## Uniplanar Algebra: Being Part I of a Prop©¡deutic to the Higher Mathematical Analysis |

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multiplication 70 61. Conjugate and reciprocal 71 62. Agenda: Properties of cis <f

> 72 63. The imaginary unit 72 64.

geometric ...

**Commutative**and associative**laws**for geometric addition 69 60. Geometricmultiplication 70 61. Conjugate and reciprocal 71 62. Agenda: Properties of cis <f

> 72 63. The imaginary unit 72 64.

**Commutative**and associative**laws**forgeometric ...

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... or, observing the law of signs (Art. 13), this is b / a= / aXb; and if b carry with it

its proper factor sign, the more explicit form of statement contained in this

equation is X b / a = / aX b. The formula for the

and ...

... or, observing the law of signs (Art. 13), this is b / a= / aXb; and if b carry with it

its proper factor sign, the more explicit form of statement contained in this

equation is X b / a = / aX b. The formula for the

**commutative law**in multiplicationand ...

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The formula for the

be written and this is easily extended to products of three or more factors. In this

formula, the same sign attaches to a or b on both sides of the equation. 19.

The formula for the

**commutative law**in multiplication and division may thereforebe written and this is easily extended to products of three or more factors. In this

formula, the same sign attaches to a or b on both sides of the equation. 19.

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The second factor may also consist of two or more terms ; for + X (c + d) = a X (c +

d) + b X (f + d), by the distributive law, = 0+ d) X a + (c + d) X b, by the

+ ...

The second factor may also consist of two or more terms ; for + X (c + d) = a X (c +

d) + b X (f + d), by the distributive law, = 0+ d) X a + (c + d) X b, by the

**commutative law**, and again, by the distributive and**commutative laws**, (c + d)X<i+ ...

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64. The Associative and

the amplitudes of a, /}, y respectively ; that is, a a • cis <jb, (i = 6 'tis y = ^,cisx.

Then, by the law of geometric multiplication, "X 03 X y) = (a-cistf) X (p-cis£| X O'

cisx]) ...

64. The Associative and

**Commutative Laws**. Let a, b, c be the tensors, <f>, ip, xthe amplitudes of a, /}, y respectively ; that is, a a • cis <jb, (i = 6 'tis y = ^,cisx.

Then, by the law of geometric multiplication, "X 03 X y) = (a-cistf) X (p-cis£| X O'

cisx]) ...

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