Definition: When the square root of a negative number is taken, the result is an imaginary number. The imaginary number i is defined as the square root of -1:
Complex numbers are numbers that have a real part and an imaginary part and are written in the form
a + bi
where a is real and bi is imaginary.
Complex conjugates are complex numbers that have equal and opposite imaginary parts. For example 1 + 2i would have a complex conjugate of 1 – 2i.
a) Find i³.
b) Add: (2 + 3i) + (1 – 2i)
c) Multiply: (1 + 2i) x (3 – 4i)
d) Divide: (1 + i) / (2 – i)
a) To simplify imaginary powers, find pairs of (i * i), simplify them to -1, and multiply:
b) To add complex numbers, add the real parts first, then add the imaginary parts separately:
c) To multiply complex numbers, use the FOIL method:
d) To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator using the FOIL method. Simplify:
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