Complex Numbers Explained: iota, a+bi Form and Modulus
Complex numbers can seem strange at first, but they solve a real problem: some equations have no answer among ordinary (real) numbers. Introduced in Class 11, they are essential for JEE and for advanced topics like quadratic roots and vectors. This guide explains iota, the a + bi form, basic operations, and the modulus with worked examples.
Why do complex numbers exist?
Consider the equation x² + 1 = 0. This gives x² = −1, but no real number squared is negative. To handle such cases, mathematicians defined a new number called iota, written i, with the property i² = −1. This single idea lets us solve every quadratic equation, which is why complex numbers were invented. If you want a refresher first, revisit how to solve quadratic equations.
The a + bi form
A complex number is written as z = a + bi, where a and b are real numbers. Here a is the real part and b is the imaginary part. For example, in z = 3 + 4i, the real part is 3 and the imaginary part is 4. If b = 0 the number is purely real, and if a = 0 it is purely imaginary.
Powers of iota
The powers of i repeat in a cycle of four, which is very handy for calculations.
- i¹ = i
- i² = −1
- i³ = −i (because i² × i = −i)
- i⁴ = 1 (because i² × i² = 1)
So to find i⁷, divide 7 by 4 to get a remainder of 3, giving i⁷ = i³ = −i.
Basic operations
Addition and subtraction: combine real parts and imaginary parts separately. If z₁ = 2 + 3i and z₂ = 4 + 5i, then z₁ + z₂ = (2 + 4) + (3 + 5)i = 6 + 8i.
Multiplication: expand as usual and replace i² with −1. (2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i² = 8 + 22i − 15 = −7 + 22i.
Conjugate: the conjugate of a + bi is a − bi. Multiplying a number by its conjugate removes i: (3 + 4i)(3 − 4i) = 9 − 16i² = 9 + 16 = 25. This trick is used to divide complex numbers.
Modulus of a complex number
The modulus of z = a + bi measures its distance from the origin on the complex plane and is written |z| = √(a² + b²).
Example: For z = 3 + 4i, |z| = √(3² + 4²) = √(9 + 16) = √25 = 5. Notice this uses the same idea as Pythagoras' theorem. These concepts appear across Class 11 Maths and are tested repeatedly in JEE Maths, so practising numeric examples pays off in the exam.
Frequently asked questions
What is the value of iota (i)?
Iota is defined by i = √(−1), which means i² = −1. It is not a real number; it is the basic imaginary unit used to build all complex numbers.
Is every real number a complex number?
Yes. A real number a can be written as a + 0i, so every real number is also a complex number with imaginary part zero. Complex numbers are a larger set that contains the reals.
Can the modulus of a complex number be negative?
No. The modulus is a distance, calculated as √(a² + b²), so it is always zero or positive. It equals zero only when the complex number itself is 0 + 0i.