Combining 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 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, i.e., i = √-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 (a + bi)(c + di)?

Back

The product is given by (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

Front

What is the square of a complex number (a + bi)^2?

Back

The square is given by (a + bi)^2 = a^2 + 2abi + (bi)^2 = (a^2 - b^2) + 2abi.

Front

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

Back

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

Front

How do you multiply complex numbers in the form of (a + bi)(c - bi)?

Back

Use the distributive property: (a + bi)(c - bi) = ac - abi + bci - b^2i^2 = (ac + b^2) + (bc - ab)i.

Front

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

Back

The modulus is given by |a + bi| = √(a^2 + b^2).

Front

How do you simplify the expression (1 - 7i)^2?

Back

(1 - 7i)^2 = 1^2 - 2(1)(7i) + (7i)^2 = 1 - 14i - 49 = -48 - 14i.