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Mathematics

Properties of complex numbers

1.Given complex numbers a = 1+2i , b = 2 – i. Determine a + b, a – b, a·b, a / b, |a|, further

Determine whether the number

a=45−i35​

is a complex unit.

Solution:

vlastnosti-komplexnych-cisel-2r

number a is a complex unit

2.Write the number a=1+i3a = 1 + i\sqrt{3}

  • a) in trigonometric form
  • b) in exponential form
  • c) calculate a5
  • d) calculate √a

Solution:

vlastnosti-komplexnych-cisel-3r

3.Calculate:

vlastnosti-komplexnych-cisel-4z.gif

Solution:

vlastnosti-komplexnych-cisel-4r

4.Calculate:

vlastnosti-komplexnych-cisel-5z

Solution:

vlastnosti-komplexnych-cisel-5r

5.Find real numbers x , y such that:

(3 – 2i)·x + (5 – 7i)·y = 1 + 3i

Solution:

 vlastnosti-komplexnych-cisel-6

6.Calculate:

vlastnosti-komplexnych-cisel-7z

Solution:

vlastnosti-komplexnych-cisel-7r

7. A square has its center at the origin of the Gaussian plane, one vertex is the image of the complex number a=4 + 3i.

  • a)    Which complex numbers represent the remaining vertices
  • b)    Determine the area of this square

Solution:

a) The required complex numbers must satisfy these conditions:

  1. Their absolute value is equal to ∣a∣|a|

  2. Each lies in a different quadrant of the Gaussian plane

∣a∣=42+32=25=5|a| = \sqrt{4^{2} + 3^{2}} = \sqrt{25} = 5

The complex numbers that satisfy the required conditions are:
a=4+3i, b=−3+4i, c=−4−3i, d=3−4ia = 4 + 3i,\; b = -3 + 4i,\; c = -4 - 3i,\; d = 3 - 4i

b) Area of the square

S=u22=2∣a∣22=(2⋅5)22=1022=1002=50S = \frac{u^{2}}{2} = \frac{2|a|^{2}}{2} = \frac{(2 \cdot 5)^{2}}{2} = \frac{10^{2}}{2} = \frac{100}{2} = 50S=50 j2S = 50\,j^{2

8.Calculate the length of the median tc of triangle ΔABC, if its vertices A,B,C are the images of the complex numbers a = –1 –i , b = –5 + 7i , c = 9 + 8i.

Solution:

The length of the median tc is the length of the segment CS, where S is the midpoint of segment AB

vlastnosti-komplexnych-cisel-9 

The median tc has length 13 u.

9.A regular hexagon ABCDEF has center S at the origin of the Gaussian plane and vertex A at the image of the complex unit on the real axis. Determine the complex numbers whose images are at the other vertices of the hexagon.

Solution:

Vertex A : a = 1

Vertex D : d = –1

The other vertices lie in individual quadrants of the Gaussian plane.

Triangle Δ SAB is equilateral SA = SB = AB = 1

vlastnosti-komplexnych-cisel-10