Number Theory: An Introduction to Mathematics

46 I The Expanding Universe of Numbers and the relationeize−iz=1 implies that cos^2 z+sin^2 z= 1. From the power series forezwe ...

5 Complex Numbers 47 Supposeeix ′ =1forsomex′∈Rand choosen∈Zso thatn≤x′/ 2 π<n+1. Ifx=x′− 2 nπ,theneix=1and0≤x< 2 π.Ifx=0 ...

48 I The Expanding Universe of Numbers It remains to show thatπhas its usual geometric significance. Since the continu- ously di ...

6 Quaternions and Octonions 49 points of a plane. The definition of quaternions adopted here will be analogous to our definition ...

50 I The Expanding Universe of Numbers where thenorm n(A)andtrace t(A)are both real: n(A)=aa ̄+bb ̄, t(A)=a+ ̄a. Moreover,n(A)&g ...

6 Quaternions and Octonions 51 A quaternionAis said to bepureifA ̄=−A. Thus any quaternion can be uniquely represented as the su ...

52 I The Expanding Universe of Numbers If we write x=ξ 1 i+ξ 2 j+ξ 3 k, then Tx=y=η 1 i+η 2 j+η 3 k, whereημ= ∑ 3 v= 1 βμvξvfor ...

6 Quaternions and Octonions 53 represented, then−Vis not. On the other hand, supposeuis a pure quaternion, so that α 0 =0. Thenu ...

54 I The Expanding Universe of Numbers (ij)ε=kε=( 0 ,k), i(jε)=i( 0 ,j)=( 0 ,−k). It is for this reason that we defined octonion ...

7 Groups 55 Consequently the norm is multiplicative:for allα,β∈O, n(αβ)=n(α)n(β). For, puttingγ=αβ,wehave n(γ)α ̄=(αγ) ̄ γ ̄=(α( ...

56 I The Expanding Universe of Numbers group. For supposea,b,care distinct elements ofA,letf:A→Abe the bijective map defined by ...

7 Groups 57 We now show that a nonemptyfinitesubsetHof a groupGis a subgroup ofGif it is closed under multiplication only. For, ...

58 I The Expanding Universe of Numbers The relation is certainly reflexive, sincee∈H. It is also symmetric, since ifc= ba−^1 ∈H, ...

7 Groups 59 Ifg:G′→G′′is a homomorphism of the groupG′into a groupG′′, then the composite mapg◦f:G→G′′is also a homomorphism. Th ...

60 I The Expanding Universe of Numbers subgroupgeneratedbyS. ClearlyS⊆〈S〉and〈S〉is contained in every subgroup of Gwhich contains ...

8 Rings and Fields 61 It may seem inconsistent to require that addition is commutative, but not multipli- cation. However, the c ...

62 I The Expanding Universe of Numbers manner just described. It was proved by Stone (1936) that every Boolean ring may be obtai ...

8 Rings and Fields 63 In a ring with no divisors of zero, the additive order of any nonzero elementais the same as the additive ...

64 I The Expanding Universe of Numbers A mapping f:R→R′of a ringRinto a ringR′is said to be a (ring)isomor- phismif it is both b ...

9 Vector Spaces and Associative Algebras 65 As another example, the setC(I)of all continuous functionsf :I→R,where Iis an interv ...