Complex Numbers Flashcards
How would you get the eqn of the perpendicular bisector to the complex numbers:z = 2 and z = 4i
Rewrite as |z-2| = |z - 4i| - replace z with x + iy and solve.
How do you get the 4th root of unity?
z^(4) = 1 plot (1.0i) on argand diagram: angle = 0 and |r| = 1.Write in complex number form:z^4 = cosΘ + isinΘRoots are: Z = cos 1/4 (Θ + 2kΠ ) +isin1/4 (Θ + 2kΠ ) where k = 0,1,2,3
What happens on an Argand diagram when you multiply a complex number by i?
Rotation of the point by 90 degrees
What do you do to the modulus and arguments when you multiply polar numbers together?
Multiply the modulus and add the arguments.
What do you do to the modulus and arguments when you divide polar numbers?
Divide the modulus and subtract the arguments.
If one of the roots of a function is Z=a+ib describe how you would find the other roots.
Other roots is Z=a-ib.1. Multiply both complex roots together to form quadratic fn.2. Use algebraic long division – divide function by quadratic function.3. Factorise remainder.
How do you prove is a complex number is a root to a function?
Sub number into function and show that it equals zero
How would you do Z1÷ Z2
Z1 x cong(Z2)Z2 cong(Z2)
How do you solve: √5 + 12i
√5 + 12i = a + ib5 + 12i = (a + ib)2Square out and using substitution methods find a and b
How would you get cos5Θ in terms of cos?
- Expand using Binomial2. Expand using De Moivre3. Equate real (or imag parts depending on whats asked for!)
How do you get the roots of z^5 = 2 + 3i
- Plot point on argand diagram2. Get |r| and Θ3. Re-write in polar form.5. Use de-Moivre thereom for Z = r[cos 1/5 (Θ + 2kΠ ) +isin1/5 (Θ + 2kΠ )] where k = 0,1,2,3,5
Given Z = 2 + i how do you get z^4 in complex number form?
- Plot point on argand diagram2. Get |r| and Θ3. Re-write in polar form.4. Use de-Moivre thereom for Z = r[cos 4Θ +isin4Θ] 5. Calculate solns for rcos 4Θ and r isin4Θ]
