The Modulus of a Complex Number

The length of the arrow

On the Argand diagram, 3 + 4i is the point 3 across and 4 up, and the arrow from the origin to it has a length. That length, the distance from the origin to the point, is the modulus of 3 + 4i, written |3 + 4i|.

Pythagoras gives it

The arrow goes 3 across and 4 up. Those two steps and the arrow make a right-angled triangle, with legs 3 and 4 and the arrow as its hypotenuse.

By Pythagoras, $|3+4i{|}^2=3^2+4^2=9+16=25$|3 + 4i|² = 3² + 4² = 9 + 16 = 25, so |3 + 4i| = 5. For any complex number a + bi, the legs are a and b, and |a + bi$|=\sqrt{a^2+b^2}$| = √(a² + b²).

The gold arrow from the origin to 3 + 4i. The dashed lines are its steps, 3 across and 4 up, and the arrow is the hypotenuse of that triangle: its length is 5.

The same triangle on its own, with legs 3 and 4. Its hypotenuse is $\sqrt{3^2+4^2}=\sqrt{25}=5$√(3² + 4²) = √25 = 5.

Signs square away

A negative part gives a step left or down, and its length is the same. Squaring removes the sign: $|3-4i|=\sqrt{9+16}=5$|3 − 4i| = √(9 + 16) = 5, $|-6+8i|=\sqrt{36+64}=10$|−6 + 8i| = √(36 + 64) = 10 and $|5-12i|=\sqrt{25+144}=13$|5 − 12i| = √(25 + 144) = 13.

The modulus need not be a whole number. $|1+i|=\sqrt{2}\approx 1.414$|1 + i| = √2 ≈ 1.414 and $|2-3i|=\sqrt{4+9}=\sqrt{13}\approx 3.606$|2 − 3i| = √(4 + 9) = √13 ≈ 3.606.

For a real number the modulus is the absolute value: $|-7|=\sqrt{49+0}=7$|−7| = √(49 + 0) = 7, the distance of −7 from 0. For an imaginary number, |4i| = 4.

Equal moduli lie on a circle

Every number at distance 5 from the origin has modulus 5: 5, 5i, 3 + 4i, −4 + 3i, −3 − 4i and 4 − 3i among them. Together, all such numbers make up the circle of radius 5 centered at the origin.

Six numbers with modulus 5. Each is 5 from the origin, so all six lie on the circle of radius 5.

The modulus and the conjugate

A number times its conjugate is $a^2+b^2$a² + b², which is the modulus squared: $\left(3+4i\right)\left(3-4i\right)=9+16=25=5^2$(3 + 4i)(3 − 4i) = 9 + 16 = 25 = 5².

So dividing by c + di, which multiplies the bottom by its conjugate, always leaves |c + di${|}^2$|² as the new denominator. And a number and its conjugate have the same modulus, since one is the reflection of the other in the real axis.

The distance between two numbers

The distance between two complex numbers z and w is |z − w|. For z = 5 + 7i and w = 2 + 3i, z − w = 3 + 4i, so the distance is 5. It is the distance formula: across 5 − 2 = 3, up 7 − 3 = 4, and $\sqrt{9+16}=5$√(9 + 16) = 5.

The arrow from 2 + 3i to 5 + 7i goes 3 across and 4 up, the same steps as the arrow for 3 + 4i, so the distance between them is |3 + 4i| = 5.

Moduli multiply

The modulus of a product is the product of the moduli. $\left(3+4i\right)\left(1+i\right)=3+3i+4i+4i^2=-1+7i$(3 + 4i)(1 + i) = 3 + 3i + 4i + 4i² = −1 + 7i, and $|-1+7i|=\sqrt{1+49}=\sqrt{50}=5\sqrt{2}$|−1 + 7i| = √(1 + 49) = √50 = 5√2, which is $|3+4i|\times |1+i|=5\times \sqrt{2}$|3 + 4i| × |1 + i| = 5 × √2. This gives a check on any multiplication.

The usual mistakes

Adding the legs. |3 + 4i| is not 3 + 4 = 7: the arrow is the hypotenuse, so square the legs, add, then take the square root.

Multiplying the legs. 3 × 4 = 12 is the area of a rectangle, not a length.

Squaring the i as well. In $\sqrt{a^2+b^2}$√(a² + b²), b is the real number 4, not 4i. Using ${\left(4i\right)}^2=-16$(4i)² = −16 would give $\sqrt{9-16}$√(9 − 16), the square root of a negative number, for a length.

Stopping before the square root. $3^2+4^2=25$3² + 4² = 25 is the modulus squared; the modulus is 5.