Core Pure - Complex Numbers (2) Flashcards
What is the exponential form to writing a complex number?
How do you convert from exponential form to a+ib form?
Substitute into the complex argument form of a complex number
How do you multiply 2 complex numbers in exponential form?
X the moduli
+ the arguments
How do you divide 2 complex numbers in exponential form?
Divide the moduli
- the arguments
When asked to do questions about dividing complex numbers in exponential form what may be in a tricky question?
You need the arguments to be the same on the denominator and the nominator
So you can + or - 2Pi to get them the same
But:
What is Demoivre’s theorem?
Generally what is it really helpful to do when using complex numbers?
Put it in exponential form
What are the 2 types or trig expansion do you need to know to do?
What is the basis step for type 1 of trig expansion?
What is the method for type 1 of trig expansion?
Expand: (cosθ+isinθ)^n where in terms of A level n=3,4,5,6,7 using binomial expansion
Then you know that cos(nθ) = all real terms and sin(nθ) = all imaginary terms
What are the 2 trig identities that you need to know?
But what needs to be true?
ModZ = 1
What can you infer from the trig identities?
Why is this useful?
This is useful for finding powers of cos or sin in term of its smaller angles where you will need to reverse the formula
What is the basis step for type 2 questions on trig expansion?
What is the method for type 2 questions on trig expansion?
From the basis step find the cos^n(θ) by expanding the LHS using binomial expansion and then reversing the trig expansion formulas to get it in terms of its small angles
What is often a mistake when doing questions in type 2 trig expansion questions?
You forget that there is a RHS where you have (2cosθ)^n. So you need to divide your LHS by 2^n (or (2i)^n for sin)
What may question on trig expansion look like?
What are the 2 key things when finding the nth root of a number?
Convert it into exponential form (for positive real numbers you can skip this and just work it out the first answer on your calculator (because θ=0))
All roots will be (2π)/n spaced apart)
How do you work out the nth root of a positive real number?
What year 1 concept that I always forget?
You need to use the principle argument
What are the roots of unity?
The roots of 1 eg squared root, cubed root, forth root…
What do you know about the 5th roots of unity?
The first answer will be 1 then they will be spaced out 2π/5 around the argand diagram
What do you know is true about the
What do you know will be true when working out the nth roots of real numbers that is not true when working out the nth roots of complex numbers?
nth roots of real numbers will come in congregate pairs
nth roots of complex numbers won’t be in congregate pairs
How do you take the nth root if a negative real number or a complex number?
(What is the change)
Put it into exponential form then the method is the same
Just don’t forget to take the nth root of r as well
What will the argument of a negative real number always be?
π
What do you know about the roots of unity?
They will form a regular polygon with n sides centred around the origin
The sum of the roots of unity = 0
So:
What do you know is true about the nth roots of complex numbers?
It will form a regular polygon (with n sides) with its centre at the origin
What is a good tip when expanding de Moivre’s theorem?
Let c=cosx and v=sinx to save time and simplify the expanding
How can you do geometric problems but keep it in exact form?
Do the complex number multiplied by the e bit
