Working with i, modulus and argument of complex numbers
A complex number has the form z = a + ib, where a and b are real and i is defined by i² = −1. Here a is the real part and b is the imaginary part. Complex numbers are added by combining real and imaginary parts separately, and multiplied like brackets using i² = −1, so (2 + 3i)(1 − i) = 5 + i. The conjugate of z is z̄ = a − ib, and z × z̄ = a² + b², a real number, which is why multiplying by the conjugate removes i from a denominator.
On the Argand plane, z = a + ib is the point (a, b). Its modulus |z| = √(a² + b²) is its distance from the origin, and its argument θ is the angle it makes with the positive real axis, giving the polar form z = r(cos θ + i sin θ). In this form, moduli multiply and arguments add when complex numbers are multiplied. Quadratic equations with a negative discriminant have complex roots in conjugate pairs: x² + 2x + 5 = 0 has roots −1 ± 2i. The powers of i repeat in a cycle of four: i, −1, −i, 1.
Why it matters
Complex numbers make every quadratic equation solvable and are essential in electrical engineering, where alternating currents and impedances are written as complex quantities. They are also used in signal processing, control systems and quantum physics. In Class 11 they complete the study of quadratic equations and supply regular entrance-exam questions on modulus and argument.
Key terms
- Imaginary unit
- The number i, defined so that its square equals −1.
- Conjugate
- For z = a + ib, the number a − ib obtained by changing the sign of the imaginary part.
- Modulus
- The distance of a complex number from the origin on the Argand plane.
- Argument
- The angle a complex number makes with the positive real axis.