Complex numbers and sines and cosines

Complex Numbers and Sines and Cosines

Complex numbers were first studied by mathematicians in the 18th century. This is a system which contains the real number system but is algebraically complete in the sense that all polynomial equations have solutions. The real numbers do not have that property as, for example, x²+1 = 0 has no solution in the reals. By introducing a new number i with the property i² = -1, and the set of all numbers of the form a+bi where a and b are real numbers we obtain the set of complex numbers . For each complex number z = a+bi, we say that a is the real part and b is the imaginary part of z. As with real number, we may perform arithmetic operations with complex numbers.

Sum: (a+bi) + (c+di) = (a+c) + (b+d)i
Difference: (a+bi) - (c+di) = (a+c) - (b+d)i
Product: (a+bi) * (c+di) = (ac - bd) + (ad + bc)i
Quotient: , provided c² + d² ≠ 0

These operations behave similarly to those for the reals. (The complex numbers is a field.)

The complex conjugate of a complex number z = a + bi, denoted by, is defined by= a - bi.

The absolute value or modulus of z= a + bi, denoted by |z|, is defined to be |z| = .

What does this have to do with sine and cosine functions? Exponents are defined for complex numbers as well as for the reals and the mathematician Euler proved that e^iθ = cos(θ) + i sin(θ). As we may represent the complex number z = x + yi by the point (x,y) on the Cartesion plane,

we may then represent the complex number z = x + yi by z = |z|( cos(θ)+ isin(θ) ) = |z|e^iθ, called the phasor form, where |z| = √(x²+y²) and θ = Arctan( y/x ).

Graphical rule
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