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My teacher said that $\Bbb R \subset \Bbb R^2$, but I don't understand how this is possible since the elemnts of $\Bbb R$ are single numbers and the elements of $\Bbb R^2$ are pairs of numbers. Help please.

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  • $\begingroup$ Do you believe $\mathbb R\subset \mathbb C$? $\endgroup$ – John Douma Nov 10 '15 at 4:00
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    $\begingroup$ You can claim that $\mathbb R$ is a subspace of $\mathbb R^2$ if you assume a canonical interpretation for $\vec x \in \mathbb R$ where $\vec x = [x_1, 0]^T$ and $x_1$ is a real number. $\endgroup$ – Axoren Nov 10 '15 at 4:02
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    $\begingroup$ In the strict sense of set containment $\mathbb{R}$ is NOT a subset of $\mathbb{R}^2$. However an isomorphic copy of $\mathbb{R}$ sits inside $\mathbb{R}^2$. $\endgroup$ – Anurag A Nov 10 '15 at 4:04
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As with many cases, there is an identification to be made. In this case, the canonical identification is to see $x\in \mathbb R$ as $(x,0)\in \mathbb R^2$.

This happens all the time. For example, integers are not fractions, but you never doubted that $2$ is rational. That is because you identify each integer $n$ with the rational number $\frac{n}1$.

Or, as mentioned by John above, you see reals as complex numbers by identifying $x$ with $x+0i$.

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