# Between the real numbers and the complex numbers, which of them is bigger, and what is the reason why? [closed]

Topic: Numbers

Am a fresher so I have absolutely no idea what so ever

• Without you specifying what "bigger" means, the question is pretty much unanswerable (except by listing various choices for what "bigger" means so you can pick your favorite). Feb 1 at 2:40

## 2 Answers

The reals $$\mathbb{R}$$ and complex numbers $$\mathbb{C}$$ have the same cardinality i.e. the same "size" : see also Do the real numbers and the complex numbers have the same cardinality?

One thing way can initially observe is that $$\mathbb{R}\subset\mathbb{C}$$, the set of real numbers is a (proper) subset of the set of complex numbers. This tells us that $$\mathbb{R}$$ cannot be "bigger" than $$\mathbb{C}$$, and suggests that it is smaller. However, there is not a single, universal way of quantifying the "sizes" of sets in mathematics.

One way to describe the size of a set, as lorenzo pointed out, is the cardinality, in which case one would say that $$\mathbb{R}$$ and $$\mathbb{C}$$ are the same size $$(\lvert\mathbb{R}\rvert=\lvert\mathbb{C}\rvert=\beth_1)$$. Intuitively, this describes the number of elements in the set. However, when it comes to infinite sets, such as real and complex numbers, this may not always be the best description of size. For example, the intervals $$[0,1]$$ and $$[0,2]$$ have the same cardinality, yet intuitively, it seems like $$[0,2]$$ would be bigger.

This apparent discrepancy leads us to look for other descriptions of the sizes of sets, such as measures. One particular measure of note is the Hausdorff measure, as well as the associated Hausdorff dimension. Going back to the example above, we find that the $$1$$-dimensional Hausdorff measure of $$[0,2]$$ is $$2$$ times that of $$[0,1]$$, as we might intuitively expect. The Hausdorff measure is useful here because it easily generalises to arbitrary dimensions and is used to define the Hausdorff dimension. Using this, we find that the $$1$$-dimensional Hausdorff measures of both $$\mathbb{R}$$ and $$\mathbb{C}$$ are infinite, whereas the $$2$$-dimensional Hausdorff measures of $$\mathbb{R}$$ and $$\mathbb{C}$$ are $$0$$ and $$\infty$$, respectively. We can also find that $$\mathbb{R}$$ has Hausdorff dimension $$1$$, while $$\mathbb{C}$$ has Hausdorff dimension $$2$$. In other words, using this description of the size of a set, we find that $$\mathbb{C}$$ is bigger than $$\mathbb{R}$$.