A7 Key Properties - Vector and Complex Numbers:

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Algebraic Properties

Below Z, W and V stand for complex numbers, or the result of calculations that yield complex numbers. Proofs of the following properties were given above, or they are immediately consequences of the rectangular and polar coordinate representation of points in the plane, alias complex numbers

Commutative Law for Addition: Z + W = W + Z

Commutative Law for multiplication: Z W = W Z

Additive Identity Exists: The zero vector 0 = 0 + i 0 has the property 0 + Z = Z

Multiplicative Identity Exist: The real number 1 = 1 + i 0 has length 1 and angle 0. So it has the property that 1 Z = Z.

Reciprocals (Multiplicative Inverses) Exist for nonzero complex numbers: If Z = (r, -q) has length r> 0 and angle q then WZ = 1 if W = (1/r, -q) with length (1/r) and angle -q

Negatives (Additive Inverses) Exist for all complex numbers: If Z = a + ib = [a, b] then W = (-a) + i(-b) = [-a, -b] has the property that W+ Z = 0

NonZero Product Law: If Z and W have lengths r and s both greater than 0 then their product has length rs > 0 (By the methods of decimal arithmetic with, the product of two positive numbers or length is positive. Alternative this follows from assuming the same law for real numbers.)

From logic, the equivalent, contrapositive form of the nonzero product law is as follows:

Zero Product Law: If the product WZ = 0 for a pair of factors W and Z given by real or complex numbers, then at least one of the factors must be zero.

This observation, an implication rule, is employed to find complex or real solutions of equations. The solution of quadratic equations, one at a time or all at once by the quadratic formula, follows from this zero product law.

Distributive Laws

Below Z, W and V again stand for complex numbers. The left and right Distributive Laws says

Z ( W + V ) = Z W + ZV (left distributive law)
( W + V ) Z = WZ + V Z (right distributive law)

Because multiplication commutes (that is, AB = BA), the left and right forms of the distributive law are equivalent. Each implies the other. So a proof of one provides a proof of the other. Because they are equivalent, we the adjectives left and right may some be omitted, and we may talk about a distributive law instead of distributive laws, a harmless variation in language.

In this quick introduction to and development of the arithmetic properties of complex numbers in the plane, we will assume these distributive laws and from it obtain several consequences. Longer introduction or developments have to make alternative assumption which imply the distributive law instead of assuming it. So this assumption of the distributive law in its left and right forms provides a shortcut for a quick description complex numbers.

But each form of the distributive law can used in two ways. For example, the left distributive law

Z ( W + V ) = Z W + ZV

allows the computation of an expression Z ( W + V ) to be replaced by Z W + ZV. This process is called expansion or distribution of multiplication over addition. But the left distributive law

Z ( W + V ) = Z W + ZV

allows the computation of an expression Z W + ZV to be replaced by Z ( W + V ). This process is called factorization. When an expression is factored, the nonzero product rule can be used to say when that expression will be zero or not. From factorization comes solution method for equations and inequalities. Similar remarks hold for the right distributive law.

Multiplication with Components (optional)

Suppose Z and W are arrows in the plane with tails or initial points at the origin. The product ZW has been defined by means of the Polar coordinate multiplication rule: add their angles, multiply their lengths. Before showing or deriving how to multiply these factors together using the rectangular coordinates of their head locations, we will show how to compute the product using horizontal and vertical components. To this end

Let Z = A+B where A = horizontal component of Z and where B = the vertical component of Z.
Let W = C +D where C = horizontal component of W and where D = the vertical component of W

Now observe

ZW = (A+B)W = AW + BW by the distributive law.

AW = A(C+D) = AC+ AD by the distributive law.

BW = B(C+D) = BC + BD by the distributive law.

Now the polar multiplication rule implies a horizontal * horizontal and vertical* vertical gives a horizontal result, while horizontal*vertical and vertical*horizontal gives a vertical result. Therefore, the component addition method implies the horizontal component of the product ZW = AC + BD while the vertical component of ZW = BC + AD.

Therefore

ZW = (AC+BD) + (BC+AD).

With some duplication of work, the case where Z= z = a + ib = [a,b] and W= w = c + id = [c,d] is considered next. The components in this case are A = a, B = ib, C = c and D = id.