Complex numbers in geometry: loci and transformations · Grade 12 · 5 units
In Complex numbers in geometry: loci and transformations (Grade 12 · 5 units) you practise 4 skills: Laws of the modulus and the conjugate · Loci in the Gauss plane · Translations and quarter turns · Stretches and turns. The modulus of the difference of two complex numbers is the distance between their points in the Gauss plane. Several laws follow: the modulus of a product equals the product of the moduli, a number and its conjugate have the same modulus, and the sum of a number and its conjugate is always real. The modulus of a sum is never more than the sum of the moduli: this is the triangle inequality.
Distances describe loci. All the points at a fixed distance from a fixed point form a circle. All the points at the same distance from two points lie on the perpendicular bisector of the segment between them. When the sum of the distances from two points is fixed, the locus is an ellipse, and when one distance is a fixed multiple of the other, other than one, it is a circle. Operations on complex numbers are transformations of the plane. Adding a fixed number is a translation, multiplying by the imaginary unit is a quarter turn anticlockwise about the origin, and multiplying by minus one is a half turn. Multiplying by any complex number turns through its argument and stretches by its modulus: for example, multiplying by one plus the imaginary unit turns through forty five degrees and stretches by the square root of two.
Tip: To find which operation sends a point to its image, substitute the point in each of the options.
What you learn
Laws of the modulus and the conjugate
For example: Which of these statements is true?
- For all complex numbers z and w , |z + w| = |z| + |w| .
- For every complex number z , z̅ = −z .
- The number z − z̅ is real for every complex number z .
- The number z + z̅ is real for every complex number z .
Answer and solution
The answer: The number z + z̅ is real for every complex number z .
If z = a + bi , then z + z̅ = 2a , and the imaginary part cancels. So the true statement is: The number z + z̅ is real for every complex number z .
Loci in the Gauss plane
For example: The locus of the points z in the Gauss plane with |z − (−4 + 5i)| = 2 is a circle. What are the coordinates of its centre?
Answer and solution
The answer: (−4, 5)
These are all the points whose distance from the point (−4, 5) is 2 , so the centre is (−4, 5) and the radius is 2 .
Translations and quarter turns
For example: Every point z of the Gauss plane is sent to the point z + w , with w = −3 − i . What are the coordinates of the image of the point that represents −4 + 5i ?
Answer and solution
The answer: (−7, 4)
Add: (−4 + 5i) + (−3 − i) is −7 + 4i , and the point is (−7, 4) .
Stretches and turns
For example: Every point z of the Gauss plane is sent to the point 2z . What are the coordinates of the image of the point that represents 5 − 4i ?
Answer and solution
The answer: (10, −8)
Double each part: that gives 10 − 8i , and the point is (10, −8) .
Not on the site yet
Parts of this topic that are not built yet: Proofs in several ways · The geometric spiral.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).