Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers.

Q1. Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$ dense in rational points, i.e., does $S$ include a dense set of rational points?

This is certainly true for $d=2$, rational points on the unit circle.

Q2. If (as I suspect) the answer to Q1 is Yes, is there a sense in which the rational coordinates are becoming arithmetically more complicated with larger $d$, say in terms of their height?

If $x= a/b$ is a rational number in lowest terms (i.e., gcd$(a,b)=1$), then the height of $x$ is $\max \lbrace |a|,|b| \rbrace$.

This is far from my expertise. No doubt this is known, in which case a pointer would suffice. Thanks!

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(Added, *22Mar13*). I just found this reference. (Misleading remark removed.)

Klee, Victor, and Stan Wagon. Old and new unsolved problems in plane geometry and number theory. No. 11. Mathematical Association of America, 1996. p.135.

 ![KleeWagonFig10.8][2]