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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. Where the curvature of the graph changes
has a solution.
Integers
point of inflection
How to multiply complex nubers(2+i)(2i-3)

2. We see in this way that the distance between two points z and w in the complex plane is
x-axis in the complex plane
|z| = mod(z)
subtracting complex numbers
|z-w|

3. 3
-1
i^3
v(-1)
point of inflection

4. Numbers on a numberline
integers
cos iy
point of inflection
Complex numbers are points in the plane

5. The complex number z representing a+bi.
Complex Exponentiation
i^4
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Affix

6. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Irrational Number
Liouville's Theorem -
How to find any Power
Integers

7. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
Square Root
Absolute Value of a Complex Number
rational

8. x + iy = r(cos? + isin?) = re^(i?)
the distance from z to the origin in the complex plane
Field
Complex Number
Polar Coordinates - z

9. Rotates anticlockwise by p/2
Imaginary Unit
a + bi for some real a and b.
Polar Coordinates - Multiplication by i
Any polynomial O(xn) - (n > 0)

10. When two complex numbers are divided.
Complex Division
rational
Imaginary number
point of inflection

11. A complex number may be taken to the power of another complex number.
multiply the numerator and the denominator by the complex conjugate of the denominator.
Rational Number
Complex Exponentiation
cosh²y - sinh²y

12. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
Integers
z + z*
transcendental
subtracting complex numbers

13. Starts at 1 - does not include 0
has a solution.
cosh²y - sinh²y
The Complex Numbers
natural

14. Written as fractions - terminating + repeating decimals
rational
i^1
z + z*
sin iy

15. 1
x-axis in the complex plane
i^4
point of inflection
Polar Coordinates - Arg(z*)

16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - Multiplication
Imaginary number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Roots of Unity

17. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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18. ½(e^(iz) + e^(-iz))
Real and Imaginary Parts
Polar Coordinates - Multiplication by i
cos iy
cos z

19. All numbers
Complex Conjugate
z + z*
conjugate pairs
complex

20. E^(ln r) e^(i?) e^(2pin)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos iy
De Moivre's Theorem
e^(ln z)

21. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Complex Multiplication
sin z
v(-1)

22. ? = -tan?
Polar Coordinates - Arg(z*)
'i'
radicals
Real and Imaginary Parts

23. Derives z = a+bi
Argand diagram
Euler Formula
a real number: (a + bi)(a - bi) = a² + b²
Absolute Value of a Complex Number

24. A subset within a field.
i^3
Integers
(cos? +isin?)n
Subfield

25. I^2 =
How to find any Power
-1
a + bi for some real a and b.
Field

26. x / r
Polar Coordinates - cos?
How to solve (2i+3)/(9-i)
|z-w|
z - z*

27. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
|z-w|
Liouville's Theorem -
transcendental
Field

28. 1st. Rule of Complex Arithmetic
rational
i^2 = -1
adding complex numbers
Polar Coordinates - Multiplication

29. 5th. Rule of Complex Arithmetic
Polar Coordinates - z?¹
cos iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers

30. A+bi
complex numbers
Complex Number Formula
real
Polar Coordinates - cos?

31. To simplify a complex fraction
Complex Multiplication
Liouville's Theorem -
Absolute Value of a Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.

32. Any number not rational
irrational
Every complex number has the 'Standard Form': a + bi for some real a and b.
Roots of Unity
Square Root

33. V(x² + y²) = |z|
cos z
Polar Coordinates - Multiplication by i
Polar Coordinates - r
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

34. Has exactly n roots by the fundamental theorem of algebra
integers
Any polynomial O(xn) - (n > 0)
Rules of Complex Arithmetic
Subfield

35. I
Complex Multiplication
(a + c) + ( b + d)i
v(-1)
Integers

36. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Complex Number
Irrational Number
rational
complex

37. The square root of -1.
Imaginary Unit
i^2 = -1
Square Root
Rules of Complex Arithmetic

38. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Affix
Every complex number has the 'Standard Form': a + bi for some real a and b.
Liouville's Theorem -
Absolute Value of a Complex Number

39. 1
i^2
Field
cos z
sin z

40. 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
conjugate pairs
imaginary
four different numbers: i - -i - 1 - and -1.

41. y / r
irrational
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^3
Polar Coordinates - sin?

42. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
has a solution.
Irrational Number
Real and Imaginary Parts
ln z

43. Not on the numberline
The Complex Numbers
'i'
subtracting complex numbers
non-integers

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

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45. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
Any polynomial O(xn) - (n > 0)
How to solve (2i+3)/(9-i)
Rational Number
adding complex numbers

46. Root negative - has letter i
imaginary
non-integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers

47. 1
cosh²y - sinh²y
Polar Coordinates - z
i^0
irrational

48. 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
cos iy
the complex numbers
Polar Coordinates - z?¹
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

49. For real a and b - a + bi =
i^3
How to solve (2i+3)/(9-i)
0 if and only if a = b = 0
radicals

50. (e^(iz) - e^(-iz)) / 2i
Complex Number Formula
Polar Coordinates - sin?
Roots of Unity
sin z