Complex Number Operations

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 the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, and it is used to extend the real number system to include complex numbers.

Front

How do you simplify powers of 'i'?

Back

Powers of 'i' cycle through four values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. For any integer n, i^n can be simplified by finding 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, where you add the real parts and the imaginary parts separately.

Front

How do you subtract complex numbers?

Back

To subtract 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, obtained by using the distributive property and combining 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 (or absolute value) of a complex number a + bi is given by √(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).