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

Complex Numbers

The complex numbers are the combination of the real numbers with the imaginary numbers. They are written as the real part plus the imaginary part. For example:

2 + 3i

The imaginary number i is defined as the square root of minus one, so i² = -1. Multiplication of complex numbers follows the same rules as the real numbers, you just have to keep track of each part of the multiplication:

(2 + 3i) × ( 4 + 5i)
= 2 × 4 + 2 × 5i + 3i × 4 + 3i × 5i
= 8 + 10i + 12i - 15
= -7 + 22i

We can add, subtract, multiply and divide any two complex numbers and get a complex number^* back (except dividing by zero). And, unlike the real numbers, we can raise any complex number to any other complex number without restriction.

For example:

3² = 9
3^-2 = 1/9 = 0.11111111...
3^(1/2) = √3 = 1.73205...
(-3)^(1/2) = √(-3) = 1.73205i
3²ⁱ = e^2ln(3)i = cos(2ln(3)) + i sin(2ln(3)) = -0.5863 + 0.8101i
3²⁺²ⁱ = 3² × 3²ⁱ = 5.2763 + 7.2911i

The complex numbers are given the symbol: ℂ

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* the complex numbers include all real numbers. So 2 is the complex number 2 + 0i. Even though some complex multiplications give answers that are part of the real numbers, those answers are still complex numbers. For example (2 +3i) × (2 - 3i) = 13 (a real number) = 13 + 0i (a complex number).

Imaginary Numbers

The imaginary numbers are needed to fill in the last gap in exponentiation. The real numbers do not allow some negative numbers to be raised to fractional powers. For example they do not allow (-1)^(1/2) = √(-1).

To get around this problem we can just make up an answer. We'll call this answer i. It can't be a real number, so it is outside the real numbers.

Using i allows us to find roots for all negative numbers. For example

√(-2) = √(2 × -1) = √2 × √(-1)= (√2)i = 1.414i

Both negative and positive numbers have two square roots. Just as both 4² and (-4)² are 16, both (4i)² and (-4i)² are -16.

When we look at cube roots, thing get more complicated. Now every number has three cube roots, not just one, like it did with the real numbers. So not only does 3³ equal 27, but so does (-3/2 + (3√3/2)i)³ and (-3/2 + (3√3/2)i)³

For more on this see the complex numbers.