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Let us assume that $\mathbb{Q}$ is defined in such a way that $(\mathbb{Q},+,\cdot)$ is a commutative ring without doubts.
Let us assume that $(a,b\in\mathbb{Q},a\neq b\Rightarrow a>b\vee a<b)$ is clear too. Let us define $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.

Definition 1 - Cauchy convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ of rational numbers is a Cauchy sequence if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q_m|\leq\varepsilon$ as soon as $n,m\geq N$. In layman's terms: a Cauchy sequence is a sequence whose terms "stick together" arbitrarily close from some point on.

Definition 2 - (metric) convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ is (metrically) convergent to $q$ if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q|\leq\varepsilon$ as soon as $n\geq N$. In layman's terms: a convergent sequence is a sequence whose terms "stick arbitrarily close to something" from some point on.

Since not every Cauchy sequence in $\mathbb{Q}$ is metrically convergent to a rational number (you may take $q_n=\frac{F_{n+1}}{F_n}$ as an example), it makes sense to enrich $\mathbb{Q}$ in order to make every Cauchy sequence (in the new space) convergent (to something belonging to the new space). Let us consider the set $S$ of Cauchy sequences in $\mathbb{Q}$, and let us equip this space with an equivalence relation $\sim$ considering $\{q_n\}_{n\geq 1}$ and $\{p_n\}_{n\geq 1}$ as the same object when $\{p_n-q_n\}_{n\geq 1}$ is (metrically) convergent to zero.

$$\mathbb{R}\stackrel{\text{Def}}{=} S/\sim.$$ Since $p_n\to p$ (shorthand notation for "$\{p_n\}_{n\geq 1}$ is metrically convergent to $p$") and $q_n\to q$ imply $p_n+q_n\to p+q$ and $p_n\cdot q_n\to pq$, the ring structure of $\mathbb{Q}$ is inherited by $\mathbb{R}$ through metric completion.

In other terms: continuous maps ($+,\cdot$) defined over dense subsets ($\mathbb{Q}\subset\mathbb{R}$) have unique continuous extensions.

I believe that students' issues with complex numbers, in the majority of cases, can be easily explained. At least in my place we are accustomed to introducing the imaginary unit $i$ as a/"the" square root of $-1$, without properly justifying why we are allowed to introducing such monstrosity. In fact, the construction of $\mathbb{C}$ from $\mathbb{R}$ shares many elements with the construction of $\mathbb{R}$ from $\mathbb{Q}$:

1. We take the original space, $\mathbb{R}$
2. We take a huge superset, $\mathbb{R}[x]$, the ring of polynomials with rational coefficients
3. We put an equivalence relation over such superset and define the new space as a quotient: $\mathbb{C}=\mathbb{R}[x]/(x^2+1)$
4. We check the new space is a field and a vector space of dimension $2$ over $\mathbb{R}$
5. We prove the new space is algebraically closed.

Or, at least, we should.

Let us assume that $\mathbb{Q}$ is defined in such a way that $(\mathbb{Q},+,\cdot)$ is a commutative ring without doubts.
Let us assume that $(a,b\in\mathbb{Q},a\neq b\Rightarrow a>b\vee a<b)$ is clear too. Let us define $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.

Definition 1 - Cauchy convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ of rational numbers is a Cauchy sequence if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q_m|\leq\varepsilon$ as soon as $n,m\geq N$. In layman's terms: a Cauchy sequence is a sequence whose terms "stick together" arbitrarily close from some point on.

Definition 2 - (metric) convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ is (metrically) convergent to $q$ if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q|\leq\varepsilon$ as soon as $n\geq N$. In layman's terms: a convergent sequence is a sequence whose terms "stick arbitrarily close to something" from some point on.

Since not every Cauchy sequence in $\mathbb{Q}$ is metrically convergent to a rational number (you may take $q_n=\frac{F_{n+1}}{F_n}$ as an example), it makes sense to enrich $\mathbb{Q}$ in order to make every Cauchy sequence (in the new space) convergent (to something belonging to the new space). Let us consider the set $S$ of Cauchy sequences in $\mathbb{Q}$, and let us equip this space with an equivalence relation $\sim$ considering $\{q_n\}_{n\geq 1}$ and $\{p_n\}_{n\geq 1}$ as the same object when $\{p_n-q_n\}_{n\geq 1}$ is (metrically) convergent to zero.

$$\mathbb{R}\stackrel{\text{Def}}{=} S/\sim.$$ Since $p_n\to p$ (shorthand notation for "$\{p_n\}_{n\geq 1}$ is metrically convergent to $p$") and $q_n\to q$ imply $p_n+q_n\to p+q$ and $p_n\cdot q_n\to pq$, the ring structure of $\mathbb{Q}$ is inherited by $\mathbb{R}$ through metric completion.

In other terms: continuous maps ($+,\cdot$) defined over dense subsets ($\mathbb{Q}\subset\mathbb{R}$) have unique continuous extensions.

