adding subtracting multiplying and dividing 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 i² = -1.

Front

How do you add complex numbers?

Back

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

Front

How do you subtract complex numbers?

Back

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

Front

What is the product of two complex numbers?

Back

The product of two complex numbers (a + bi)(c + di) is calculated using the distributive property: ac + adi + bci + bdi², which simplifies to (ac - bd) + (ad + bc)i.

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) to eliminate the imaginary part in the denominator.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is used in division to eliminate the imaginary part from the denominator.

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 to the point (a, b) 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 real axis, calculated as θ = arctan(b/a). It represents the direction of the complex number in the complex plane.

Front

Simplify: (-7 + 9i) + (2 - 10i)

Back

-5 - i

Front

Simplify: (2 + 3i)²

Back

-5 + 12i