A complex number is a number that has two parts - a real part and an imaginary part. It is written in the form a + bi, where a is the real part and bi is the imaginary part with the letter i representing the square root of -1.
To add or subtract complex numbers, you simply add or subtract the real parts separately and then add or subtract the imaginary parts separately. For example, (3 + 4i) + (2 + 5i) = (3 + 2) + (4 + 5)i = 5 + 9i.
The conjugate of a complex number a + bi is a - bi. For example, the conjugate of 5 + 2i is 5 - 2i. This means that the conjugate of a complex number has the same real part but the opposite sign for the imaginary part.
To multiply complex numbers, you use the FOIL method (First, Outer, Inner, Last). For example, (2 + 3i)(4 + 5i) = 2(4) + 2(5i) + 3i(4) + 3i(5i) = 8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i.
The complex conjugate theorem states that when you multiply a complex number by its conjugate, the result is a real number. In other words, (a + bi)(a - bi) = a^2 - b^2i^2 = a^2 + b^2, where i^2 = -1.