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 the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, and it is used to express complex numbers.

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 (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 multiply a complex number by a real number?

Back

To multiply a complex number by a real number, multiply both the real and imaginary parts by that real number. For example, k(a + bi) = ka + kbi.

Front

What is the quadratic formula?

Back

The quadratic formula is used to find the solutions of a quadratic equation ax^2 + bx + c = 0, and is given by x = (-b ± √(b² - 4ac)) / (2a).

Front

What does it mean if the discriminant of a quadratic equation is positive?

Back

If the discriminant (b² - 4ac) is positive, the quadratic equation has two distinct real solutions.

Front

What does it mean if the discriminant of a quadratic equation is zero?

Back

If the discriminant (b² - 4ac) is zero, the quadratic equation has exactly one real solution (a repeated root).