Editing 2028: Complex Numbers
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| title = Complex Numbers | | title = Complex Numbers | ||
| image = complex_numbers.png | | image = complex_numbers.png | ||
β | | titletext = I'm trying to prove that mathematics forms a meta-abelian group, which would finally confirm my suspicions that | + | | titletext = I'm trying to prove that mathematics forms a meta-abelian group, which would finally confirm my suspicions that algebraic geometry and geometric algebra are the same thing. |
}} | }} | ||
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: <math>(a,\ b) \cdot (c,\ d) = (ac - bd,\ ad + bc)</math> | : <math>(a,\ b) \cdot (c,\ d) = (ac - bd,\ ad + bc)</math> | ||
β | As such, they can be modeled as two-dimensional {{w|Euclidean vector|vectors}}, with | + | As such, they can be modeled as two-dimensional {{w|Euclidean vector|vectors}}, with an interesting rule for multiplication. The justification for this rule is to consider a complex number as an expression of the form <math>a+bi</math>, where <math>i^2 = -1</math>, i.e. ''i'' is the square root of negative 1. Applying the common rules of algebra and the definition of ''i'' yields rules for addition and multiplication above. |
Regular two-dimensional vectors are pairs of values, with the same rule for addition, and no rule for multiplication. | Regular two-dimensional vectors are pairs of values, with the same rule for addition, and no rule for multiplication. |