Core Pure - Complex Numbers (2)

Question 1

What is the exponential form to writing a complex number?

Answer 1

Question 2

How do you convert from exponential form to a+ib form?

Answer 2

Substitute into the complex argument form of a complex number

Question 3

How do you multiply 2 complex numbers in exponential form?

Answer 3

X the moduli
+ the arguments

Question 4

How do you divide 2 complex numbers in exponential form?

Answer 4

Divide the moduli
- the arguments

Question 5

When asked to do questions about dividing complex numbers in exponential form what may be in a tricky question?

Answer 5

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:

Question 6

What is Demoivre’s theorem?

Answer 6

Question 7

Generally what is it really helpful to do when using complex numbers?

Answer 7

Put it in exponential form

Question 8

What are the 2 types or trig expansion do you need to know to do?

Answer 8

Question 9

What is the basis step for type 1 of trig expansion?

Answer 9

Question 10

What is the method for type 1 of trig expansion?

Answer 10

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

Question 11

What are the 2 trig identities that you need to know?
But what needs to be true?

Answer 11

ModZ = 1

Question 12

What can you infer from the trig identities?
Why is this useful?

Answer 12

This is useful for finding powers of cos or sin in term of its smaller angles where you will need to reverse the formula

Question 13

What is the basis step for type 2 questions on trig expansion?

Answer 13

Question 14

What is the method for type 2 questions on trig expansion?

Answer 14

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

Question 15

What is often a mistake when doing questions in type 2 trig expansion questions?

Answer 15

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)

Question 16

What may question on trig expansion look like?

Answer 16

Question 17

What are the 2 key things when finding the nth root of a number?

Answer 17

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)

Question 18

How do you work out the nth root of a positive real number?

Answer 18

Question 19

What year 1 concept that I always forget?

Answer 19

You need to use the principle argument

Question 20

What are the roots of unity?

Answer 20

The roots of 1 eg squared root, cubed root, forth root…

Question 21

What do you know about the 5th roots of unity?

Answer 21

The first answer will be 1 then they will be spaced out 2π/5 around the argand diagram

Question 22

What do you know is true about the

Question 23

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?

Answer 23

nth roots of real numbers will come in congregate pairs
nth roots of complex numbers won’t be in congregate pairs

Question 24

How do you take the nth root if a negative real number or a complex number?
(What is the change)

Answer 24

Put it into exponential form then the method is the same
Just don’t forget to take the nth root of r as well

Question 25

What will the argument of a negative real number always be?

Answer 25

π

Question 26

What do you know about the roots of unity?

Answer 26

They will form a regular polygon with n sides centred around the origin
The sum of the roots of unity = 0
So:

Question 27

What do you know is true about the nth roots of complex numbers?

Answer 27

It will form a regular polygon (with n sides) with its centre at the origin

Question 28

What is a good tip when expanding de Moivre’s theorem?

Answer 28

Let c=cosx and v=sinx to save time and simplify the expanding

Question 29

How can you do geometric problems but keep it in exact form?

Answer 29

Do the complex number multiplied by the e bit