A question on notation - R^1, R^2, ,R^n

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Hello everyone,

What does my professor mean when he says x belongs in R^{n}. What is R^{n}?

Does R^{1} mean 1-variable? or 2-dimension (just a line)? or both?

What about R^{2}? is this 2 variable? or 3-dimension? both?


Thank you for your help.

M
 
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Well R usually just denotes the real plane as opposed to C which is the complex plane. if you have R^{2} and R^{3} it could be different sets in set theory but I am not sure what you are working on right now. Depending on the subject of study it could be completely different than that.
 
michonamona said:
Hello everyone,

What does my professor mean when he says x belongs in R^{n}. What is R^{n}?

Does R^{1} mean 1-variable? or 2-dimension (just a line)? or both?

What about R^{2}? is this 2 variable? or 3-dimension? both?


Thank you for your help.

M
R or R1 is a one-dimensional space of real numbers - the real number line. A single coordinate suffices to locate a point on a line. You are apparently confused about the dimension of a line.

R2 or R X R is a two-dimensional space of pairs of real numbers - the real plane. A point or vector in R2 has two coordinates.

R3 or R X R X R is a three-dimensional space of triples of real numbers. A point or vector in R3 has three coordinates.

Rn is an n-dimensional space of n-tuples of real numbers. A point or vector in Rn has n coordinates. Although we have a hard time imagining spaces of more than three dimensions, most of the concepts we understand from one-, two-, or three-dimensional space extend naturally to a space of n dimensions.
 
Asphyxiated said:
Well R usually just denotes the real plane
No, R does not represent the real plane.
Asphyxiated said:
as opposed to C which is the complex plane. if you have R^{2} and R^{3} it could be different sets in set theory but I am not sure what you are working on right now. Depending on the subject of study it could be completely different than that.
 
real number line/set of real numbers. sorry
 
Excellent, thanks guys.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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