Twice as large. The act of defining things in quantities is giving them numerical value. The description of this is with the "things" we create called numbers.
And in just the same way that there are different types of almost anything, there are different types of numbers. Each of these number groupings below have different rules and systems in place that say what's included and what's not included. Think of it like a group or a club. In order to be part of that club each number has to meet certain rules to be included in that club of numbers. Yet, a key rule in all of these systems is that each number is unique and has a very precise quantity that it represents. For example, "many" is not a number. How many would many be? Two? Twenty? Two thousand? See how when I say the number two you know exactly how many that is? That's a key property of numbers - they are unique in their numerical value.
Another particular note is that the definition of different types of numbers has been primarily shaped by our ability to communicate these "things" in written form. This art of writing numbers - the notation of numbers - plays a key role in our definition of different types of numbers. The difficulty or ease of writing a number has led to the creation of different number groups and notations to communicate these ideas and groupings below.
The defining of new clubs of numbers and the rules involved is called a "number system". The word system is used to indicate that rules for allowing and rejecting numbers is important. If you're interested in a deep dive on these rules including base numbers and place value, you can check out this other article on what is a number system. Other than this brief description here, we'll turn our attention to the different types of numbers you're likely to encounter.
Counting numbers are the club of numbers that any of us would normally use in our day to day life. We look around and we see one tree. Two houses. Three forks. Fifteen dogs. These numbers are also known as the natural numbers because we naturally count the things in our world.
Whole numbers are all the counting numbers and the number zero. It sounds weird to think, but the number zero had to be invented. Even though people knew that there could be no apples, the number zero and representing it with the symbol "0" is a rule that we created. That rule is what separates counting numbers from whole numbers. Whole numbers included zero into the club.
Integers are the group of numbers that includes counting numbers, whole numbers, and negative numbers. Negative numbers are like zero in that you can't see them in the world. You can't see negative two apples in the same way you can't see zero apples. But where negative numbers become really useful is when we talk about letting someone borrow something from you. I might have five toys, but if I let my brother play with three of them, I only have two toys in my room. And my brother, who has five of his own toys, now has eight toys because of the three I let him borrow. Even though he has eight toys in his room, he owes me three of them. To represent this idea of "owing" we use negative numbers. Who owes what to each other and how we keep track of this is why negative numbers are really useful.
Fractions are a notation created to represent numbers that are in between integers. For example, 1⁄2 is halfway between zero and one. The notational tool of " ⁄ " is used to represent a fraction and is called a fraction bar. The number above the fraction bar is called the numerator and the number below the fraction bar is called the denominator. The numerator relative to the denominator is the same as saying the relative part to the whole. 1⁄2 is like cutting an apple in two pieces and keeping one piece. 1⁄2 is like cutting that same apple into three and keeping one piece.
Decimals are another notation created to represent numbers in between integers. The notational rule used is to put a decimal point and then any digits written to the right of the decimal point indicates the decimal number. Decimals can be represented as fractions where every denominator is some multiple of ten. For example, .5 is equvalent to 5⁄10 and .79 is equivalent to 79⁄100.
Mixed numbers are the combination of integers and fractions. 31⁄3 is a mixed number because it's an integer mixed with a fraction. They are called mixed numbers because any integer can be written as a fraction whereby the numerator is the integer you want and the denominator is 1. For example, 3 written as a fraction is 3⁄1. The fraction 3⁄1 is not considered a mixed number because it's written completely as a fraction. This is also why an integer with a decimal is not considered a mixed number - because there's no way to represent an integer with a decimal.
Rational numbers are the club that includes counting numbers, whole numbers, integers, fractions, and decimals. Mathematicians have gone further to define rational numbers with their fractional and decimal numbers having three specific qualities. 1) Fractions cannot have a denominator of 0 as it's impossible to have a partial something of nothing. 2) Decimal numbers must have a terminating number of digits. 3) If the decimal doesn't terminate but instead has numbers that repeat.
Rational numbers are the first set of numbers we're encountering that seem to have some unexpected rules. The reason for this definition is primarily in contrast to numbers that don't fit this definition - irrational numbers. If rational numbers are still a little confusing it may be helpful to see them as the definition of numbers that are not irrational.
