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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
real
How to find any Power
Integers
Complex Numbers: Add & subtract

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

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3. ½(e^(iz) + e^(-iz))
natural
cos z
'i'
The Complex Numbers

4. For real a and b - a + bi =
Complex Numbers: Add & subtract
For real a and b - a + bi = 0 if and only if a = b = 0
0 if and only if a = b = 0
'i'

5. 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
z1 ^ (z2)
Liouville's Theorem -
Subfield

6. The product of an imaginary number and its conjugate is
Complex Number Formula
a real number: (a + bi)(a - bi) = a² + b²
Subfield
cos z

7. All numbers
complex
a real number: (a + bi)(a - bi) = a² + b²
Argand diagram
(cos? +isin?)n

8. ? = -tan?
Polar Coordinates - sin?
Polar Coordinates - Arg(z*)
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

9. R?¹(cos? - isin?)
Roots of Unity
Complex Numbers: Add & subtract
the complex numbers
Polar Coordinates - z?¹

10. Starts at 1 - does not include 0
Complex Numbers: Multiply
natural
Complex Exponentiation
For real a and b - a + bi = 0 if and only if a = b = 0

11. x + iy = r(cos? + isin?) = re^(i?)
natural
Polar Coordinates - z
Polar Coordinates - Division
Complex Numbers: Add & subtract

12. All the powers of i can be written as
a real number: (a + bi)(a - bi) = a² + b²
real
i^3
four different numbers: i - -i - 1 - and -1.

13. E ^ (z2 ln z1)
Imaginary number
Euler's Formula
i^3
z1 ^ (z2)

14. 1
x-axis in the complex plane
i^0
imaginary
i²

15. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z
Polar Coordinates - Multiplication
non-integers

16. A subset within a field.
(cos? +isin?)n
Complex numbers are points in the plane
Subfield
Complex Number Formula

17. Divide moduli and subtract arguments
complex
Complex Division
cos z
Polar Coordinates - Division

18. 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
Real and Imaginary Parts
i²
x-axis in the complex plane
cosh²y - sinh²y

19. R^2 = x
Square Root
Every complex number has the 'Standard Form': a + bi for some real a and b.
For real a and b - a + bi = 0 if and only if a = b = 0
Field

20. I
radicals
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Any polynomial O(xn) - (n > 0)
v(-1)

21. (e^(iz) - e^(-iz)) / 2i
Affix
Real and Imaginary Parts
sin z
Square Root

22. Multiply moduli and add arguments
Real Numbers
Polar Coordinates - Multiplication
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^4

23. Derives z = a+bi
imaginary
Euler Formula
Polar Coordinates - z
i^3

24. 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
Complex Subtraction
conjugate pairs
cos z
Rules of Complex Arithmetic

25. Numbers on a numberline
Square Root
integers
How to solve (2i+3)/(9-i)
Any polynomial O(xn) - (n > 0)

26. 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
subtracting complex numbers
the vector (a -b)
Argand diagram
(a + c) + ( b + d)i

27. When two complex numbers are subtracted from one another.
Complex Subtraction
the vector (a -b)
Euler Formula
cos iy

28. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
Imaginary number
v(-1)
How to find any Power
has a solution.

29. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
The Complex Numbers
Polar Coordinates - z?¹
Field

30. Not on the numberline
non-integers
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
|z| = mod(z)

31. 2a
Polar Coordinates - sin?
radicals
The Complex Numbers
z + z*

32. Where the curvature of the graph changes
(a + c) + ( b + d)i
i^3
point of inflection
Subfield

33. When two complex numbers are divided.
Subfield
standard form of complex numbers
Complex Multiplication
Complex Division

34. 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
Complex Numbers: Add & subtract
z + z*
Every complex number has the 'Standard Form': a + bi for some real a and b.
Any polynomial O(xn) - (n > 0)

35. 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.
How to find any Power
Complex Numbers: Multiply
i^3
Real Numbers

36. Any number not rational
z + z*
Imaginary Unit
irrational
the complex numbers

37. V(x² + y²) = |z|
cos z
x-axis in the complex plane
-1
Polar Coordinates - r

38. 1
cos iy
Polar Coordinates - Arg(z*)
cos z
i²

39. x / r
zz*
cos z
Polar Coordinates - cos?
We say that c+di and c-di are complex conjugates.

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

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41. 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
How to multiply complex nubers(2+i)(2i-3)
Any polynomial O(xn) - (n > 0)
the complex numbers
Polar Coordinates - Arg(z*)

42. A complex number and its conjugate
sin iy
conjugate pairs
adding complex numbers
Rational Number

43. A complex number may be taken to the power of another complex number.
multiplying complex numbers
four different numbers: i - -i - 1 - and -1.
Complex Exponentiation
ln z

44. (a + bi) = (c + bi) =
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + c) + ( b + d)i
i^1
ln z

45. Have radical
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
radicals
Polar Coordinates - z

46. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Complex numbers are points in the plane
i^0
ln z
Polar Coordinates - z?¹

47. 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
The Complex Numbers
Polar Coordinates - Multiplication by i
Subfield
adding complex numbers

48. Given (4-2i) the complex conjugate would be (4+2i)
Complex numbers are points in the plane
Polar Coordinates - Multiplication by i
Complex Conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.

49. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex numbers are points in the plane
Absolute Value of a Complex Number
multiplying complex numbers
Polar Coordinates - Multiplication

50. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
i^2
Affix
Rational Number