Imaginary and Complex Numbers

Front

What is an imaginary number?

Back

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where i is defined as the square root of -1.

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.

Front

How do you multiply two complex numbers?

Back

To multiply two complex numbers (a + bi) and (c + di), use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2. Remember that i^2 = -1.

Front

What is the conjugate of a complex number?

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 complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, to divide (a + bi) by (c + di), multiply by (c - di)/(c - di).

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by the formula |a + bi| = √(a² + b²). 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 x-axis in the complex plane, calculated as θ = arctan(b/a).

Front

How do you simplify the expression \( \frac{10-10i}{2+2i} \)?

Back

To simplify \( \frac{10-10i}{2+2i} \), multiply the numerator and denominator by the conjugate of the denominator: \( \frac{(10-10i)(2-2i)}{(2+2i)(2-2i)} = -5i \).

Front

What is the square root of a negative number?

Back

The square root of a negative number can be expressed using imaginary numbers. For example, \( \sqrt{-64} = 8i \).

Front

What is the standard form of a circle's equation?

Back

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.