16 1. From the Real Numbers to the Complex Numbers
identify the point (x,y) with the arrow from the origin (0, 0) to the point (x,y).
We know how to add vectors; hence we deﬁne
(12) (x,y) + (a,b) = (x + a,y + b).
This formula amounts to adding vectors in the usual geometric manner. See Figure
1.5. More subtle is our deﬁnition of multiplication
(13) (x,y) * (a,b) = (xa - yb,xb + ya).
Let us temporarily write 0 for (0, 0) and 1 for (1, 0). We claim that the opera-
tions in equations (12) and (13) turn R2 into a ﬁeld.
We must ﬁrst verify that both addition and multiplication are commutative
and associative. The veriﬁcations are rather trivial, especially for addition:
(x,y) + (a,b) = (x + a,y + b) = (a + x,b + y) = (a,b) + (x,y),
((x,y) + (a,b)) + (s,t) = (x + a,y + b) + (s,t) = (x + a + s,y + b + t)
= (x,y) + (a + s,b + t) = (x,y) + ((a,b) + (s,t)) .
Here are the computations for multiplication:
(x,y) * (a,b) = (xa - yb,xb + ya) = (ax - by,ay + bx) = (a,b) * (x,y),
((x,y) * (a,b)) * (s,t) = (xa - by,xb + ya) * (s,t)
= (xas - bys - txb - tya,xbt + yat - (xas - bys)) = (x,y) * ((a,b) * (s,t)) .
We next verify that 0 and 1 have the desired properties.
(x,y) + (0, 0) = (x,y),
(x,y) * (1, 0) = (x1 - y0,x0 + y1) = (x,y).
The additive inverse of (x,y) is easily checked to be (-x, -y). When (x,y) 6 =
(0, 0), the multiplicative inverse of (x,y) is easily checked to be
x2 + y2
x2 + y2
Checking the distributive law is not hard, but it is tedious and left to the reader in
These calculations provide the starting point for discussion.
Theorem 4.1. Formulas (12) and (13) make
into a ﬁeld.
The veriﬁcation of the ﬁeld axioms given above is rather dull and uninspired.
We do note, however, that (-1, 0) is the additive inverse of (1, 0) = 1 and that
(0, 1) * (0, 1) = (-1, 0). Hence there is a square root of -1 in this ﬁeld.
The ordered pair notation for elements is a bit awkward. We wish to give two
alternative deﬁnitions of C where things are more elegant.
What have we done so far? Our ﬁrst deﬁnition of C as pairs of real numbers
gave an unmotivated recipe for multiplication; it seems almost a fluke that we
obtain a ﬁeld using this deﬁnition. Furthermore computations seem clumsy. A
more appealing approach begins by introducing a formal symbol i and deﬁning C