Algebra of complex numbers

What is the algebra of complex numbers?

In this Section, we shall develop the algebra of complex numbers.

NCERT Mathematics (Class 11), Section 4.3, page 77

The four operations work on a + ib exactly as on an expression in a letter i, with one extra rule, i² = −1. The results are always again complex numbers (except that we never divide by 0).

Adding and subtracting complex numbers

In plain words: add the real parts, and add the imaginary parts. NCERT’s examples (page 77): (2 + i3) + (−6 + i5) = −4 + i8, and (6 + 3i) − (2 − i) = 4 + 4i, while (2 − i) − (6 + 3i) = −4 − 4i. Subtraction is not commutative.

The addition of complex numbers satisfy the following properties:

NCERT Mathematics (Class 11), Section 4.3.1, page 77

These are the same laws as for real numbers:

• closure: the sum of two complex numbers is a complex number;
• commutative: z₁ + z₂ = z₂ + z₁;
• associative: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃);
• additive identity 0 = 0 + i0, with z + 0 = z;
• additive inverse −z = −a + i(−b), with z + (−z) = 0.

Multiplying complex numbers

Multiply out as for ordinary brackets and replace i² by −1: (a + ib)(c + id) = ac + iad + ibc + i²bd. This is NCERT’s definition (Section 4.3.3, page 78):

For example, (3 + i5)(2 + i6) = (3 × 2 − 5 × 6) + i(3 × 6 + 5 × 2) = −24 + i28 (NCERT page 78).

The multiplication of complex numbers possesses the following properties, which we state without proofs.

NCERT Mathematics (Class 11), Section 4.3.3, page 78

Multiplication is closed, commutative and associative, distributes over addition, and has the identity 1 = 1 + i0. Every non-zero complex number also has a multiplicative inverse:

This works whenever a² + b² ≠ 0, that is, whenever a and b are not both 0. NCERT (page 78, and again in the chapter summary on page 87) writes the condition as “a ≠ 0, b ≠ 0”, which is too strong: z = 2, with b = 0, has the inverse 1/2, and z = 3i, with a = 0, has the inverse −i/3. The correct condition is z ≠ 0. NCERT Section 4.4 rewrites this inverse using the conjugate a − ib.

Dividing complex numbers

For z₂ ≠ 0, z₁/z₂ = z₁ × (1/z₂) (NCERT Section 4.3.4, page 78). In practice, multiply the top and the bottom by the bottom with the sign of its imaginary part changed; the bottom then becomes a real number:

NCERT works this example (pages 78–79) from the definition, multiplying 6 + 3i by 1/(2 − i) = (2 + i)/5, and gets (1/5)(9 + 12i), that is 9/5 + (12/5)i.

Powers of i

Since i² = −1 (NCERT Section 4.3.5, page 79), each extra factor of i moves one step round the cycle 1 → i → −1 → −i → 1. If a + ib is drawn as the point (a, b) (the Argand plane of NCERT Section 4.5), then i(a + ib) = −b + ia is the point (−b, a): the same distance from 0, turned a right angle (π/2 radian) anticlockwise about 0 (see the figure below). In symbols:

• i³ = i² × i = −i and i⁴ = (i²)² = 1;
• i⁵ = i, i⁶ = −1, and so on: the powers repeat every 4;
• negative powers too: i⁻¹ = 1/i = −i, i⁻² = −1, i⁻³ = i, i⁻⁴ = 1.

To find iⁿ, divide n by 4 and keep the remainder: i³⁵ = i^{32 + 3} = −i, and i⁻³⁵ = i^{−36 + 1} = i.

Square roots of negative numbers

Both i and −i square to −1, so −1 has the two square roots ±i; the symbol √−1 means i only (NCERT Section 4.3.6, page 79). In the same way the square roots of −3 are ±√3 i, and √−3 means √3 i. In general, for a positive real number a, √−a = √a i.

The rule √a × √b = √(ab) holds for positive a and b, and also when one of them is negative, but not when both are negative (NCERT pages 79–80). If it did, then i² = √−1 × √−1 = √((−1)(−1)) = √1 = 1, which contradicts i² = −1.

Identities

Because addition and multiplication obey the usual laws, the algebraic identities for real numbers hold for complex numbers too (NCERT Section 4.3.7, page 80):

• (z₁ + z₂)² = z₁² + 2z₁z₂ + z₂² and (z₁ − z₂)² = z₁² − 2z₁z₂ + z₂²;
• (z₁ ± z₂)³ = z₁³ ± 3z₁²z₂ + 3z₁z₂² ± z₂³;
• z₁² − z₂² = (z₁ + z₂)(z₁ − z₂).

