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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
How to find any Power
Complex Numbers: Add & subtract
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Multiplication by i

2. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
v(-1)
cosh²y - sinh²y
Complex Exponentiation

3. V(zz*) = v(a² + b²)
conjugate pairs
|z| = mod(z)
standard form of complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

4. A+bi
standard form of complex numbers
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y
Complex Number Formula

5. A number that can be expressed as a fraction p/q where q is not equal to 0.
-1
De Moivre's Theorem
non-integers
Rational Number

6. 1
cosh²y - sinh²y
Polar Coordinates - cos?
x-axis in the complex plane
natural

7. Starts at 1 - does not include 0
Complex Multiplication
a + bi for some real a and b.
cos iy
natural

8. z1z2* / |z2|²
z1 / z2
We say that c+di and c-di are complex conjugates.
point of inflection
Imaginary number

9. Root negative - has letter i
imaginary
cos z
Complex Conjugate
Complex numbers are points in the plane

10. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
How to add and subtract complex numbers (2-3i)-(4+6i)
Rules of Complex Arithmetic
ln z
i^4

11. Has exactly n roots by the fundamental theorem of algebra
Complex Numbers: Add & subtract
Any polynomial O(xn) - (n > 0)
How to solve (2i+3)/(9-i)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

12. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition
Subfield
conjugate

13. A plot of complex numbers as points.
We say that c+di and c-di are complex conjugates.
Polar Coordinates - sin?
Argand diagram
(a + bi) = (c + bi) = (a + c) + ( b + d)i

14. Like pi
Complex Addition
real
transcendental
For real a and b - a + bi = 0 if and only if a = b = 0

15. We see in this way that the distance between two points z and w in the complex plane is
Liouville's Theorem -
Polar Coordinates - cos?
|z-w|
four different numbers: i - -i - 1 - and -1.

16. Equivalent to an Imaginary Unit.
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - z
Imaginary number
Liouville's Theorem -

17. A subset within a field.
v(-1)
multiplying complex numbers
Polar Coordinates - r
Subfield

18. Every complex number has the 'Standard Form':
|z| = mod(z)
We say that c+di and c-di are complex conjugates.
a + bi for some real a and b.
Any polynomial O(xn) - (n > 0)

19. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
has a solution.
v(-1)
integers

20. 1
i^2
Rules of Complex Arithmetic
|z-w|
i^1

21. Have radical
Polar Coordinates - Division
radicals
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

22. I
non-integers
How to solve (2i+3)/(9-i)
i^1
Complex Numbers: Add & subtract

23. 3
the complex numbers
i^3
For real a and b - a + bi = 0 if and only if a = b = 0
non-integers

24. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Complex Numbers: Multiply
x-axis in the complex plane
Complex Division
z1 / z2

25. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

26. Any number not rational
natural
Complex Conjugate
z1 / z2
irrational

27. R^2 = x
Square Root
Polar Coordinates - cos?
a + bi for some real a and b.
zz*

28. A complex number may be taken to the power of another complex number.
four different numbers: i - -i - 1 - and -1.
Affix
the distance from z to the origin in the complex plane
Complex Exponentiation

29. When two complex numbers are divided.
Polar Coordinates - cos?
(a + c) + ( b + d)i
Complex Division
We say that c+di and c-di are complex conjugates.

30. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Complex Number Formula
Square Root
i^4

31. 1
subtracting complex numbers
cosh²y - sinh²y
i²
Polar Coordinates - Division

32. A² + b² - real and non negative
i²
Affix
Complex Addition
zz*

33. Rotates anticlockwise by p/2
real
Polar Coordinates - Multiplication by i
Roots of Unity
How to solve (2i+3)/(9-i)

34. x + iy = r(cos? + isin?) = re^(i?)
(a + c) + ( b + d)i
Polar Coordinates - z
complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.

35. In this amazing number field every algebraic equation in z with complex coefficients
Complex Subtraction
has a solution.
Complex numbers are points in the plane
z1 / z2

36. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
How to find any Power
Subfield
sin z

37. I = imaginary unit - i² = -1 or i = v-1
Field
complex numbers
Imaginary Numbers
cosh²y - sinh²y

38. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary number
multiply the numerator and the denominator by the complex conjugate of the denominator.
ln z

39. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
sin z
the complex numbers
i^3
'i'

40. I
Rational Number
Complex Numbers: Multiply
(cos? +isin?)n
v(-1)

41. When two complex numbers are multipiled together.
the vector (a -b)
z1 / z2
Complex Multiplication
four different numbers: i - -i - 1 - and -1.

42. Divide moduli and subtract arguments
How to multiply complex nubers(2+i)(2i-3)
i^2 = -1
Polar Coordinates - Division
i^2

43. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

44. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
ln z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

45. (e^(iz) - e^(-iz)) / 2i
non-integers
a + bi for some real a and b.
i²
sin z

46. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Polar Coordinates - r
the complex numbers
sin iy

47. 2a
radicals
Complex Addition
z + z*
0 if and only if a = b = 0

48. Written as fractions - terminating + repeating decimals
v(-1)
conjugate
rational
How to add and subtract complex numbers (2-3i)-(4+6i)

49. A complex number and its conjugate
conjugate pairs
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
transcendental
a real number: (a + bi)(a - bi) = a² + b²

50. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0