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→Unique square root of -1 necessary for complex analysis: fix formatting
##It is commutative
##It is commutative
##It is assoiciatve
##It is assoiciatve
##There exists an element of <math>\mathbb{C}</math>, namely <math>0 := (0, 0)</math>, such that for all <math>a \in \mathbb{C}</math>,
##There exists an element of <math>\mathbb{C}</math>, namely <math>0 := (0, 0)</math>, such that for all <math>a \in \mathbb{C}</math>, <math>a + 0 = a = 0 + a</math>
<math>a + 0 = a = 0 + a</math>
##For every element <math>a \in \mathbb{C}</math> there exists an additive inverse <math>-a</math> such that <math>a + (-a) = 0 = (-a) + a</math>
##For every element <math>a \in \mathbb{C}</math> there exists an additive inverse <math>-a</math> such that
<math>a + (-a) = 0 = (-a) + a</math>
##Therefore, <math>\mathbb{C}</math> is an [[abelian group]] under addition.
##Therefore, <math>\mathbb{C}</math> is an [[abelian group]] under addition.
#We verify the following about <math>\cdot</math>:
#We verify the following about <math>\cdot</math>:
##It is associative
##It is associative
##It has the distributive property over addition
##It has the distributive property over addition
##There exists an element of <math>\mathbb{C}</math>, namely <math>1 := (1, )</math>, such that for all <math>a \in \mathbb{C}</math>,
##There exists an element of <math>\mathbb{C}</math>, namely <math>1 := (1, )</math>, such that for all <math>a \in \mathbb{C}</math>, <math>a \cdot 1 = a = 1 \cdot a</math>
<math>a \cdot 1 = a = 1 \cdot a</math>
##For every nonzero element <math>a \in \mathbb{C}</math> there exists a multiplicative inverse (reciprocal) <math>a^{-1}</math> such that <math>a \cdot a^{-1} = 1 = a^{-1} \cdot a</math>
##For every nonzero element <math>a \in \mathbb{C}</math> there exists a multiplicative inverse (reciprocal) <math>a^{-1}</math> such
that <math>a \cdot a^{-1} = 1 = a^{-1} \cdot a</math>
##Therefore, <math>\mathbb{C}</math> is a [[field]].
##Therefore, <math>\mathbb{C}</math> is a [[field]].
#We consider the subset <math>S = \{(a, 0)|a \in \mathbb{R}\} \subseteq \mathbb{C}</math> We note that there is a natural [[isomorphism]]
#We consider the subset <math>S = \{(a, 0)|a \in \mathbb{R}\} \subseteq \mathbb{C}</math> We note that there is a natural [[isomorphism]] from <math>S</math> to <math>\mathbb{R}</math>. Thus, we can associate elements of <math>\mathbb{C}</math> with zero second coordinate with real numbers.
from <math>S</math> to <math>\mathbb{R}</math>. Thus, we can associate elements of <math>\mathbb{C}</math> with zero second coordinate with real numbers.
#As notation, we write the ordered pair <math>(a, b)</math> as <math>a + bi</math>. This is just notation. We note that it does not constitute an [[abuse of notation]] since <math>(a, b) = (a, 0) + (0, b)</math>, so the complex number notated by <math>a + bi</math> is really the sum of the complex numbers <math>a + 0i</math> and <math>0 + bi</math>.
#As notation, we write the ordered pair <math>(a, b)</math> as <math>a + bi</math>. This is just notation. We note that it does not constitute an [[abuse of notation]] since <math>(a, b) = (a, 0) + (0, b)</math>, so the complex number notated by <math>a + bi</math> is really the sum
#We can also use natural shorthand for the <math>a + bi</math> notation. In particular, we can write <math>i</math> to represent the element <math>(0, 1) \in \mathbb{C}</math>. Note that <math>i^2 = (0, 1)\cdot(0, 1) = (-1, 0) = -1</math>.
of the complex numbers <math>a + 0i</math> and <math>0 + bi</math>.
#We can also use natural shorthand for the <math>a + bi</math> notation. In particular, we can write <math>i</math> to represent the element
<math>(0, 1) \in \mathbb{C}</math>. Note that <math>i^2 = (0, 1)\cdot(0, 1) = (-1, 0) = -1</math>.
This is how complex numbers are typically derived. (One could of course, define them to be the cosets in the quotient field <math>\mathbb{R}[x]/(x^2 + 1)</math>; this is clearly equivalent to the derivation above.) Please let me know where you see a fault in this derivation.
This is how complex numbers are typically derived. (One could of course, define them to be the cosets in the quotient field <math>\mathbb{R}[x]/(x^2 + 1)</math>; this is clearly equivalent to the derivation above.) Please let me know where you see a fault in this derivation.