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<b>[*]</b> Associated with the real number system are familiar rules that will be extended to apply to the new number system about to be developed.<br>
&nbsp;&nbsp;&nbsp;(1a) Numbers in any system can be added together. Their sum is in the same number system.<br>
&nbsp;&nbsp;&nbsp;(1b) The same  sum is obtained by adding the numbers in any order. (Commutative law for addition)<br>
&nbsp;&nbsp;&nbsp;(1c) There is an additive identity 0:&nbsp;&nbsp;&nbsp; x + 0 = 0.<br>
&nbsp;&nbsp;&nbsp;(1d) Each number has in its system an additive inverse (negative) such that the sum of the two numbers is the identity 0:&nbsp;&nbsp;&nbsp;x + (-x) = 0.<br>
&nbsp;&nbsp;&nbsp;(2a) Numbers in any system can be multiplied together. Their product is in the same number system.<br>
&nbsp;&nbsp;&nbsp;(2b) The same product is obtained by multiplying the numbers in any order. (Commutative law for multiplication)<br>
&nbsp;&nbsp;&nbsp;(2c) There is a multiplicative identity 1: &nbsp;&nbsp;&nbsp; 1x = x.<br>
&nbsp;&nbsp;&nbsp;(2d) Each number, except zero, in its system has a multiplicative inverse (reciprocal) such that the product of the two numbers is identity 1:&nbsp;&nbsp;&nbsp;x(1/x) = 1.&nbsp;&nbsp; (x is not 0).<br> 
&nbsp;&nbsp;&nbsp;(3) The distributive law relates multiplication and addition: x(a + b) = xa + xb.<br>
From (3), (1c) and (2b) the rule that zero times any number is zero can be proved:&nbsp;&nbsp;&nbsp;0x = 0.