Complex Numbers Review

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).

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.

To simplify the square root of a negative number, factor out the negative sign: $$\text{√}(-36) = \text{√}(36) \times \text{√}(-1) = 6i$$ .

The product is calculated as follows: (4 – 3i)(-7 – 2i) = 4*(-7) + 4*(-2i) - 3i*(-7) - 3i*(-2i) = -28 - 8i + 21i + 6 = -34 + 13i.

The result is: 2i * -3i = -6i^2. Since i^2 = -1, this simplifies to -6 * -1 = 6.

The simplified form is: -6i * 3i = -18i^2. Since i^2 = -1, this simplifies to 18.

To calculate, distribute the real number: -4(13 + 5i) = -4*13 + (-4)*5i = -52 - 20i.

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

To add, combine the real parts and the imaginary parts: (3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i.

The modulus of a complex number a + bi is given by the formula: |a + bi| = √(a^2 + b^2).