I believe that students' issues with complex numbers, in the majority of cases, can be easily explained. At least in my place we are accustomed to introducing the imaginary unit $i$ as a/"the" square root of $-1$, without properly justifying why we are allowed to introduce such monstrosity among numbers. In fact, the construction of $\mathbb{C}$ from $\mathbb{R}$ shares many elements with the construction of $\mathbb{R}$ from $\mathbb{Q}$:

1. We take the original space, $\mathbb{R}$
2. We take a huge superset, $\mathbb{R}[x]$, the ring of polynomials with real coefficients
3. We put an equivalence relation over such superset and define the new space as a quotient: $\mathbb{C}=\mathbb{R}[x]/(x^2+1)$
4. We check the new space is a field and a vector space of dimension $2$ over $\mathbb{R}$
5. We prove the new space is algebraically closed.

Or, at least, we should.

Let us assume that $\mathbb{Q}$ is defined in such a way that $(\mathbb{Q},+,\cdot)$ is a commutative ring without doubts.
Let us assume that $(a,b\in\mathbb{Q},a\neq b\Rightarrow a>b\vee a<b)$ is clear too. Let us define $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.

Definition 1 - Cauchy convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ of rational numbers is a Cauchy sequence if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q_m|\leq\varepsilon$ as soon as $n,m\geq N$. In layman's terms: a Cauchy sequence is a sequence whose terms "stick together" arbitrarily close from some point on.

Definition 2 - (metric) convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ is (metrically) convergent to $q$ if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q|\leq\varepsilon$ as soon as $n\geq N$. In layman's terms: a convergent sequence is a sequence whose terms "stick arbitrarily close to something" from some point on.

Since not every Cauchy sequence in $\mathbb{Q}$ is metrically convergent to a rational number (you may take $q_n=\frac{F_{n+1}}{F_n}$ as an example), it makes sense to enrich $\mathbb{Q}$ in order to make every Cauchy sequence (in the new space) convergent (to something belonging to the new space). Let us consider the set $S$ of Cauchy sequences in $\mathbb{Q}$, and let us equip this space with an equivalence relation $\sim$ considering $\{q_n\}_{n\geq 1}$ and $\{p_n\}_{n\geq 1}$ as the same object when $\{p_n-q_n\}_{n\geq 1}$ is (metrically) convergent to zero.

$$\mathbb{R}\stackrel{\text{Def}}{=} S/\sim.$$ Since $p_n\to p$ (shorthand notation for "$\{p_n\}_{n\geq 1}$ is metrically convergent to $p$") and $q_n\to q$ imply $p_n+q_n\to p+q$ and $p_n\cdot q_n\to pq$, the ring structure of $\mathbb{Q}$ is inherited by $\mathbb{R}$ through metric completion.

In other terms: continuous maps ($+,\cdot$) defined over dense subsets ($\mathbb{Q}\subset\mathbb{R}$) have unique continuous extensions.

Let us assume that $\mathbb{Q}$ is defined in such a way that $(\mathbb{Q},+,\cdot)$ is a commutative ring without doubts.
Let us assume that $(a,b\in\mathbb{Q},a\neq b\Rightarrow a>b\vee a<b)$ is clear too. Let us define $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.

Definition 1 - Cauchy convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ of rational numbers is a Cauchy sequence if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q_m|\leq\varepsilon$ as soon as $n,m\geq N$. In layman's terms: a Cauchy sequence is a sequence whose terms "stick together" arbitrarily close from some point on.

Definition 2 - (metric) convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ is (metrically) convergent to $q$ if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q|\leq\varepsilon$ as soon as $n\geq N$. In layman's terms: a convergent sequence is a sequence whose terms "stick arbitrarily close to something" from some point on.

Since not every Cauchy sequence in $\mathbb{Q}$ is metrically convergent to a rational number (you may take $q_n=\frac{F_{n+1}}{F_n}$ as an example), it makes sense to enrich $\mathbb{Q}$ in order to make every Cauchy sequence (in the new space) convergent (to something belonging to the new space). Let us consider the set $S$ of Cauchy sequences in $\mathbb{Q}$, and let us equip this space with an equivalence relation $\sim$ considering $\{q_n\}_{n\geq 1}$ and $\{p_n\}_{n\geq 1}$ as the same object when $\{p_n-q_n\}_{n\geq 1}$ is (metrically) convergent to zero.

$$\mathbb{R}\stackrel{\text{Def}}{=} S/\sim.$$ Since $p_n\to p$ (shorthand notation for "$\{p_n\}_{n\geq 1}$ is metrically convergent to $p$") and $q_n\to q$ imply $p_n+q_n\to p+q$ and $p_n\cdot q_n\to pq$, the ring structure of $\mathbb{Q}$ is inherited by $\mathbb{R}$ through metric completion.

In other terms: continuous maps ($+,\cdot$) defined over dense subsets ($\mathbb{Q}\subset\mathbb{R}$) have unique continuous extensions.

