C7 Nb Imaginaires

Question 1

opération dans C

z + z’

Answer 1

(a + a’) + i(b + b’)

Question 2

opération dans C

zz’

Answer 2

(aa’ - bb’) + i(ab’ + a’b)

Question 3

opération dans C

z = z’

Answer 3

[ a = a’ ⋀ b = b’]

Question 4

conjugué

⏤(z+z’)

Answer 4

⏤z + ⏤z’

Question 5

conjugué

⏤(z×z’)

Answer 5

⏤z × ⏤z’

Question 6

conjugué

⏤(1/z)

Answer 6

1/⏤z

Question 7

conjugué

⏤(z/z’)

Answer 7

⏤z / ⏤z’

Question 8

conjugué

z + ⏤z

Answer 8

2 Re (z)

Question 9

conjugué

z - ⏤z

Answer 9

2 i Im (z)

Question 10

conjugué

z ∈ ℝ

Answer 10

Im (z) = 0

Question 11

z ∈ iℝ

Answer 11

Re (z) = 0

Question 12

z ∈ ℝ

Answer 12

⏤z = z

Question 13

z ∈ iℝ

Answer 13

⏤z = − z

Question 14

module

⎥⏤z⎪= ⎥z⎪ = ⎥−z⎪

Question 15

module

z×⏤z

Answer 15

⎥z⎪²

Question 16

module

⎥z×z’⎪

Answer 16

⎥z⎪×⎥z’⎪

Question 17

module

⎥1/z⎪

Answer 17

1/⎥z⎪

Question 18

module

⎥z/z’⎪

Answer 18

⎥z⎪/⎥z’⎪

Question 19

module

⎥z⎪ = 0

Answer 19

z = 0

Question 20

module

⎥z⎪

avec Im et Re

Answer 20

⎥z⎪≤ ⎥Re(z)⎪+ ⎥Im(z)⎪

Question 21

module

⎥Re(z)⎪
⎥Im(z)⎪

Answer 21

≤⎥z⎪

Question 22

module

⎥zⁿ⎪

Answer 22

⎥z⎪ⁿ

récurrence

Question 23

module

Inégalité triangulaire

Answer 23

⎥z + z’⎪≤ ⎥z⎪+⎥z’⎪

Question 24

cercle et disque

⎥z - a⎪= r

Answer 24

le cercle de centre A et de rayon r

Question 25

cercle et disque

⎥z - a⎪≤ r

Answer 25

le disque fermé de centre A et de rayon r

Question 26

cercle et disque

⎥z - a⎪< r

Answer 26

le disque ouvert de centre A et de rayon r

Question 27

Ensemble 𝕌

forme trigo

Answer 27

z = cos θ + i sin θ

Question 28

Ensemble 𝕌

Formule d’Euler

Answer 28

cos θ = (e^iθ + e^-iθ) / 2
sin θ = (e^iθ - e^-iθ) / 2i

Question 29

Ensemble 𝕌

e^iθ × e^iθ’

Answer 29

e^i(θ + θ’)

Question 30

Ensemble 𝕌

e^iθ / e^iθ’

Answer 30

e^i(θ - θ’)

Question 31

Ensemble 𝕌

Question 32

arguments

z = z’

Answer 32

[(⎪z⎥=⎪z’⎥) ⋀ (arg z = arg z’ [2π])]

Question 33

arguments

arg (zz’)

Answer 33

arg z + arg z’ [2π]

Question 34

arg (1/z)

Answer 34

− arg(z) [2π]

Question 35

arguments

arg (z’/z)

Answer 35

arg (z’) - arg(z) [2π]

Question 36

argument

arg(⏤z)

Answer 36

− arg (z) [2π]

Question 37

argument

z ∈ℝ*

Answer 37

arg(z) = 0 [π]!!

Question 38

argument

z ∈ i ℝ*

Answer 38

arg (z) = π/2 [π]

Question 39

argument

arg (zⁿ)

Answer 39

n arg z