Saturday, April 18, 2009

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:

natural numbersyesonly larger number minus smaller or equal numberyesonly if it divides evenly; can't divide by zeroyes
integersyesyesyesonly if it divides evenly; can't divide by zeroonly positive and zero powers
rational numbersyesyesyescan't divide by zerointeger powers; some fractional powers
real numbersyesyesyescan't divide by zeronot allow some fractional powers of negative numbers e.q. (-1)(1/2)
complex numbersyesyesyescan't divide by zeroyes
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.

complex numbersyesyesyescan't divide by zeroyes
quaternionsyesyesnot commutativecan't divide by zeroyes
octonionsyesyesnot commutative, not associativecan't divide by zeroyes
sedenionsyesyesnot commutative, not associative, not alternativecan't divide by zeroyes

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