(a) The commutative law and the associative law are closely related, and it is best to establish each law for the case of two numbers before proceeding to the general case.
Here each member is a number, and the equation may, by the commutative law for multiplication, be written 2(x+I) - 4(x-2) This means that, whatever unit A we take, 2(x+ I) A Sand 5 4(x-2) A are equal.
May be regarded as resulting from the commutative law for addition and subtraction.
This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, - which merely means that-1 is commutative with all other quantities.
This is due to the fact that there are really two kinds of subtraction, respectively involving counting forwards (complementary addition) and counting backwards (ordinary subtraction); and it suggests that it may be wise not to use the one symbol - to represent the result of both operations until the commutative law for addition has been fully grasped.
(i.) The logical result of the commutative law, applied to a succession of additions and subtractions, is to produce a negative quantity-3s.
Multiplication may or may not be commutative, and in the same way it may or may not be associative.
For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' and jizy', the term would vanish of itself when the factorlines are coplanar provided ij = - ji, for it would then assume the form ij(yz' - zy').
Clifford makes use of a quasi-scalar w, commutative with quaternions, and such that if p, q, &c., are quaternions, when p-I-wq= p'+wq', then necessarily p= p', q = q'.
If Q and Q' are commutative, that is, if QQ' = Q'Q, then Q and Q' have the same centre and the same radius.
Division.-From the commutative law for multiplication, which shows that 3 X 4d.
(v) Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g.
In each case the grouping system involves rearrangement, which implies the commutative law, while the counting system requires the expression of a quantity in different denominations to be regarded as a notation in a varying scale (§§ 17, 3 2).
This, If We Regard 3.4 2 7 As 4274 2742 7, Is A Direct Consequence Of The Commutative Law For Addition (§ 58 (Iii)), Which Enables Us To Add Separately The Hundreds, The Tens And The Ones.
To Multiply 4273 By 200, We Use The Commutative Law, Which Gives 200.4273 = 2 X100 X4273 2X4273X100=8546X100=854600; And Similarly For 30.4273.
When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical.
By (A+a) (B+b) (C+c) ..., so that no two terms are the same; the " like " -ness which determines the placing of two terms in one group being the fact that they become equal (by the commutative law) when B, C,.
They are (a+b)-?-c=a+(b+c) (A) (aXb)Xc=aX(bXc) (A') a+b=b+a (c) aXb=bXa (c') a(b c) =ab-Fac (D) (a - b)+b=a (I) (a=b)Xb=a (I') These formulae express the associative and commutative laws of the operations + and X, the distributive law of X, and the definitions of the inverse symbols - and =, which are assumed to be unambiguous.
The types of linear associative algebras, not assumed to be commutative, have been enumerated (with some omissions) up to sextuple algebras inclusive by B.