Complex Numbers Unit Review

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^2 = -1.

Front

How do you find the distance between two complex numbers z1 = a + bi and z2 = c + di?

Back

The distance is given by the formula: \( |z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2} \)

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, i.e., \( i = \sqrt{-1} \). It is used to extend the real number system to complex numbers.

Front

How do you represent a complex number graphically?

Back

A complex number a + bi can be represented as a point (a, b) in the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

Front

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

Back

The conjugate of z is given by \( \overline{z} = a - bi \). It reflects the complex number across the real axis.

Front

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

Back

The modulus is given by \( |z| = \sqrt{a^2 + b^2} \), representing the distance from the origin to the point (a, b) in the complex plane.

Front

How do you add two complex numbers?

Back

To add two complex numbers z1 = a + bi and z2 = c + di, simply add their real and imaginary parts: \( z_1 + z_2 = (a + c) + (b + d)i \).

Front

How do you multiply two complex numbers?

Back

To multiply z1 = a + bi and z2 = c + di, use the distributive property: \( z_1 z_2 = (ac - bd) + (ad + bc)i \).

Front

What is the difference of squares?

Back

The difference of squares is a factoring technique that states: \( a^2 - b^2 = (a - b)(a + b) \).

Front

How do you factor the expression x^2 + 81?

Back

The expression can be factored as \( (x + 9i)(x - 9i) \), using the difference of squares with imaginary numbers.