Chapter 1: Complex Numbers.

Question 1

What is the form of a complex number?

Answer 1

z = a + bi
where a,b ∈ ℝ.

Question 2

What is the name for z*?

Answer 2

The Complex Conjugate.

Question 3

If z = a + bi, what does z* equal?

Answer 3

z* = a - bi.

Question 4

What is the Cartesian form of a complex number?

Answer 4

z = a + bi.

Question 5

What is the Modulus Argument Form of a complex number?

Answer 5

z = r(cosθ + isinθ).

Question 6

What is the Polar Form of a complex number?

Answer 6

(r, θ).

Question 7

What does ‘ℕ’ mean? Give examples.

Answer 7

• ℕ = Natural Numbers.
• These include: 0,1,2,3… or 1,2,3…

Question 8

What does ‘ℤ’ mean? Give examples.

Answer 8

• ℤ = Integers.
• These include: …-3, -2, -1, 0, 1, 2, 3…

Question 9

What does ‘ℚ’ mean? Give examples.

Answer 9

• ℚ = Rational Numbers.
• These include: ‘a/b’ where ‘a’ and ‘b’ are integers and b ≠ 0.

Question 10

What does ‘ℝ’ mean? Give examples.

Answer 10

• ℝ = Real Numbers.
• These include: All natural, integers and rational numbers. (-1, 0, 1, 3/2).

Question 11

What does ‘ℂ’ mean? Give examples.

Answer 11

• ℂ = Complex Numbers.
• These include: a + bi where ‘a’ and ‘b’ are real numbers and ‘i’ is the imaginary number √-1.

Question 12

What letter is an imaginary number used by? What is this equal to?

Answer 12

i = √-1
or
i² = -1.

Question 13

What does i³ equal?

Answer 13

i³ = √-1 x √-1 x √-1 = -i.

Question 14

What does i⁴ equal?

Answer 14

i⁴ = √-1 x √-1 x √-1 x √-1 = 1.

Question 15

What does i⁷ equal?

Answer 15

i⁷ = (√-1)⁷ = -i.

Question 16

What is the periodic cycle involving ‘i’?

Answer 16

i, -1, -i, 1…
(Repeats every 4).

Question 17

How do you add 2 complex numbers?

Answer 17

Add the Real Parts and add the imaginary parts.

Question 18

How do you multiply 2 complex numbers?

Answer 18

Expand out the brackets and simplify, rememebering that i² = -1.

Question 19

If z = 3+2i and w = 1-4i, what does w-z equal?

Answer 19

(1-4i) - (3+2i) = -2-6i.

Question 20

If z = 3+2i and w = 1-4i, what is zw?

Answer 20

(3+2i)(1-4i) = 3-12i+2i-8i² = 3-10i+8 = 11-10i.

Question 21

What is the complex conjugate of 3+5i?

Answer 21

3-5i.

Question 22

What is the complex conjugate of -7-8i?

Answer 22

-7+8i.

Question 23

What formula can you use to Solve a Quadratic Equation with complex roots?

Answer 23

Quadratic Formula (-b±√b²-4ac / 2a).-

Question 24

What alternative way can be used to determine the quadratic equation if given both complex roots?

Answer 24

x² - (sum of roots)x + (product of roots) = 0.

Question 25

For Argand Diagrams, which axis is the imaginary axis and which is the real axis?

Answer 25

• Imaginary = y-axis (vertical).
• Real = x-axis (horizontal).

Question 26

What is the formula for the modulus of z?

Answer 26

r = |z| = √x²+y².

Question 27

What is the formula for the argument of z?

Answer 27

• arg(z) = θ.
• tan(θ) = y/x.

Question 28

What is the formula for the modulus-argument form?

Answer 28

z = r(cosθ + isinθ), where
r =|z|and θ = arg(z).

Question 29

What is the formula for Loci in the complex plane?

Answer 29

|z - (z1)|= r, where r is the radius and z1 is the circle centre.

e.g. |z-2+3i| = 2 –> |z-(2-3i)| = 2, where the centre is (2,-3) and the radius is 2.

Question 30

What is the formula for the locus of points of a half line?

Answer 30

arg(z-(z1)) = θ, where z1 is the point it begins at and angle of θ to the real positive axis.