Question

Explain, in complete sentences and in your own words, the answers to the following questions relating to complex numbers.
What are the characteristics of a complex number?
What is the relationship between a complex number and its conjugate?
Describe the usefulness of the conjugate and its effect on other complex numbers.

Answer 1

CAN SOMEONE PLEASE JUST DESCRIBE THIS ALL FOR ME PLEASE

Answer 2

1.A complex number is just a representation. It is a ordered pair of two numbers. The first one is called the real part, the second one the imaginary part.
Graphically, in a argand plane. A complex number has magnitude and amplitude.

Answer 3

2. If x+iy is a complex number, x-iy is the conjugate.

Answer 4

3. Conjugate is useful in several situations. Like, dividing two complex numbers, in geometrical approach(equation of lines,complex slopes etc,.).....
What do you mean by effect on other complex numbers?

Answer 5

uhmm im not really sure. like how conjugates effect complex numbers ?

Answer 6

hmm.. I have a idea which may be what you want..

So, the conjugate effects the complex number by reflecting it with respect to the axis of reals(X-axis).

Answer 7

What is a complex number?
Consider the real numbers. In short, they include any number you can find on the number line. What about numbers like \[\sqrt{-1} = i?\] They don't belong in the real field, and are considered imaginary numbers.
Complex number is a clash of the field of real and imaginary numbers. It's like Batman met Superman. So a complex number consists of a real part and an imaginary part.
We write this as \[z = a+ib\] where both a and b are real numbers and i is the sqrt of -1.
a is called the real part and b is called the imaginary part of the complex number.
The conjugate is similar to for surds.
If z =a+ib, then its conjugate is a - ib, often denoted with a dash above the top of z.
As with surds, the conjugate is used not to rationalise the denominator, but to realise it.
(i.e. make the denominator real)
If we decide we want to divide one complex number by another, it is useful since the [roduct of a complex number and its conjugate is real.
\[z \times z(conjugate) = (a+ib)(a-ib) = a^2 -(ib)^2 = a^2 -(-1)b^2 = a^2 + b^2 \]
Notice how these two complex numbers multiply to give a real number (has no i in it!).
How we divide one complex number by another is shown below:
\[\frac{ a +ib }{ c+id } = \frac{ a+ib }{ c+id } \times \frac{ c - id }{ c-id } = \frac{ (ac+bd)+(bc-ad)i }{ c^2 +d^2 }\]

Also if you add a complex and its conjugate, you get 2Re(z) i.e. two times the real part of z, another real number.
\[z + z(conjugate) = (a+ib) + (a-ib) = 2a = 2R e(z)\]
This can be helpful as well to evaluate powers of cosines (based on De Moivre's Theorem)