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Its magnitude is its length, and its direction is the direction the arrow points.[[1](https://en.wikipedia.org/wiki/Dot_product#Geometric_definition)]\\n\","," \"\""," ]"," },"," {"," \"cell_type\": \"markdown\","," \"metadata\": {},"," \"source\": ["," \"> A vector can be pictured as a transaction record. Its magnitude is its price, and its direction is the username receiving the value. \\n\","," \"\\n\","," \"||||\\n\","," \"|---|---|---|\\n\","," \"|John|Mary|5.00|\""," ]"," },"," {"," \"cell_type\": \"markdown\","," \"metadata\": {},"," \"source\": ["," \"replacing numerical coordinates with strings sets them adrift from the axis. however, string **recurrence** remains measurable:\\n\","," \"\\n\","," \"||||\\n\","," \"|---|---|---|\\n\","," \"|John|Mary|5.00|\\n\","," \"|Mary|Isaac|3.25|\\n\","," \"\\n\","," \"> 1. \\\"Mary\\\" account recurred 1*1=1 (once)\""," ]"," },"," {"," \"cell_type\": \"markdown\","," \"metadata\": {},"," \"source\": ["," \"now, add signs to strings by dividing their recurrence across 2 positive and negative dimensions extending a 3d space, and add \\\"buyer\\\" (-) and \\\"seller\\\"(+) labels to the new dimensions:\\n\","," \"\\n\","," \"|buyer|seller|price|\\n\","," \"|---|---|---|\\n\","," \"|John|Mary|5.00|\\n\","," \"|Mary|Isaac|3.25|\\n\","," \"\\n\","," \"> 1. \\\"Mary\\\" account occurred once as a \\\"seller\\\" (+) \\n\","," \"> 2. \\\"Mary\\\" account occurred once as a \\\"buyer\\\" (-) \""," ]"," },"," {"," \"cell_type\": \"markdown\","," \"metadata\": {},"," \"source\": ["," \"orienting strings between positive and negative directions and scaling per their signed recurrence restores distance. parameterizing the signed string's recurrence by time extends a conservation law to economic events:\\n\","," \"\\n\","," \"|buyer|seller|price|time|\\n\","," \"|---|---|---|---|\\n\","," \"|John|Mary|5.00|2016-11-23T20:18:35.253Z|\\n\","," \"|Mary|Isaac|3.25|2016-11-22T20:12:01.003Z|\\n\","," \"\\n\","," \"> 1. \\\"Mary\\\" account balance changed +1.75 after earning 5.00, then spending 3.25 \\n\","," \"> 2. \\\"Mary\\\" account not capable of spending 5.00 if previous earning less than 5.00 \""," ]"," },"," {"," \"cell_type\": \"markdown\","," \"metadata\": {},"," \"source\": ["," \"finally, expressing mathematical objects through data structures eliminates the learning curve to mathematical physics. embedding \\\"a change in ownership of 3d value with respect to time\\\" in the data empowers its consumer to articulate the behavior and influence of a currency in an *information-conserving*, *free* market without studying the recurring event structure between debitors and creditors across time, the 2^n dimensional space it creates in the data model, or the square root of the quadratic form sealing certainty between independent events:\\n\","," \"\\n\","," \"\\\\begin{equation}\\n\","," \"i\\\\equiv {\\\\begin{bmatrix}0&-Mary\\\\\\\\Mary&0\\\\end{bmatrix}},\\n\","," \"\\\\end{equation}\\n\","," \"\\n\","," \"|buyer|seller|price|time|item|\\n\","," \"|---|---|---|---|---|\\n\","," \"|John|Mary|5.00|2016-11-23T20:18:35.253Z|bread|\\n\","," \"|Mary|Isaac|3.25|2016-11-22T20:12:01.003Z|milk| \\n\","," \"\\n\","," \"> 1. recurring \\\"Mary\\\" account reflected across time axis with sign change \\n\","," \"> 2. \\\"bread\\\" moved in opposite direction of \\\"Mary\\\" account when balance changed -5.00, or *sold bread* \\n\","," \"> 3. \\\"milk\\\" moved in \\\"Mary\\\" account direction when balance changed -3.25, or *bought milk* \""," ]"," },"," {"," \"cell_type\": \"markdown\","," \"metadata\": {},"," \"source\": ["," \"## [continuity](https://en.wikipedia.org/wiki/Continuity_equation) as an abstract data type\""," ]"," },"," {"," \"cell_type\": \"markdown\","," \"metadata\": {},"," \"source\": ["," \"let t = time (e.g. seconds) \\n\","," \"let a(t) = value measured by seller @t \\n\","," \"let b(t) = value measured by buyer @t \\n\","," \"let w = value/transaction, where the difference in value measured between buyers and sellers in a transaction @t = a(t) - 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