Download Discrete mathematics I. practice

Discrete mathematics I. practice - Complex numbers
Emil Vatai
February 21, 2017
(b) 1 − i;
1. Express the following numbers in Cartesian form:
(a) (3 + i)(2 + 3i); (b) (1 − 2i)(5 + i) ; (c) (2 − 5i)2 ;
(d) (1 − i)3 .
(c) 4i;
(d) −3;
10
(e) √
;
3−i
2 + 3i
(f)
;
5+i
(g) 3 − 4i;h) −2 + i.
2. Simplify the following expressions: (a) i3 ; (b) i5 ;
1
1
1
(c) i8 ; (d) 2 ; (e) ; (f) 3 ; (g) (1+i)2 ; (h) (1+i)2013 ;
i
i
i
(1 + i)2013
(i)
.
(1 − i)2013
3. Solve the following quadratic functions over the set of
complex numbers:
9. Calculate the values of the following expressions using
the polar form:
(a) x2 + x + 1 = 0;
(1 + i)9
;
(1 − i)7
√
24
3−i
.
• 1−
2
•
(b) x2 + 2x + 2 = 0;
(c) x2 + 2ix − 1 = 0.
4. Illustrate the following sets on the complex plane:
10. Solve the following cubic equations:
• {z : Re(z + 2i) ≤ 0};
• {z : Re(z + 1) ≥ Im(z − 3i)};
• x3 − 7x + 6 = 0;
• {z : |z − i − 1| ≤ 3};
• x3 − 13x − 12 = 0;
• {z : |z − 3 + 2i| = |z + 4 − i|};
11. Write a program to solve cubic equations!
• {z : z = 1/z};
12. To which transformations of the two dimensional
plane do the following maps correspond to: z 7→
3z + 2; z 7→ (1 + i)z; z 7→ 1/z.
• {z : z + z = 0};
• {z : |z| = iz}.
5. Express the following numbers in Cartesian form:
(a)
(b)
(c)
(d)
(e)
(f)
13. Let z 6= w be two complex numbers! Using z and
w, express the midpoint of the line segment between
them, also express the third points (angles?) of the
two regular triangles defined by z and w and balance
point of these triangles.
3 + 4i
;
1 − 2i
√
3−i
√
;
3+i
1
;
(1 + i)2
1
;
(2 − i)(1 + 2i)
1
1
+
;
2 + 3i 2 − 3i
1
1
+
.
3 + i 1 + 7i
14. What is the square root of the following numbers:
(a) 3 − 4i;
(b) 2i;
(c) −7 − 24i;
(d) 8 + 6i.
15. Solve the following quadratic equation: (2 + i)x2 −
(5 − i)x + (2 − 2i) = 0.
√
16. Calculate the fifth roots of z = −16 · 3 + 16i!
6. What is the value of the real numbers a and b if:
(a) (a + bi)(2 − i) = a + 3i;
17. Calculate all the solutions of the following equations:
(b) (a + i)(1 + bi) = 3b + ai.
(a) x3 = 1;
5
2
+
= 1, where x and y are real numx + yi 1 + 3i
bers. What are the values of x and y?
(b) x3 = 2 + 2i;
√
(c) x8 = 3 − i;
7. Let
(d) x6 = 1 + i;
8. What is the polar form of the following numbers:
√
(a) 3 + i;
18. What is the fourth root of:
1
−4
.
(2 + i)3
√
19. Take √
the following numbers:
1, −1,√i, 1+i, (1+i)/
√
√ 2,
(1 + 3i)/2, (−1 + 3i)/2, cos( 2π) + i sin( 2π),
cos(π/361) + i sin(π/361). Which of them are roots
of unity, what is their order, for which n will they be
n -th roots of unity, or primitive n -th roots of unity?
20. Show that if ε4 = i, then 4 | o(ε).
21. If o(ε) = 128, then what is o(i · ε) =?
22. Prove that the set of (integer) exponents of a primitive n -th root of unity is the set of n -th roots of
unity.
23. Prove that a primitive n -th root of unity is a k -th
root of unity, if and only if n | k.
24. Express the value of sin(nα) and cos(nα) using sin(α),
cos(α)!
Solutions
1. HW
2. HW
3. HW
4. HW
5. HW
6. HW
7. Multiply everything with the two denominators:
5 · (1 + 3i) + 2 · (x + yi) = (x + yi) · (1 + 3i)
5 + 15i + 2x + 2yi = x − 3y + i(y + 3x)
Then Re(lhs) = 5 + 2x = x − 3y = Re(rhs) and
Im(lhs) = 15 + 2y = y + 3x = Im(rhs). So x =
−5 − 3y, and y = −15 + 3x. Substituting the x into
the second equation, y = −15 − 15 − 9y, which gives
y = −3 and x = 4.
8. HW
9. HW
10. HW
11. HW
12. HW
13. The midpoint halfway between z and w is m =
z+w
2 .
The other question is a bit more tricky. The vector from z to w is d = w − z. From geometry you
should know that√the height of an equaliteral
√ (regular) triangle is 3/2 times the side, so 3/2 · d
has just the √
right length, but there are two things
to be fixed: 3/2 · d is not pointing to the right direction, the height is perpendicular to the side of a
triangle, so we need √
to rotate it 90 degrees left and
right,
and
this
is
±i
·
3/2 · d. The other trick is, that
√
±i 3/2 · d starts from the 0, but we can easily fix
that, by adding it to the midpoint (where
it should
√
√
3
start) i.e. m ± i · 3/2 · d = z+w
±
i
(w
−
z) is the
2
2
answer. You should draw the steps!
2