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

Rational Numbers

We grouped the negative numbers with the natural numbers to get the integers. Now we increase our set of numbers further to get the rational numbers.

The rational numbers include all of the fractions made by dividing one integer by another, except that you can't divide by zero. So positive and negative fractions, fractions smaller than one, fractions larger than one, and all the integers (you can have 1 on the bottom your fraction) make up the rational numbers.

You can remember that the RATIOnals are made up from RATIOs.

Rational numbers do not have to be written as a fraction, but can be in decimal form - they are still rational.

Some rational numbers:
1/2, 6, -4/5, 10/3, 54 3/4, 1.75, -0.33333

With the inclusion of fractions in the rational numbers we are better off for division that we were with the integers. We can now divide any two numbers and get another rational number, with only one exception. We cannot divide by zero.

Like the integers, we can add, subtract and multiply without restriction. But we still have to be careful about exponentiation.

In most cases we can only raise rationals to the power of an integer. 3² is OK, and, unlike the integers, we can now do 3^-2, but we cannot do 3^1/2 as that is outside of the rational numbers. We can only raise numbers to a fraction when the answer lies in the rational numbers. For example 25^(1/2) = 5 or (8/27)^(1/3) = 2/3.

The rational numbers are given the symbol: ℚ