Complex numbers

Question 1

What is i?

Answer 1

The square root of negative 1

Question 2

What is an imaginary number ?

Answer 2

Any multiple of i

Question 3

What is a complex number ?

Answer 3

An imaginary number and a real number

Question 4

How is the equality different for complex numbers?

Answer 4

Question 5

What is a conjugate ?

Answer 5

A change in the sign in the middle of two terms

Question 6

What do you know about the roots of a quadratic equation that are non- real complex numbers?

Answer 6

The roots are conjugate.
If f(z)=0 (a+bi)
Then f(z*)=0 (a-bi)

formula
z^2 - (2a)+ a^2b^2

Question 7

Answer 7

Question 8

what always must be done to complex numbers before they can be manipulated

when in fraction form

Answer 8

1. check the denominator
2. if i is present in the denominator then rationalise by multiplying by the conjugate
3. once rationalised (no conjugate in the denominator) then manipulate normally

Question 9

how would you determine a quadratic equation given one complex root?

considering co efficents are real numbers

Answer 9

1. know that the other root must be the conjuagte
2. if root in the form (a+bi)
3. -2a is the b term
4. a squared + b square is the c term

Question 10

Answer 10

Question 11

What is the Argand diagram?

Question 12

Plot the points

Answer 12

Question 13

How would two conjugate pairs be represented geometrically? (On the Angrand diagram)

Answer 13

Reflection on the x axis

Question 14

why are complex numbers used?

Answer 14

to represent roots of quadratics that we previously couldn’t represent with real numbers

Question 15

what are the possible root outcomes for a cubic?

Answer 15

• 3 real roots
• 1 real root, 2 conjuagtes

must have at least 1 real root

Question 16

what are the possible roots for a quartic?

Answer 16

• 4 real roots
• 4 complex roots
• 2 real roots, 2 complex roots

Question 17

how would you solve a cubic/ quartic equation given one complex root?

Answer 17

1. turn complex and it’s conjugate into a quadratic
2. look at a term
3. look at last term
4. using intuition solve for a second quadratic
5. solve this

Question 18

Answer 18

Question 19

Answer 19

Make sure to highlight the a , b c … terms to make sure you don’t make a sign error

Question 20

More determining second bracket by intuition questions

Answer 20

Question 21

when does a complex number ever =0

Answer 21

when both real and imaginary parts =0

Question 22

what is the x axis and the y axis known as in the argand diagram?

Answer 22

• x axis= real axis
• y axis= imaginary axis

Question 23

what are some mark scheme must dos?

Answer 23

• show full expansion/working out with complex numbers
• show i^2 = -1
• when plotting points on argand diagram label Re and Im
• must show method of obtaining roots from quadratics (quadratic formula)