I believe that students' issues with complex numbers, in the majority of cases, can be easily explained. At least in my place we are accustomed to introducing the imaginary unit $i$ as a/"the" square root of $-1$, without properly justifying why we are allowed to introducing such monstrosity. In fact, the construction of $\mathbb{C}$ from $\mathbb{R}$ shares many elements with the construction of $\mathbb{R}$ from $\mathbb{Q}$:

1. We take the original space, $\mathbb{R}$
2. We take a huge superset, $\mathbb{R}[x]$, the ring of polynomials with rational coefficients
3. We put an equivalence relation over such superset and define the new space as a quotient: $\mathbb{C}=\mathbb{R}[x]/(x^2+1)$
4. We check the new space is a field and a vector space of dimension $2$ over $\mathbb{R}$
5. We prove the new space is algebraically closed.

Or, at least, we should.

Let us assume that $\mathbb{Q}$ is defined in such a way that $(\mathbb{Q},+,\cdot)$ is a commutative ring without doubts.
Let us assume that $(a,b\in\mathbb{Q},a\neq b\Rightarrow a>b\vee a<b)$ is clear too. Let us define $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.

Definition 1 - Cauchy convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ of rational numbers is a Cauchy sequence if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q_m|\leq\varepsilon$ as soon as $n,m\geq N$. In layman's terms: a Cauchy sequence is a sequence whose terms "stick together" arbitrarily close from some point on.

Definition 2 - (metric) convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ is (metrically) convergent to $q$ if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q|\leq\varepsilon$ as soon as $n\geq N$. In layman's terms: a convergent sequence is a sequence whose terms "stick arbitrarily close to something" from some point on.

Since not every Cauchy sequence in $\mathbb{Q}$ is metrically convergent to a rational number (you may take $q_n=\frac{F_{n+1}}{F_n}$ as an example), it makes sense to enrich $\mathbb{Q}$ in order to make every Cauchy sequence (in the new space) convergent (to something belonging to the new space). Let us consider the set $S$ of Cauchy sequences in $\mathbb{Q}$, and let us equip this space with an equivalence relation $\sim$ considering $\{q_n\}_{n\geq 1}$ and $\{p_n\}_{n\geq 1}$ as the same object when $\{p_n-q_n\}_{n\geq 1}$ is (metrically) convergent to zero.

$$\mathbb{R}\stackrel{\text{Def}}{=} S/\sim.$$ Since $p_n\to p$ (shorthand notation for "$\{p_n\}_{n\geq 1}$ is metrically convergent to $p$") and $q_n\to q$ imply $p_n+q_n\to p+q$ and $p_n\cdot q_n\to pq$, the ring structure of $\mathbb{Q}$ is inherited by $\mathbb{R}$ through metric completion.

Let us assume that $\mathbb{Q}$ is defined in such a way that $(\mathbb{Q},+,\cdot)$ is a commutative ring without doubts.
Let us assume that $(a,b\in\mathbb{Q},a\neq b\Rightarrow a>b\vee a<b)$ is clear too. Let us define $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.

Definition 1 - Cauchy convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ of rational numbers is a Cauchy sequence if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q_m|\leq\varepsilon$ as soon as $n,m\geq N$. In layman's terms: a Cauchy sequence is a sequence whose terms "stick together" arbitrarily close from some point on.

Definition 2 - (metric) convergence. We say that a sequence $\{q_n\}_{n\geq 1}$ is (metrically) convergent to $q$ if for any $\varepsilon>0$ there is a natural number $N=N(\varepsilon)$ ensuring $|q_n-q|\leq\varepsilon$ as soon as $n\geq N$. In layman's terms: a convergent sequence is a sequence whose terms "stick arbitrarily close to something" from some point on.

Since not every Cauchy sequence in $\mathbb{Q}$ is metrically convergent to a rational number (you may take $q_n=\frac{F_{n+1}}{F_n}$ as an example), it makes sense to enrich $\mathbb{Q}$ in order to make every Cauchy sequence (in the new space) convergent (to something belonging to the new space). Let us consider the set $S$ of Cauchy sequences in $\mathbb{Q}$, and let us equip this space with an equivalence relation $\sim$ considering $\{q_n\}_{n\geq 1}$ and $\{p_n\}_{n\geq 1}$ as the same object when $\{p_n-q_n\}_{n\geq 1}$ is (metrically) convergent to zero.

$$\mathbb{R}\stackrel{\text{Def}}{=} S/\sim.$$ Since $p_n\to p$ (shorthand notation for "$\{p_n\}_{n\geq 1}$ is metrically convergent to $p$") and $q_n\to q$ imply $p_n+q_n\to p+q$ and $p_n\cdot q_n\to pq$, the ring structure of $\mathbb{Q}$ is inherited by $\mathbb{R}$ through metric completion.

In other terms: continuous maps ($+,\cdot$) defined over dense subsets ($\mathbb{Q}\subset\mathbb{R}$) have unique continuous extensions.