Modulus and conjugate of a complex number

Modulus and conjugate

Let z = a + ib be a complex number.

NCERT Mathematics (Class 11), Section 4.4, page 81

In plain words: the modulus measures the size of z, and the conjugate is z with i replaced by −i. NCERT’s examples (page 81):

• |3 + i| = √(3² + 1²) = √10 and |2 − 5i| = √(2² + (−5)²) = √29;
• the conjugates of 3 + i, 2 − 5i and −3i − 5 are 3 − i, 2 + 5i and 3i − 5.

For a real number a = a + i0, the modulus is √(a² + 0²) = √(a²), the usual absolute value |a|, so the modulus extends the modulus function of real numbers.

The product of z and its conjugate

Multiplying z by its conjugate always gives a non-negative real number:

(a + ib)(a − ib) = a² − (ib)² = a² + b².

This is why dividing by a complex number works: multiplying the top and bottom of a fraction by the conjugate of the bottom makes the bottom real.

Multiplicative inverse

NCERT (page 81) observes that the multiplicative inverse of a non-zero complex number z = a + ib is z⁻¹ = 1/(a + ib) = (a − ib)/(a² + b²), that is:

NCERT Example 5 (page 82): for z = 2 − 3i, $\bar{z}$ = 2 + 3i and |z|² = 2² + (−3)² = 13, so z⁻¹ = (2 + 3i)/13 = 2/13 + (3/13)i. Check: (2 − 3i)(2 + 3i)/13 = 13/13 = 1.

Properties

Furthermore, the following results can easily be derived.

NCERT Mathematics (Class 11), Section 4.4, page 81

For any complex numbers z₁ and z₂ (NCERT page 81):

• |z₁z₂| = |z₁| |z₂|, and |z₁/z₂| = |z₁|/|z₂| when z₂ ≠ 0;
• the conjugate of z₁z₂ is $\bar{z}_1\bar{z}_2$, and the conjugate of z₁ ± z₂ is $\bar{z}_1$ ± $\bar{z}_2$;
• the conjugate of z₁/z₂ is $\bar{z}_1$/$\bar{z}_2$, when z₂ ≠ 0.

Two more facts are used constantly: |$\bar{z}$| = |z|, and the conjugate of $\bar{z}$ is z again. Moduli are real, so unlike complex numbers they can be compared; the modulus of a sum is not the sum of the moduli: |3 + 4i| = 5, but |3| + |4i| = 7. In general |z₁ + z₂| ≤ |z₁| + |z₂| (the triangle inequality), with equality when one is a non-negative real multiple of the other.

Solved examples

Common mistakes to avoid

Quick revision summary

• |z| = √(a² + b²) ≥ 0 and $\bar{z}$ = a − ib for z = a + ib.
• z $\bar{z}$ = |z|², so z⁻¹ = $\bar{z}$/|z|² for z ≠ 0; to divide, multiply the top and bottom by the conjugate of the bottom.
• |z₁z₂| = |z₁| |z₂|; the conjugate of a sum, product or quotient is the sum, product or quotient of the conjugates.

Questions students ask

What is the difference between the modulus and the conjugate of a complex number?

The modulus of a complex number is a real number, its size, while the conjugate is another complex number: for z = a + ib, |z| = √(a² + b²) and the conjugate is a − ib. For z = 6 − 8i, the modulus is √(36 + 64) = 10 and the conjugate is 6 + 8i. The two are linked, since z times its conjugate is |z|², here (6 − 8i)(6 + 8i) = 100. In the Argand plane the modulus is the distance from 0 and the conjugate is the mirror image in the real axis.

Is the modulus of a complex number always positive?

The modulus of a complex number is never negative, but it is not always positive: |z| = √(a² + b²) ≥ 0, and |z| = 0 only for z = 0, because a² + b² = 0 forces a = b = 0. Every non-zero complex number has a positive modulus; a purely imaginary number ib has modulus |b|, so |−7i| = 7 and |i| = 1. So |z| > 0 is the same condition as z ≠ 0, which is exactly when the inverse 1/z exists.

Why do we take the conjugate of a complex number?

The conjugate is used mainly because a complex number times its conjugate is real, (a + ib)(a − ib) = a² + b², so it clears i from a denominator in division and gives the inverse 1/z. It also picks out parts and tests numbers: z plus its conjugate is 2 Re z, z equals its conjugate exactly when z is real, and z equals minus its conjugate exactly when z is purely imaginary. For example, if |z| = 1 then 1/z is the conjugate of z, so z + 1/z = 2 Re z is real.

What is the multiplicative inverse of 3 + 4i?

The multiplicative inverse of 3 + 4i is 3/25 − (4/25)i: divide its conjugate 3 − 4i by |3 + 4i|² = 9 + 16 = 25. Check: (3 + 4i)(3 − 4i) = 9 + 16 = 25, so (3 + 4i) × (3 − 4i)/25 = 1. The same rule, the conjugate divided by |z|², gives the inverse of every non-zero z; for z = 1 + i it is (1 − i)/2.

Practice questions

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

1. Q1. Find |z| and $\bar{z}$ for z = −5 + 12i.

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   |z| = √(25 + 144) = 13; $\bar{z}$ = −5 − 12i.

2. Q2. Express i⁻³⁵ in the form a + ib (NCERT Example 6 (ii), page 82).

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   i⁻³⁵ = 1/i³⁵ = 1/((i²)¹⁷ i) = 1/(−i) = i, that is 0 + 1i.

3. Q3. Express (3 + i√5)(3 − i√5)/((√3 + √2 i) − (√3 − i√2)) in the form a + ib (NCERT Exercise 4.1, Q14).

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   Top: 9 − (i√5)² = 9 + 5 = 14. Bottom: 2√2 i. So 14/(2√2 i) = 7/(√2 i) = −7i/√2 = 0 − (7√2/2)i.

4. Q4. Find the modulus of (1 + i)/(1 − i) − (1 − i)/(1 + i) (NCERT Miscellaneous Exercise, Q9).

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   (1 + i)/(1 − i) = (1 + i)²/2 = i, and (1 − i)/(1 + i) = −i. The difference is 2i, whose modulus is 2.