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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. 1
|z| = mod(z)
multiply the numerator and the denominator by the complex conjugate of the denominator.
cosh²y - sinh²y
Euler's Formula

2. 2ib
irrational
z - z*
i^0
Polar Coordinates - r

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

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4. Root negative - has letter i
imaginary
Rational Number
real
Polar Coordinates - cos?

5. Imaginary number

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6. A complex number and its conjugate
four different numbers: i - -i - 1 - and -1.
conjugate pairs
conjugate
Liouville's Theorem -

7. When two complex numbers are multipiled together.
Subfield
Complex Multiplication
Argand diagram
i^3

8. xpressions such as ``the complex number z'' - and ``the point z'' are now
cos z
sin z
interchangeable
cos iy

9. 2a
Polar Coordinates - Arg(z*)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z + z*
Polar Coordinates - r

10. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
real
i^1

11. ? = -tan?
subtracting complex numbers
Polar Coordinates - Arg(z*)
'i'
Complex numbers are points in the plane

12. We see in this way that the distance between two points z and w in the complex plane is
z + z*
transcendental
(a + c) + ( b + d)i
|z-w|

13. I
v(-1)
e^(ln z)
sin z
Square Root

14. 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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Add & subtract
Imaginary number
ln z

15. I
i^3
radicals
i^1
conjugate pairs

16. 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
i^0
Polar Coordinates - z?¹
subtracting complex numbers
non-integers

17. All numbers
Subfield
complex
a real number: (a + bi)(a - bi) = a² + b²
Affix

18. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
(a + bi) = (c + bi) = (a + c) + ( b + d)i
We say that c+di and c-di are complex conjugates.
i^1
has a solution.

19. 2nd. Rule of Complex Arithmetic

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20. A number that cannot be expressed as a fraction for any integer.
Any polynomial O(xn) - (n > 0)
Irrational Number
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.

21. Real and imaginary numbers
complex numbers
Complex Multiplication
Polar Coordinates - sin?
Imaginary Numbers

22. 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
cos z
Liouville's Theorem -
adding complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

23. Not on the numberline
Imaginary Numbers
Argand diagram
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
non-integers

24. 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
zz*
has a solution.
a + bi for some real a and b.

25. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Multiplication
z - z*
multiplying complex numbers
irrational

26. The modulus of the complex number z= a + ib now can be interpreted as
(cos? +isin?)n
the vector (a -b)
the distance from z to the origin in the complex plane
imaginary

27. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
How to solve (2i+3)/(9-i)
The Complex Numbers
Integers

28. 1
i^4
x-axis in the complex plane
interchangeable
De Moivre's Theorem

29. Have radical
radicals
ln z
Complex Division
Imaginary Numbers

30. 1
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Add & subtract
i^0
i^2 = -1

31. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
i^4
Absolute Value of a Complex Number
complex numbers
Polar Coordinates - z

32. 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.'
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number
i^4
0 if and only if a = b = 0

33. Multiply moduli and add arguments
Integers
Polar Coordinates - Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - cos?

34. To simplify a complex fraction
i^4
standard form of complex numbers
(a + c) + ( b + d)i
multiply the numerator and the denominator by the complex conjugate of the denominator.

35. Every complex number has the 'Standard Form':
z1 ^ (z2)
the complex numbers
Argand diagram
a + bi for some real a and b.

36. The square root of -1.
(cos? +isin?)n
Imaginary Unit
Absolute Value of a Complex Number
i^1

37. 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
We say that c+di and c-di are complex conjugates.
z1 ^ (z2)
Real and Imaginary Parts
transcendental

38. (a + bi) = (c + bi) =
0 if and only if a = b = 0
ln z
(a + c) + ( b + d)i
Real Numbers

39. Where the curvature of the graph changes
We say that c+di and c-di are complex conjugates.
a + bi for some real a and b.
subtracting complex numbers
point of inflection

40. All the powers of i can be written as
Complex Multiplication
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - z
z + z*

41. Starts at 1 - does not include 0
v(-1)
|z-w|
Euler's Formula
natural

42. Written as fractions - terminating + repeating decimals
rational
Polar Coordinates - Arg(z*)
Complex Numbers: Add & subtract
z + z*

43. (a + bi)(c + bi) =
Subfield
Polar Coordinates - Multiplication by i
four different numbers: i - -i - 1 - and -1.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

44. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
irrational
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication by i
standard form of complex numbers

45. ½(e^(-y) +e^(y)) = cosh y
How to find any Power
multiplying complex numbers
We say that c+di and c-di are complex conjugates.
cos iy

46. A+bi
transcendental
Polar Coordinates - Multiplication by i
|z| = mod(z)
Complex Number Formula

47. y / r
i^4
i^2 = -1
Polar Coordinates - sin?
cos iy

48. When two complex numbers are divided.
v(-1)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
Field

49. (e^(-y) - e^(y)) / 2i = i sinh y
e^(ln z)
sin iy
i^2
rational

50. 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.
How to find any Power
interchangeable
i^4
zz*