 In all but the last example, the integers (other than the integer \(n\) giving the size of the matrices) may be replaced by any of the rationals, the reals, or the complex numbers. When replacing the integers with the reals the fourth example becomes simply the ring of reals because even if \(b\) is zero \(a\) can be any real. However when replacing with the rational numbers the ring includes the rationals, but is more than that because \(\sqrt{2}\) is irrational, yet it does not contain for example \(\sqrt{3}\).