In fact, many other identities which are true for all real numbers, can be proved to be true for all complex numbers.

NCERT Mathematics (Class 11), Section 4.3.7, page 80

NCERT proves the first one step by step: (z₁ + z₂)(z₁ + z₂) = z₁² + z₂z₁ + z₁z₂ + z₂² by the distributive law, and z₂z₁ = z₁z₂ by the commutative law.

Solved examples

Common mistakes to avoid

Quick revision summary

• (a + ib) ± (c + id) = (a ± c) + i(b ± d); (a + ib)(c + id) = (ac − bd) + i(ad + bc), using i² = −1.
• 1/(a + ib) = (a − ib)/(a² + b²) for a + ib ≠ 0; to divide by c + id, multiply the top and the bottom by c − id.
• i^4k = 1, i^{4k + 1} = i, i^{4k + 2} = −1, i^{4k + 3} = −i.
• √−a = √a i for a > 0; √a √b = √(ab) fails when a and b are both negative.

Questions students ask

Is there a formula for dividing complex numbers?

Yes, the formula for dividing complex numbers is (a + ib)/(c + id) = [(ac + bd) + i(bc − ad)]/(c² + d²), for c + id ≠ 0. It comes from multiplying the top and the bottom by c − id, the conjugate of the bottom: the bottom becomes the real number c² + d², and expanding (a + ib)(c − id) gives the top. For example, (2 + 3i)/(1 + 2i) = (2 + 3i)(1 − 2i)/5 = (8 − i)/5 = 8/5 − (1/5)i.

Is the square root of a negative number undefined?

The square root of a negative number is undefined among the real numbers but defined among the complex numbers: no real number squares to a negative number, while for any real a > 0 the two complex numbers ±√a i square to −a, and √−a means √a i. For example, the square roots of −49 are 7i and −7i, since (7i)² = 49i² = −49, and √−49 = 7i. NCERT’s historical note (page 87) records that Mahavira (850), in his Ganitasara Sangraha, already stated that a negative quantity has no square root because it is not a square.

How do you find the square root of a complex number such as 3 + 4i?

To find the square roots of a + ib, write a root as x + iy with x, y real, equate parts in (x + iy)² = a + ib and add x² + y² = |a + ib| from the moduli; solve for x² and y², giving x and y the same sign if b > 0 and opposite signs if b < 0. For 3 + 4i, x² = 4 and y² = 1 with like signs, so the square roots are 2 + i and −2 − i; indeed (2 + i)² = 3 + 4i. The method is beyond the 2026–27 NCERT text and is not named in the JEE Main 2027 syllabus.

What is the difference between the additive and multiplicative inverse of a complex number?

The additive inverse of z = a + ib is −z = −a − ib, which adds to z to give 0, while the multiplicative inverse is 1/z = (a − ib)/(a² + b²), which multiplies z to give 1. Every complex number has an additive inverse, but only a non-zero one has a multiplicative inverse. For z = 1 − 2i, the additive inverse is −1 + 2i and the multiplicative inverse is (1 + 2i)/5, since (1 − 2i)(1 + 2i) = 5.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Express in the form a + ib: (i) (5i)(−3i/5); (ii) i⁹ + i¹⁹; (iii) i⁻³⁹ (NCERT Exercise 4.1, Q1–Q3).

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   (i) −3i² = 3 + 0i; (ii) i + i³ = i − i = 0 + 0i; (iii) i⁻³⁹ = i^{−40 + 1} = i, that is 0 + 1i.

2. Q2. Express (1 − i) − (−1 + i6) in the form a + ib (NCERT Exercise 4.1, Q5).

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   (1 + 1) + i(−1 − 6) = 2 − 7i.

3. Q3. Express (1/5 + (2/5)i) − (4 + (5/2)i) in the form a + ib (NCERT Exercise 4.1, Q6).

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   (1/5 − 4) + i(2/5 − 5/2) = −19/5 − (21/10)i.

4. Q4. Express (−2 − i/3)³ in the form a + ib (NCERT Exercise 4.1, Q10).

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   (−2 − i/3)³ = −(2 + i/3)³. By the cube of a sum, (2 + i/3)³ = 8 + 3 × 4 × i/3 + 3 × 2 × i²/9 + i³/27 = 8 + 4i − 2/3 − i/27 = 22/3 + (107/27)i. So (−2 − i/3)³ = −22/3 − (107/27)i.