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Cardano (1545): use the quadratic formula formally with negative square roots, then the solutions formally work in the quadratic equation.

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Using the cubic formula for solving cubic equations, there are examples where plugging in the coefficients into the formula gives imaginary numbers at an intermediate step but in the end get a real result that can be verified.

Use Euler's formula for complex exponentials to derive trigonometric identities.

In subsequent centuries, mathematicians often found that using imaginary numbers enabled shortcuts in problems involving real variables.

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DesCartes (1600s): disparaged such expressions as "imaginary solutions"•

Brief History of Complex Variables:

Makes Fundamental Theorem of Algebra have an elegant form•Much of these useful formulas were worked out by Euler•

Wessel, Argand (1797, 1806): offered a solution (geometric formulation in the complex plane) for how to interpret imaginary numbers, but not noticed

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Gauss was developing his complex variable theory in his thesis in 1799○

Gauss (geometric), Hamilton (arithmetic) (1831, 1837): provided a solid mathematical foundation for complex variable theory, which became mainstream

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The more advanced applications of complex analysis followed in the later 19th century.

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We will develop complex variables axiomatically. One reason to do this is that some operations are a bit subtle, so we should approach them carefully.

For example, we can't carry over

Axiomatics for Complex VariablesMonday, August 26, 20132:07 PM

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This doesn't work for negative numbers:

We will develop complex variables from a standpoint that doesn't start with worrying about the interpretation of square root of negative numbers, but we will find that we arrive at the same place.

Let's view a complex variable as 2-dimensional vector in the complex plane.

We will call the complex number

Following convention, we can represent the complex number as

Add, multiply, divide (by nonzero elements), subtract•Associative, distributive properties•

The goal is to endow the complex numbers with a field structure.

The interesting part of this is how to define a field of two-dimensional vectors for which division is well-defined. The trick is to find a definition of "addition" and "multiplication" of complex numbers that enable a good

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of "addition" and "multiplication" of complex numbers that enable a good field structure to result (i.e., so that division by nonzero vectors can be well-defined).

For complex numbers:

Addition:

(this is clearly desirable if we want the complex field to also be vector space with respect to the real coordinates)

Multiplication:

(This is of course what you would get by formally multiplying and treating , but we could completely avoid reference to questions about square roots of negative numbers in making this definition).

These definitions will give us a field structure, as we shall shortly demonstrate. But let's first observe that this is not at all a trivial consequence. Other natural ways to multiply two-dimensional vectors together that would not give a field structure.

(can't divide by some nonzero entries like

Scalar product, vector (cross) product don't give two-dimensional results, so those won't work.

Commutative addition and multiplication•

How does one check that the above definitions do give us an algebraic field of complex numbers:

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Commutative addition and multiplication•Associativity for addition and multiplication•Distributive property•Additive identity:•Multiplicative identity: •Additive inverse (able to subtract): •

What would this require?

Multiplicative inverse (able to divide):•

Solve this as a linear system for (u,v):

This gives a well-defined multiplicative inverse provided

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