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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. When two complex numbers are divided.
Polar Coordinates - Arg(z*)
Complex Division
i²
i^1

2. 3rd. Rule of Complex Arithmetic
i^0
Argand diagram
For real a and b - a + bi = 0 if and only if a = b = 0
Absolute Value of a Complex Number

3. Like pi
Complex Numbers: Add & subtract
transcendental
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex numbers are points in the plane

4. A complex number may be taken to the power of another complex number.
Rules of Complex Arithmetic
conjugate
The Complex Numbers
Complex Exponentiation

5. Any number not rational
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z
Polar Coordinates - cos?
irrational

6. (a + bi)(c + bi) =
sin z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
irrational
Polar Coordinates - z?¹

7. 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.
sin iy
complex
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply

8. 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
Polar Coordinates - Division
Complex Subtraction
Complex Numbers: Add & subtract
Complex Multiplication

9. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
complex numbers
How to solve (2i+3)/(9-i)
Complex Numbers: Multiply
Argand diagram

10. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
ln z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
e^(ln z)

11. 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
Liouville's Theorem -
Any polynomial O(xn) - (n > 0)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

12. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
De Moivre's Theorem
How to add and subtract complex numbers (2-3i)-(4+6i)
natural
(cos? +isin?)n

13. I = imaginary unit - i² = -1 or i = v-1
0 if and only if a = b = 0
the distance from z to the origin in the complex plane
Imaginary Numbers
standard form of complex numbers

14. V(zz*) = v(a² + b²)
real
Euler's Formula
non-integers
|z| = mod(z)

15. 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.
'i'
complex numbers
Complex numbers are points in the plane
Imaginary Numbers

16. A complex number and its conjugate
a real number: (a + bi)(a - bi) = a² + b²
Complex Addition
z - z*
conjugate pairs

17. 4th. Rule of Complex Arithmetic
i^2 = -1
the complex numbers
z + z*
(a + bi) = (c + bi) = (a + c) + ( b + d)i

18. A plot of complex numbers as points.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Rules of Complex Arithmetic
Argand diagram
|z-w|

19. y / r
Polar Coordinates - sin?
Complex Numbers: Add & subtract
subtracting complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
i²
Absolute Value of a Complex Number
cos z
Polar Coordinates - cos?

21. 2ib
sin iy
z - z*
irrational
For real a and b - a + bi = 0 if and only if a = b = 0

22. I^2 =
-1
cosh²y - sinh²y
Complex Numbers: Add & subtract
complex numbers

23. Derives z = a+bi
Euler Formula
v(-1)
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i

24. z1z2* / |z2|²
Integers
the complex numbers
z1 / z2
interchangeable

25. When two complex numbers are subtracted from one another.
non-integers
Complex Subtraction
the vector (a -b)
point of inflection

26. 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
Liouville's Theorem -
adding complex numbers
Rules of Complex Arithmetic
Complex Numbers: Add & subtract

27. (e^(iz) - e^(-iz)) / 2i
Complex Number Formula
Imaginary Unit
sin z
Polar Coordinates - cos?

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.
Affix
adding complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to find any Power

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

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30. When two complex numbers are added together.
the complex numbers
imaginary
adding complex numbers
Complex Addition

31. 1st. Rule of Complex Arithmetic
(a + c) + ( b + d)i
i^2 = -1
De Moivre's Theorem
conjugate pairs

32. E ^ (z2 ln z1)
irrational
z1 ^ (z2)
Absolute Value of a Complex Number
z - z*

33. 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
Polar Coordinates - Multiplication by i
Complex Numbers: Add & subtract
the complex numbers
Complex Exponentiation

34. R?¹(cos? - isin?)
Affix
Polar Coordinates - z?¹
0 if and only if a = b = 0
has a solution.

35. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex Multiplication
interchangeable
transcendental

36. ? = -tan?
Polar Coordinates - Arg(z*)
transcendental
the vector (a -b)
z - z*

37. When two complex numbers are multipiled together.
irrational
Complex Multiplication
imaginary
Liouville's Theorem -

38. The complex number z representing a+bi.
standard form of complex numbers
Polar Coordinates - Arg(z*)
Affix
imaginary

39. xpressions such as ``the complex number z'' - and ``the point z'' are now
i^2 = -1
interchangeable
conjugate pairs
Liouville's Theorem -

40. Cos n? + i sin n? (for all n integers)
z1 ^ (z2)
Real and Imaginary Parts
sin z
(cos? +isin?)n

41. 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.

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42. Not on the numberline
Rules of Complex Arithmetic
non-integers
|z| = mod(z)
How to add and subtract complex numbers (2-3i)-(4+6i)

43. R^2 = x
has a solution.
Square Root
'i'
Polar Coordinates - Multiplication

44. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
adding complex numbers
De Moivre's Theorem
zz*

45. For real a and b - a + bi =
0 if and only if a = b = 0
i^3
x-axis in the complex plane
Imaginary number

46. All the powers of i can be written as
sin iy
The Complex Numbers
the vector (a -b)
four different numbers: i - -i - 1 - and -1.

47. Real and imaginary numbers
De Moivre's Theorem
complex numbers
cos z
Complex Division

48. A number that cannot be expressed as a fraction for any integer.
zz*
Irrational Number
multiplying complex numbers
Polar Coordinates - z

49. Have radical
radicals
Real and Imaginary Parts
Rules of Complex Arithmetic
Complex Numbers: Add & subtract

50. 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
a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.
Argand diagram
subtracting complex numbers