Complex Numbers Flashcard

Front

What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as i = √(-1).

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, i.e., i = √(-1). It is used to extend the real number system to include solutions to equations that do not have real solutions.

Front

How do you simplify powers of 'i'?

Back

Powers of 'i' cycle every four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. To simplify i^n, find n mod 4.

Front

What is the sum of two complex numbers (a + bi) and (c + di)?

Back

The sum is (a + c) + (b + d)i. You add the real parts and the imaginary parts separately.

Front

How do you subtract complex numbers?

Back

To subtract two complex numbers (a + bi) - (c + di), subtract the real parts and the imaginary parts: (a - c) + (b - d)i.

Front

What is the product of two complex numbers (a + bi) and (c + di)?

Back

The product is (ac - bd) + (ad + bc)i. Use the distributive property and combine like terms.

Front

What is the conjugate of a complex number a + bi?

Back

The conjugate of a complex number a + bi is a - bi. It is used to simplify division of complex numbers.

Front

How do you divide complex numbers?

Back

To divide (a + bi) by (c + di), multiply the numerator and denominator by the conjugate of the denominator: (a + bi)(c - di) / (c^2 + d^2).

Front

What is the modulus of a complex number a + bi?

Back

The modulus is |a + bi| = √(a^2 + b^2). It represents the distance from the origin in the complex plane.

Front

What is the argument of a complex number?

Back

The argument of a complex number a + bi is the angle θ formed with the positive real axis, calculated as θ = arctan(b/a).