The Modulus, and Distance in the Plane
Chapter Twelve
Syllabus topic Module 1, "1.1 Complex Numbers"
Pages 27 to 29 of 303
In one line
The modulus of x + iy is the square root of x squared plus y squared, which is the distance of the point from the origin.
The definition
For z = x + iy, the modulus of z, written abs(z) or the modulus of z, is defined below.
z = x + i y
abs(z) = sqrt(x^2 + y^2)
It is a real, non-negative number. That matters: the modulus is the one thing about a complex number that can be compared, added to, and put in an inequality.
On the Argand diagram it is the length of the line from the origin to the point, which is Pythagoras applied to the two parts. The real part and the imaginary part are the two short sides of a right-angled triangle, and the modulus is the hypotenuse.
Worked
abs(3 + 4i) = sqrt(9 + 16) = 5
abs(-3 + 4i) = 5
abs(5) = 5
abs(-7i) = 7
abs(1 + i) = sqrt(2)
abs(2 - 2i) = 2 sqrt(2)
The first two being equal is the point of the second line: the modulus does not care about the signs, because both parts get squared. Four different numbers, 3 + 4i and 3 minus 4i and minus 3 plus 4i and minus 3 minus 4i, all have modulus 5, and they sit one in each quadrant on a circle of radius 5.
The third and fourth lines say that for a real number the modulus is the ordinary absolute value, and for a purely imaginary number it is the size of the imaginary part with the sign thrown away.
The identity that does the work
The modulus is almost never computed from the definition in a proof. This identity is used instead.
z = x + i y
z conjugate(z) = x^2 + y^2
abs(z)^2 = z conjugate(z)
The modulus squared is z times its conjugate. That is convenient because it is a product rather than a square root, and products are easy to manipulate.
The properties, all of which are examined
abs((a + i b)(c + i d)) = abs(a + i b) abs(c + i d)
abs((3 + 4i)/(1 + 2i)) = abs(3 + 4i)/abs(1 + 2i)
abs(conjugate(3 - 5i)) = abs(3 - 5i)
abs((2 + i)^3) = abs(2 + i)^3
In words, and these are worth knowing as sentences:
The modulus of a product is the product of the moduli. This is the most used of the four. It is proved with the conjugate identity in three lines, and it is what makes De Moivre's theorem work.
The modulus of a quotient is the quotient of the moduli. The same proof.
Conjugating does not change the modulus. Obvious from the picture: reflecting in the real axis does not move a point closer to the origin.
The Modulus, and Distance in the Plane
The modulus of a power is the power of the modulus. This follows from the product rule applied repeatedly.
The triangle inequality
This one is different in kind from the four above, because it is an inequality rather than an equation, and because it is not obvious.
For any two complex numbers, the modulus of their sum is at most the sum of their moduli.
The picture is the proof. Adding two complex numbers puts their vectors nose to tail, and the sum is the third side of a triangle. No side of a triangle is longer than the other two put together, and the two sides are equal only when the triangle collapses flat, which happens when the two numbers point in the same direction.
There is a companion inequality, obtained by applying the first one cleverly: the modulus of a sum is at least the difference of the moduli. The two together say that the size of z1 plus z2 is trapped between the difference and the sum of the two sizes.
Worked, so the numbers are in front of you. Take z1 = 3 + 4i, of modulus 5, and z2 = 5 + 12i, of modulus 13.
abs((3 + 4i) + (5 + 12i)) = abs(8 + 16i) = 8 sqrt(5)
Eight root five is about 17.89. The sum of the moduli is 18, and the difference is 8. So 8 is less than 17.89, which is less than 18, and both inequalities hold. They are nearly equal to 18 because the two numbers point in nearly the same direction.
Where the modulus is used
Three places, and all three are coming.
Locus questions, now. The condition that abs(z) equals r is a circle of radius r about the origin, and the condition that abs(z minus a) equals r is a circle of radius r about the point a. Every locus question in this topic is one of those two, or a perpendicular bisector.
Polar form, two chapters from here. The modulus is the r in r(cos t + i sin t), so it is half of what you need to write a complex number in polar form.
Stability, in Module 1's second half and beyond. Whether a digital filter or a control loop settles down or blows up is a question about whether certain complex numbers have modulus less than one. That is why the unit circle matters so much in engineering.
Check yourself
| Question | Answer |
|---|---|
| abs(5 - 12i) | abs(5 - 12i) = 13 |
| abs(-8) | abs(-8) = 8 |
| abs(i) | abs(i) = 1 |
| abs((1 + i)(1 - i)) | abs((1 + i)(1 - i)) = 2 |
| abs((1 + i)^8) | abs((1 + i)^8) = 16 |
| abs(1/(3 + 4i)) | abs(1/(3 + 4i)) = 1/5 |
The Modulus, and Distance in the Plane
The fifth is the power rule: the modulus of 1 + i is root two, and root two to the eighth is sixteen. Expanding the bracket would also work and would take five minutes.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.