To head off the next obvious question; yes, there are even larger number-systems, like the “octonians“, and inventing ever higher systems is easy enough.
Physicist: Particles and sets of particles are frequently seen to be in multiple states simultaneously.
Unlike an ordinary bit, a qbit is in both states at the same time.
If you set this up several times you can get several qbits, and you can talk about larger numbers.
Finally, here’s the answer, there are a lot of (infinite) number-systems bigger than the complex numbers that contain the complex numbers in the same way that complex numbers contain the real numbers. The smallest number system that’s bigger than the complex numbers is the “quaternions”.
The real numbers can be built from “1” and then seeing what you can get from any combination of adds, multiplies, etc. Complex numbers can be built the same way, starting with “1” and “i”. i, j, and k all do basically the same thing that i does in complex numbers; .
Closed-ness is comforting to have, because it means that when you’re doing basic math, no matter how you jump you’ll always have somewhere to land. To “solve” this problem Euler decided to make up a new “number” called ““, with the property that , and complex numbers were born. may patch the problem with , but does it just give rise to a new problem when you try to figure out what is? You can check this by squaring it: Weirdly enough, there is absolutely no combination of roots/exponentiations or multiplications/divisions or additions/subtractions that can break out of complex numbers.
Where the closed-ness of real numbers fail, complex numbers hold strong.
But, a quantum computer is more than just a lot of normal computers running in parallel.If the Oracle’s answer to the number x is f(x), then as a result (never mind how), you can construct a new state: .There are other weird summations for each of the other states () but for this problem the zero state is the only important one.Physicist: Yes, but they don’t fix problems the way the complex numbers do.The nice thing about real numbers (which includes basically every number you might think of: 0, 1, π, -5/2, …) is that no matter how you add, subtract, multiply, or divide (other than 0) them together, you always get another real number. A mathematician would say “the real numbers are under addition, because any real numbers added together always give you another real number”. When you’re doing square roots the real numbers are not closed. For example, to find , you just answer the question, , and find that the answers are ? But if you try the same thing with you’ll be trying to answer the question , which doesn’t have any answers (try it).