91Ó°ÊÓ

When adding two complex numbers, you simply add their corresponding real parts and their imaginary parts separately. A complex number is often noted as a pair of numbers, where the first number represents the real component and the second number is the imaginary component multiplied by the imaginary unit 'i'. For instance, if we have two complex numbers, represented as \( x = a + bi \) and \( y = c + di \) where 'a' and 'c' are the real parts, and 'bi' and 'di' are the imaginary parts. To add \( x \) and \( y \) we combine them like this: \(
a + bi + c + di. \)
This results in \( (a+c) + (b+d)i \), a new complex number with the real parts added and the imaginary parts added.
With the given example in the exercise, \( x = -\frac{1}{3} + i \sqrt{5} \) and \( y = -\frac{1}{2} - 2i \sqrt{5} \), adding these gives us
\(-\frac{1}{3} -\frac{1}{2}+(i \sqrt{5} -2 i \sqrt{5})\) which simplifies to \( -\frac{5}{6} - i\sqrt{5} \).
It's crucial to line up real and imaginary parts properly to ensure accurate addition.

Subtracting one complex number from another is a lot like addition. This time around you subtract the real parts from each other and the imaginary parts from each other. From our earlier defined complex numbers \( x = a + bi \) and \( y = c + di \), subtraction would look like \(
a + bi - (c + di). \)
After distributing the subtraction sign, it becomes: \( (a-c) + (b-d)i \).
Returning to our exercise, for \( x = -\frac{1}{3} + i \sqrt{5} \) and \( y = -\frac{1}{2} - 2i \sqrt{5} \), the subtraction \( x - y \) becomes \( -\frac{1}{3} - (-\frac{1}{2}) + (i \sqrt{5} - (-2 i \sqrt{5})) \)
which simplifies neatly to \( \frac{1}{6} + 3i\sqrt{5} \). Ensure to distribute the negative sign to both the real and the imaginary part of the second complex number during subtraction.

To multiply complex numbers, you'll use the distributive property, also known as the FOIL method for binomials: First, Outer, Inner, Last. The multiplication of \( x = a + bi \) and \( y = c + di \) unfolds as: \(
a(c + di) + bi(c+ di) \).
When you expand, you get \( ac + adi + bci + bdi^2 \). Remember that \( i^2 = -1 \), which changes the equation to \( (ac - bd) + (ad + bc)i \), combining the real parts and the imaginary parts separately.
For our given complex numbers \( x = -\frac{1}{3} + i \sqrt{5} \) and \( y = -\frac{1}{2} - 2i \sqrt{5} \), the multiplication yields:
\((-\frac{1}{3})(-\frac{1}{2}) - (\sqrt{5})(2\sqrt{5}) + (-\frac{1}{3})(-2\sqrt{5})i + (\sqrt{5})(-\frac{1}{2})i \).
After simplifying, we find that the product is purely imaginary and equals \( i \) in this case. Multiplying complex numbers can result in real numbers, imaginary numbers, or a mix of both.

Division of complex numbers is more involved, as it requires ridding the denominator of any imaginary components. To achieve this, we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \( y = c + di \) is \( c - di \) which flips the sign of the imaginary part.
The steps for dividing \( x \) by \( y \), where \( x = a + bi \) and \( y = c + di \) are:
1. Multiply the numerator by the conjugate of the denominator.
2. Distribute both the numerators and denominators.
3. Apply \( i^2 = -1 \) and simplify the expression.
4. Separate the resultant fraction into real and imaginary parts.
For our exercise with \( x = -\frac{1}{3} + i\sqrt{5} \) and \( y = -\frac{1}{2} - 2i\sqrt{5} \), the division \( x / y \) is performed as \( \frac{-1/3 + i\sqrt{5}}{-1/2 - 2 i\sqrt{5}} \)x\( \frac{-1/2 + 2 i\sqrt{5}}{-1/2 + 2 i\sqrt{5}} \) which simplifies to \( \frac{11}{6} - \frac{i}{6} \).
Dividing complex numbers reveals the interplay between arithmetic and algebra when working with complex numbers' real and imaginary parts.