# Commutative Sentence Examples

Division.-From the

**commutative**law for multiplication, which shows that 3 X 4d.Multiplication may or may not be

**commutative**, and in the same way it may or may not be associative.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 included in the preceding, but it is simpler in that the various operations are

**commutative**.The method is similar in some respects to the treatment of non

**commutative**algebras.AdvertisementRecall that in Heisenberg's first paper, he defined a multiplication which was not necessarily

**commutative**.Recall concepts of a ring homomorphism, ideal, right (left) ideal, principal ideal in a

**commutative**ring.To transpose a term which is not the last term on either side we must first use the

**commutative**law, which involves an algebraical transformation.The

**commutative**law in arithmetic, for instance, states that adb and b+a, or ab and ba, are equal.The effect of these definitions is that the sum and the product of two quaternions are also quaternions; that addition is associative and

**commutative**; and that multiplication is associative and distributive, but not**commutative**.AdvertisementWith this notation the values of x and y may be expressed in the forms x q q /N q ', gg /Nq', which are free from ambiguity, since scalars are

**commutative**with quaternions.This idea finds fuller expression in the algebra of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of scalars, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general,

**commutative**.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.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

**commutative**law for multiplication is directly illustrated; and subdivisions or groupings of the units lead to such formulae as (a+ a) (b + 0)=.AdvertisementFor 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.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).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.AdvertisementThis 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.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.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.