Complex Numbers

Play an AI-generated podcast conversation about this lesson

What is the ordered pair representation of the imaginary unit i?

(1, 1)
(0, 1) (correct)
(1, 0)
(0, 0)

Which of the following statements about complex numbers is true?

A complex number can be written as a + bi, where a and b are real numbers. (correct)
A complex number is always an imaginary number.
A complex number is always purely imaginary.
A complex number is always a real number.

What is the condition for a complex number to be imaginary?

Its imaginary part is zero.
Its imaginary part is non-zero. (correct)
Its real part is non-zero.
Its real part is zero.

What is the relationship between the set of real numbers (R) and the set of complex numbers (C)?

R is a subset of C. (D) Signup and view all the answers

How can a complex number be represented?

As an ordered pair of real numbers. (A) Signup and view all the answers

What is the value of |z| if iz² - z = 0?

0 or 1 (B) Signup and view all the answers

If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, where x₁, x₂, y₁, y₂ ∈ R, and 0 < x₁ < x₂, y₁ = y₂, then which of the following is true?

|z₁| < |z₃| < |z₂| (A) Signup and view all the answers

If z = a + ib, what is the argument of z?

tan⁻¹(b/a) (B) Signup and view all the answers

What is the value of i²?

-1 (D) Signup and view all the answers

What is the representation of a complex number z = a + ib in the complex plane?

A point P in the complex plane, where the angle made by the line OP with the real axis is the argument of z (C) Signup and view all the answers

What is the value of i^2 in complex numbers?

-1 (A) Signup and view all the answers

Which of the following is a correct representation of a complex number?

x + iy (D) Signup and view all the answers

What is the value of -a when a is a positive real number?

ia (A) Signup and view all the answers

What is the property of the imaginary unit i?

i is neither 0, nor greater than 0, nor less than 0 (B) Signup and view all the answers

What is the set of complex numbers denoted by?

C (A) Signup and view all the answers

What is the property of addition of complex numbers that satisfies z1 + z2 = z2 + z1?

Commutative property (A) Signup and view all the answers

Which of the following is a property of multiplication of complex numbers?

Commutative property (B) Signup and view all the answers

What is the additive inverse of a complex number z = a + ib?

-z (C) Signup and view all the answers

What is the identity element for multiplication of complex numbers?

1 + 0i (D) Signup and view all the answers

What is the property of complex numbers that satisfies z1(z2 + z3) = z1z2 + z1z3?

Distributive property over addition (C) Signup and view all the answers

If z₁ = a + ib and z₂ = c + id, what is the value of z₁ + z₂?

(a + c) + i(b + d) (D) Signup and view all the answers

What is the value of z₁ - z₂ if z₁ = 2 + 3i and z₂ = 1 + 2i?

1 + i (B) Signup and view all the answers

What can be said about the imaginary part of a complex number if it is a purely imaginary number?

It is always non-zero (B) Signup and view all the answers

If z is a complex number, what is the relationship between z and its conjugate?

z is always unequal to its conjugate (A) Signup and view all the answers

What is the result of multiplying a complex number by its conjugate?

A real number (B) Signup and view all the answers

If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, where x₁, x₂, y₁, y₂ ∈ R and x₁ < x₂, y₁ > y₂, then which of the following is true?

arg(z₁) < arg(z₂) (A) Signup and view all the answers

If z = a + ib, where a, b ∈ R, then which of the following is a property of z?

z² = a² - b² (B) Signup and view all the answers

If z = x + iy, where x, y ∈ R, then the principal value of arg(z) lies in which of the following intervals?

(-π, π] (D) Signup and view all the answers

If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, where x₁, x₂, y₁, y₂ ∈ R, then which of the following is true?

z₁ + z₂ = (x₁ + x₂) + i(y₁ + y₂) (D) Signup and view all the answers

If z = x + iy, where x, y ∈ R, then which of the following is the polar form of z?

z = r(cos(θ) + i sin(θ)) (D) Signup and view all the answers