Irrational numbers are numbers that can't be written as a fraction or as decimals that terminate or repeat. These numbers usually get their own symbol to represent them. Pi (symbol) is probably the most famous irrational number in that it has an infinite number of digits that follow no discernible pattern. No fraction can exactly equal pi. Another famous irrational number is "e" - also known as euler's number after a famous mathematician - Leonhard Euler. But the most common encounter with irrational numbers involves square roots - or other root-based numbers. A square root is the number that, when multiplied by itself, equals the number you want. For example, two is the square root of four, because two times two is four. The notation used to express finding roots is the radical symbol - "√". What led to the definition of irrational numbers is trying to communicate the square root of two - √2... Which, cannot be represented with decimals or fractions. So this new set of numbers was created to represent these numbers.
The grouping of real numbers includes all of the numbers above - irrational numbers, rational numbers, integers, whole numbers, and counting numbers. Mathematicians, in their quest to create ever-simpler rules and definitions, refer to this set of numbers using the symbol "R".
Imaginary numbers are the world of real numbers multiplied by the imaginary unit "i". The origin of this number comes from an exploration of irrational numbers. Or, specifically, what would be the √-1? For hundreds of years mathematicians disregarded this question because two negative numbers, when multiplied together, always equal a positive number... But what if there was a number that, when multiplied by itself, is -1? This number is "i". The easiest way to visualize "i" is to imagine that all the real numbers lie on a horizontal number line while a perpendicular, vertical line that intersects the number 0 has "i" as the integer with coefficients of the real numbers. A bit of a mouthful, but the image below will help you visualize how to think about "i" in relation to all the numbers before. Imaginary numbers are incredibly useful when making calculations that involve two dimensions, especially in physics and engineering.
Complex numbers are the set of numbers that include real numbers and imaginary numbers. Visually, this is the plane created by the number line of real numbers and the number line of imaginary numbers. To represent any number on this plane, we use the notation (15 + 3i) or (1⁄4 + .7i) or (8π + ei). Notice how there are two parts to this number? That's what makes this a complex number. Any imaginary number or real number can be converted to a complex number by using zero as the coefficient. For example: (0 + 5i) or (15 + 0i).
Within the world of complex numbers is a subset called algebraic numbers. Algebraic numbers are a combination of the set of natural numbers, integers, rational, imaginary, and irrational numbers that are a root of a polynomial equation. A polynomial simply means "many numbers" - "poly" meaning many and "nomial" meaning numbers. An example is a complex number like (3 + 5i). The numbers that are not included are the irrationals such as π and e.
Transcendental numbers are the complement to algebraic numbers in that they are the subset of numbers that are not the root of any polynomial. The numbers π and e are transcendental (in addition to being irrational).
Hyperreal numbers are the set of numbers that include all the numbers above as well as infinitely large numbers (infinite) and infinitely small numbers (infinitesimal). This sounds a little strange but the mathematics here are really fascinating. The reciprocal of an infinite number is an infinitesimal. While we won't go any deeper than this explanation, suffice it to say that this realm of mathematics is not for the faint of heart.
Surreal numbers are all of the numbers above combined with an ordered structuring of the numbers in a recursive, regular pattern. This pattern illustrates the emergent property of negative infinity, positive infinity, roots, and exponents.
Hypercomplex numbers take the concept of dimensions introduced with imaginary numbers and complex numbers and extends to further dimensions. Quaternions are four dimensional numbers (reals, i, j, k). Further extension is octonions (8D), sedonions (16D), pathions (32D), chingons(64D), routons (128D), voudons (256D), and so on. One interesting property of extending number systems into higher dimensions is that they lose mathematical properties such as commutativity, and associativity.
This list is far from a conclusive definiton of the different numbers and number systems out there. Instead, this article is meant to expand your understanding of what a number is and how they relate to each other. My fascination with mathematics is regularly expanded with deeper explorations into the higher level mathematical concepts towards the end of this article. With these expanded concepts I believe we are gaining a deeper ability to understand and communicate the universe. Over and over throughout history these mathematical explorations yeild fascinating insights into the underlying mechanics of how the universe works.