# Commutative sentence example

commutative
• Division.-From the commutative law for multiplication, which shows that 3 X 4d.
• 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.
• With 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.
• 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.
• Recall 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.
• 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.
• 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'.