176 Euler’s theorem

π

–

2

The formula eiθ= cosθ+ isinθis today known as

Euler’s formula. It has some interesting consequences:

Setting θ= π, we obtain: eiπ= cosπ+ isinπ=

–1 + i· 0 = –1. Mathematicians often deem this

as one of the most beautiful facts of mathemat-

ics: it is a remarkably simple equation that

connects the mysterious, and pervasive, num-

bers e, π, i, and –1.

Setting θ= yields , which shows that

. Thus raising a complex

number to a complex power can yield a real

answer as a result.

Euler’s formula provides a very simple means for

deriving (and memorizing) certain identities from

TRIGONOMETRY

. For example, since

eiA · eiB = ei(A+ B)

we have:

(cosA+ isinA) · (cosB+ isinB) = cos(A+ B) + isin(A+ B)

Expanding the brackets on the left and collecting terms

that contain iand those that do not quickly yields:

cos(A+ B) = cosA· cosB– sinA· sinB

sin(A+ B) = sinA· cosB+ cosA· sinB

Similarly, the equations (eiA)2= ei(2A), (eiA)3= ei(3A), and

so forth yield double-angle and triple-angle formulae,

for example.

Euler’s formula is also used to represent complex

numbers. For example, if zis a point in the complex

plane a distance rfrom the origin, making an angle θ

with the x-axis, then its x- and y-coordinates can be

written:

x= rcosθ

y= rsinθ

and the complex number is thus:

z= x+ iy = rcosθ+ ir sinθ= reiθ

This is called the polar form of the complex number. If

one multiples two complex numbers, z= reiθand

w=seiτ, we see that z· w= rsei(θ+τ), that is:

The product of two complex numbers is a new

complex number whose distance from the ori-

gin is the product of the distances from the ori-

gin of the two original numbers, and whose

angle with the x-axis is the sum of the two

angles made by the two original numbers.

Euler’s formula makes the derivation of this fact swift

and easy.

See also

COMPLEX NUMBERS

; D

E

M

OIVRE

’

S FOR

-

MULA

;

EHYPERBOLIC FUNCTIONS

.

Euler’s theorem (Euler’s formula, Euler-Descartes for-

mula) A

GRAPH

is a collection of dots, called vertices,

connected in pairs by line segments, called edges, sub-

sequently dividing the plane into a finite number of

regions. In 1752 L

EONHARD

E

ULER

showed that if a

graph drawn on the plane has vvertices, eedges, and

divides the plane into a total of rregions (this includes

the large “outer region”), then:

v– e+ r= 1 + c

where cis the number of “connected components” of

the graph, that is, the number of distinct pieces of

which it is composed. For example, the graph shown

is composed of two “distinct pieces” (c= 2) and has

nine vertices, 13 edges, and divides the plane into

seven distinct regions, and indeed v– e+ requals 3,

one more than c.

The formula is easily proved via an

INDUCTION

argument on the number of edges: if a graph has no

edges, then it consists solely of vdisconnected points.

Thus it has c= vcomponents and divides the plane into

just one region. The formula v– e+ r= 1 + cholds

true. One checks that adding an edge either divides a

region into two (thereby increasing the value of rby

one), creates an extra region if that edge is a loop

(again increasing the value of rby 1), or connects two

disconnected components of the graph (thereby