Imaginary and Complex Numbers

Front

What is the definition of 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 the value of i^2?

Back

i^2 = -1.

Front

How do you simplify the square root of a negative number, such as √-x?

Back

To simplify √-x, you can express it as i√x.

Front

What is the result of multiplying two imaginary numbers, such as (ai)(bi)?

Back

The result is -ab, where a and b are real numbers.

Front

What is the formula for multiplying two complex numbers (a + bi)(c + di)?

Back

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

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 find the modulus of a complex number a + bi?

Back

The modulus of a complex number a + bi is given by √(a^2 + b^2).

Front

What is the result of (2i)(3i)?

Back

The result is -6.

Front

How do you simplify the expression (2 - 5i)(2 + 5i)?

Back

The expression simplifies to 29.

Front

What is the geometric representation of complex numbers on the complex plane?

Back

Complex numbers are represented as points or vectors in the complex plane, with the x-axis representing the real part and the y-axis representing the imaginary part.