Complex numbers: Operations

A complex number is a number that can be expressed in the form $$a + bi$$ , where $$a$$ is the real part, $$b$$ is the imaginary part, and $$i$$ is the imaginary unit, defined as $$i^2 = -1$$ .

The imaginary unit, denoted as $$i$$ , is defined as the square root of -1, such that $$i^2 = -1$$ .

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

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

The result is $$6 + 6i$$ .

The result is $$-5 + 12i$$ .

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

The result is $$-9 - 12i$$ .

To simplify $$\frac{1}{i}$$ , multiply the numerator and denominator by $$i$$ to get $$\frac{i}{i^2} = \frac{i}{-1} = -i$$ .

The result is $$\frac{11i}{9}$$ .