When are addition, subtraction, multiplication, division and exponentiation allowed?

Now that we've had a look at several groups of numbers let's bring together what operations are allowed for each one:

	+	-	×	÷	ab
natural numbers	yes	only larger number minus smaller or equal number	yes	only if it divides evenly; can't divide by zero	yes
integers	yes	yes	yes	only if it divides evenly; can't divide by zero	only positive and zero powers
rational numbers	yes	yes	yes	can't divide by zero	integer powers; some fractional powers
real numbers	yes	yes	yes	can't divide by zero	not allow some fractional powers of negative numbers e.q. (-1)(1/2)
complex numbers	yes	yes	yes	can't divide by zero	yes

The complex numbers are the only group that allows addition, subtraction, multiplication and exponentiation without restriction.

These increasingly larger groups of numbers can be seen as attempts to make subtraction, division and exponentiation work without restrictions.

The natural numbers allow exponentiation without restriction, but restrict subtraction and division. We can introduce negative numbers to allow subtraction (giving us the integers), but this forces restrictions on exponentiation.

We can then introduce fractions (giving us the rational numbers) to allow almost all divisions. Then adding irrational numbers (giving us the reals) allows fractional powers of all positive numbers and some powers of negative numbers.

To finally get back to having no restrictions on exponentiation, we need to include imaginary numbers leaving us with the complex numbers.

Beyond the complex numbers are the quaternions, octonians and sedenions.

	+	-	×	÷	ab
complex numbers	yes	yes	yes	can't divide by zero	yes
quaternions	yes	yes	not commutative	can't divide by zero	yes
octonions	yes	yes	not commutative, not associative	can't divide by zero	yes
sedenions	yes	yes	not commutative, not associative, not alternative	can't divide by zero	yes

Quaternions

The quaternions are an extension of the complex numbers. Instead of having just one square root of minus one: i, why not have 3: i, j and k?

We will need a way of multiplying them together. It turns out that the following works:

ij = k = -ji
jk = i = -kj
ki = j = -ki
ijk = -1

The quaternions are not commutative, which is quite strange. The order in which you multiply them matters! If you have two quaternions, x and y, then:

x × y = - y × x

Why have 3 square roots of -1 and not 2 square roots? Because with 2 the division doesn't work properly. Why not 4, 5, 6 roots etc? Well it works with 7 roots of -1. They are called the octonians. And with 15 roots you get the sedenions. The octonians are not commutative and not associative either. The sedenions are not commutative, associative or alternative.

The quaternions are given the symbol: ℍ