Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and the Yahoo Answers website is now in read-only mode. **There will be no changes to other Yahoo properties or services, or your Yahoo account.** You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.

## Trending News

# Is the set of complex number a subset of any other type of numbers?

For example, we know that the natural numbers are a subset of the real numbers

And the real numbers are a subset of the complex numbers.

So, are the complex numbers a subset of any other type of numbers?

### 1 Answer

- 7 years agoFavorite Answer
Yes. Complex numbers can be considered a subset of the quaternions. Quaternions were invented by Hamilton in the 1800s and have the following properties:

There are 3 basis numbers i,j,k for which i^2 = j^2 = k^2 = -1

ij = -ji = k

jk = -kj = i

ki = - ik = j

Notice that for quaternion bases, multiplication is anti-commutative and that it parallels the vector cross-product.

A general quaternion is written as a + bi + cj + dk where a,b,c,d are real numbers.

If we set c=d=0, then we get a + bi which is of course a regular complex number.

Also quaternions themselves are a subset of octonions (invented by Cayley).

This is a big topic I can't even begin to cover on Yahoo Answers. But you can go to wikipedia if you're interested.