Simplifying Complex Numbers and 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 i = √(-1). It is used to represent the square root of negative numbers.

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 conjugate of a complex number a + bi?

Back

The conjugate of a complex number a + bi is a - bi.

Front

How do you multiply two complex numbers (a + bi)(c + di)?

Back

To multiply, use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

Front

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

Back

The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²).

Front

How do you add two complex numbers?

Back

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

Front

How do you subtract two complex numbers?

Back

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

Front

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

Back

The result of multiplying a complex number a + bi by its conjugate a - bi is a² + b².

Front

Solve for x: x^4 + 4x^3 + 3x^2 = 0.

Back

x = 0, x = -3, x = -1.