Complex Numbers - maths topic 1

Flashcards about Complex Numbers

If y = x² - 1, what are the roots (x-intercepts where y=0)?

x = ±√1 = ±1

If y = x² + 1, what are the roots?

x = ±√-1 = ±i

What does 'i' represent in complex numbers?

i = √-1

What is another name for the complex plane?

Argand Plane

For z = 3 + 2i, what are the real and imaginary components?

Re(z) = 3, Im(z) = 2

If y = (x-2)² + 3, what are the roots?

x = 2 ± √3i

If y = x² + 6x + 11, what are the roots?

x = -3 ± √2i

If z = a + bi, what is the conjugate of z?

z̄ = a - bi

If z₁ = 1 + 2i and z₂ = 3 + i, what is z₁ + z₂?

4 + 3i

If z₁ = 1 + 2i and z₂ = 3 + i, what is z₁ - z₂?

-2 + i

If Re(z₁) = -1, Im(z₂) = 3, z₁ + z₂ = (2 + 3i), determine Z1 + Z2

z₁ + z₂ = 2 + 3i

If z₁ - z₂ = -5 + 2i, determine the real part of Z1 & Imaginary part of Z2, when Z₁ = x + 2i & Z₂ = 3 + yi

Re(z₁) = -2, Im(z₂) = 0

If z₁ = 2 - 3i and z₂ = 4 + i, what is z₁ * z₂?

11 - 10i

If z₁ = -2 + 3i and z₂ = 2 + 2i, what is z₁ * z₂?

-10 + 2i

If z₁ = x + 2i, z₂ = 3 + i, and Re(z₁ * z₂) = 0, what is the value of x?

x = 2/3

If z1(2 +3i) = z2 => (-1 + xi)(2 + 3i) = (y + 3i), solve for x

(-2 - 3x) + (2x - 3)i = y + 3i

If Re(z) = -1, Im(z₂) = 3, and z₁(2 + 3i) = z₂, and z1 = -1 +3i, what is z₂?

z₂ = -11 + 3i

Evaluate (3+2i)/(4-i)

10/17 + 11/17 i

Provide the formula for z z̄

z z̄ = a² + b²

If Z₁ = x + 3i & Z₂ = 2 + i, Determine Re(z1/z2) & Im(z1/z2)

Determine Re(z1/z2) & Im(z1/z2)

If Z₁ = 2 - i & Z₂ = x + 2i, determine Z1/Z2

-4-x/x²+4 i

What is the polar form of a complex number z where r is the modulus and θ is the argument?

z = r cis(θ)

What is the polar form of z = 2 + 2√3i?

z = 4 cis(π/3)

Convert z = 3 - 3i to polar form.

z = 3 cis(7π/4)

Convert z = -4√3 - 4i to polar form.

z = 8 cis(7π/6)

Convert z = 2 cis (π/6) into Cartesian form.

≈ 1.732 + i

Convert z = 6 cis (5π/6) into Cartesian form.

-3√3 - 3i

If z₁ = 2 cis (π/3) and z₂ = 3 cis (π), what is z₁ * z₂?

6 cis (4π/3)

If Z₁ = 4 cis (π/4) & Z₂ = 3 cis (5π/4), Determine Z1 Z2

Z1 Z2 = 12 cis (9π/4)

If mod(z₁) = 3, Z1Z2​ = 12, and arg(z1z2) = π/4, find arg(z2).

5π/4

If z = r cis θ what is Z1/Z2?

4 cis (-2π/3)

What is 2 / 3 cis (-3π/4)?

Divide Z1 by Z2

If Z₁ = 4 cis (π/3) & Z₂ = 3 cis (π), Determine Z1/Z2

Re

If mod(z₁) = 6, arg(z2) = π/2, and |z1/z2| = 2, what is mod(z₂)?

3

What is Euler form?

rcis(θ)

2 cis(1/3), 4 cis(1/3)

8 cis