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

Integers

"Dad, can you buy me those butterfly wings?"
"They are seven dollars. How about you use some of your own money?"
"But I don't have my money with me."
"If I buy the wings, you can owe me the money and give it to me when we get back home."
Shortly afterwards my daughter has the butterfly wings and -$7 in her pocket and is thus introduced to negative numbers, and spending on credit.

The first extension we can make to the natural numbers is to go backwards as well as forwards to get the integers. The natural numbers start and zero and count 1, 2, 3 and so on. The integers allow us to count backwards as well. So -1, -2, -3... are integers, but are not natural numbers.

Now we can subtract any two integers and get another integer. We are not restricted like we were for the natural numbers. We can take a larger number from a smaller one and get an answer: a negative number.

And like with the natural numbers we can add and multiply any integer and get another integer. We are still restricted with division, though. 12 divided by 3 is fine, but 13 divided by 3 is a problem for the integers, just as it was for the natural numbers.

Strangely, exponentiation (raising a number to the power of something), which was OK for all natural numbers, is a problem for the integers. If we raise an integer to the power of a negative number we do not get an integer back. What is 3^-1? or 2^-2? Not an integer, that's for sure. So we can only use zero and positive powers in the land of the integers.

The Integers are given the symbol: ℤ