Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$?

Question

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$ ? If have then what is the mapping ? Please define the mapping. They have same cardinality then it is possible to have a bijection between them.

Answer 1

Hint: first, show that $\mathbb{R} \simeq (0,1)$. Then, observe that if $(0,1) \simeq (0,1)^2$, then $\mathbb{R} \simeq \mathbb{R}^2$. For the former, one can consider a map such as $0.a_1 a_2 a_3 a_4 \dots \mapsto (0.a_1 a_3 \dots,0.a_2a_4\dots) $.