Operations with 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^2 = -1.

Front

What is the imaginary unit 'i'?

Back

Front

How do you add complex numbers?

Back

To add 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 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 + bdi^2, which simplifies to (ac - bd) + (ad + bc)i.

Front

How do you simplify the expression (5 - 6i) - (6 - 3i)?

Back

The simplified form is -1 - 3i.

Front

What is the result of multiplying (-3 + 2i)(1 - 4i)?

Back

The result is 5 + 14i.

Front

How do you square a complex number?

Back

To square a complex number (a + bi), use the formula (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi.

Front

What is the simplified form of (-4 - 8i)^2?

Back

The simplified form is -48 + 64i.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It reflects the complex number across the real axis.