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# Real coordinate spaces

Created by Sal Khan.

## Want to join the conversation?

• This may sound weird but can there be a negative dimension? Like R^-3?
• The smallest number of dimensions is zero. For example: 3 dimensions is `{x,y,z}`. 2 dimensions is `{x,y}`. 1 dimension is `{x}`. Zero dimensions is just `{}`. It is null, there is no information there, and that's as low as you can get.

You can think of dimensionality as like the distance from something. It is an absolute value. You can't be "negative 3 miles" from home, because you'd just be 3 miles from home. Similarly, you can't have "negative 3" dimensions.
• At , Sal talks about tuples. This makes me think of "quintuples," except the "tu" is pronounced differently. Is that where "tuples" comes from? Where does the term "tuples" come from?
• A tuple is a collection of objects but unlike sets tuples are always OEDERED collections of objects.
I do not know if the word originated from quintuple(does it mean ordered collection of 5 things ? )
• In the case of a complex numbers in a tuple, could you define a complex coordinate space? Like C^n?
• Yes, indeed. Such spaces are very useful in mathematical physics as well as interesting from a pure mathematics point of view.
• So let me get this straight,
if you had a vector with coordinates (5, I (<That is not a one, it is the letter i.)), it could not be defined by R^2 because I is an imaginary number whose square is -1.
• You are correct, any vector whose elements are complex numbers is outside of `ℜⁿ`, those vectors would belong to the set of complex numbers `ℂⁿ`; in your case, since your vector has 2 components, it would belong to `ℂ²`.
• what is the dimension of [0,0]..?
• R2, there are two components to your vector, which are both without imaginary components.
• how would the vector v=[0,0] be a vector if it has no direction it is not a line
• We declare <0,0> to be a vector because we want the sum of any two vectors to be a vector. <1, 1> and <-1, -1> are vectors, so we want their sum <0,0> to be a vector as well.
• can n be a decimal number?
• Not in this context. The dimension of a space is the number of vectors in its basis; we're counting something. You can't have a noninteger number of vectors, so you can't have a noninteger dimension.

There are other contexts, like when rescaling fractals, where fractional dimension makes sense (look up 'Hausdorff dimension'), but that doesn't apply here.
• If i am going to write a vector say in the 10 dimension, i just need to add numbers to the list, 10 numbers vertically?