Complex Numbers

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 = √-1.

Front

What is the imaginary unit?

Back

The imaginary unit, denoted as i, is defined as the square root of -1, i.e., i = √-1.

Front

How do you write a complex number in standard form?

Back

A complex number is written in standard form as a + bi, where a is the real part and b is the imaginary part.

Front

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

Back

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

Front

How do you add two complex numbers?

Back

To add two complex numbers, add their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

How do you subtract two complex numbers?

Back

To subtract two complex numbers, subtract their real parts and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

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. Since i^2 = -1, simplify to (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| = √(a^2 + b^2).

Front

How do you divide two complex numbers?

Back

To divide two complex numbers (a + bi) by (c + di), multiply the numerator and denominator by the conjugate of the denominator: (a + bi)/(c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)].

Front

What is the result of multiplying (4 - 3i)(7 + 2i)?

Back

The result is 34 - 13i.