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(Q1+Q2−Q3)2=4Q1Q2(1−s3)open paren cap Q sub 1 plus cap Q sub 2 minus cap Q sub 3 close paren squared equals 4 cap Q sub 1 cap Q sub 2 open paren 1 minus s sub 3 close paren Why This Matters : You never need to use a calculator for 2the square root of 2 end-root . All results are exact fractions.
with purely algebraic concepts. By avoiding irrational numbers and infinite series, it allows for exact calculations over any field, not just the real numbers. 1. Replace distance with quadrance
: Replaces the Triangle Inequality. For three points to be collinear, their quadrances must satisfy: Divine Proportions: Rational Trigonometry to Un...
), a dimensionless ratio that measures the "separation" between two lines. Unlike angles, which are circular, spread is a rational function. For a right triangle with quadrances Q1cap Q sub 1 Q2cap Q sub 2 , and hypotenuse Q3cap Q sub 3
(Q1+Q2+Q3)2=2(Q12+Q22+Q32)open paren cap Q sub 1 plus cap Q sub 2 plus cap Q sub 3 close paren squared equals 2 open paren cap Q sub 1 squared plus cap Q sub 2 squared plus cap Q sub 3 squared close paren : The rational equivalent of the Sine Law: (Q1+Q2−Q3)2=4Q1Q2(1−s3)open paren cap Q sub 1 plus cap
: Computers are much faster at adding and multiplying than calculating trigonometric series.
s1Q1=s2Q2=s3Q3the fraction with numerator s sub 1 and denominator cap Q sub 1 end-fraction equals the fraction with numerator s sub 2 and denominator cap Q sub 2 end-fraction equals the fraction with numerator s sub 3 and denominator cap Q sub 3 end-fraction By avoiding irrational numbers and infinite series, it
: These laws work in any coordinate system, including those used in Einstein's Special Relativity (Minkowski space). ✅ Answer