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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
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the distance from z to the origin in the complex plane
Complex Numbers: Add & subtract

2. 2ib
multiply the numerator and the denominator by the complex conjugate of the denominator.
z - z*
(a + bi) = (c + bi) = (a + c) + ( b + d)i
|z| = mod(z)

3. (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 Addition
How to solve (2i+3)/(9-i)
Polar Coordinates - Division
point of inflection

4. 1
Real and Imaginary Parts
transcendental
ln z
cosh²y - sinh²y

5. 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
-1
Complex Numbers: Add & subtract
subtracting complex numbers
Rules of Complex Arithmetic

6. y / r
multiplying complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
'i'
Polar Coordinates - sin?

7. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Complex Addition
transcendental
Polar Coordinates - Multiplication by i

8. 1
i^2
Complex Number
irrational
complex

9. For real a and b - a + bi =
-1
Complex Addition
0 if and only if a = b = 0
four different numbers: i - -i - 1 - and -1.

10. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

11. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
has a solution.
Complex Division
Irrational Number

12. E ^ (z2 ln z1)
real
z1 ^ (z2)
Complex Multiplication
rational

13. No i
sin z
v(-1)
real
Complex Subtraction

14. A subset within a field.
How to add and subtract complex numbers (2-3i)-(4+6i)
Subfield
How to find any Power
|z-w|

15. All the powers of i can be written as
conjugate pairs
point of inflection
Polar Coordinates - Arg(z*)
four different numbers: i - -i - 1 - and -1.

16. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Number
zz*
conjugate
non-integers

17. 1
Roots of Unity
i^0
Liouville's Theorem -
Euler's Formula

18. Where the curvature of the graph changes
Rules of Complex Arithmetic
i^3
point of inflection
i²

19. Given (4-2i) the complex conjugate would be (4+2i)
We say that c+di and c-di are complex conjugates.
Complex Conjugate
multiplying complex numbers
Polar Coordinates - Arg(z*)

20. 2a
conjugate
z + z*
For real a and b - a + bi = 0 if and only if a = b = 0
i^3

21. xpressions such as ``the complex number z'' - and ``the point z'' are now
De Moivre's Theorem
sin iy
Liouville's Theorem -
interchangeable

22. Real and imaginary numbers
cos iy
integers
-1
complex numbers

23. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
four different numbers: i - -i - 1 - and -1.
cos z
Field

24. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Division
Polar Coordinates - sin?
Polar Coordinates - z?¹
sin z

25. V(x² + y²) = |z|
|z| = mod(z)
Polar Coordinates - r
Roots of Unity
'i'

26. Has exactly n roots by the fundamental theorem of algebra
complex
Any polynomial O(xn) - (n > 0)
conjugate
Roots of Unity

27. Multiply moduli and add arguments
Polar Coordinates - Multiplication
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to find any Power
i^3

28. 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
the complex numbers
e^(ln z)
x-axis in the complex plane
Liouville's Theorem -

29. Starts at 1 - does not include 0
integers
natural
Polar Coordinates - z?¹
imaginary

30. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
For real a and b - a + bi = 0 if and only if a = b = 0
multiplying complex numbers
How to solve (2i+3)/(9-i)
z1 / z2

31. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
the vector (a -b)
Absolute Value of a Complex Number
Polar Coordinates - Multiplication
adding complex numbers

32. 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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33. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
standard form of complex numbers
ln z
i^2 = -1
-1

34. When two complex numbers are added together.
Complex Division
Complex Addition
rational
i^3

35. 1
point of inflection
Polar Coordinates - Multiplication
i^4
Euler's Formula

36. The modulus of the complex number z= a + ib now can be interpreted as
e^(ln z)
cosh²y - sinh²y
the distance from z to the origin in the complex plane
interchangeable

37. x + iy = r(cos? + isin?) = re^(i?)
multiplying complex numbers
Euler's Formula
Polar Coordinates - z
0 if and only if a = b = 0

38. Written as fractions - terminating + repeating decimals
four different numbers: i - -i - 1 - and -1.
i^0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
rational

39. Root negative - has letter i
z1 ^ (z2)
v(-1)
imaginary
rational

40. When two complex numbers are multipiled together.
Complex Number Formula
Complex Multiplication
(a + c) + ( b + d)i
Complex Number

41. z1z2* / |z2|²
i^4
Field
z1 / z2
Complex Subtraction

42. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
zz*
i^4
Liouville's Theorem -

43. A complex number may be taken to the power of another complex number.
Subfield
Complex Exponentiation
Real Numbers
natural

44. We can also think of the point z= a+ ib as
the vector (a -b)
|z| = mod(z)
Polar Coordinates - cos?
Polar Coordinates - z?¹

45. 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
imaginary
'i'
z1 ^ (z2)
subtracting complex numbers

46. 1
Complex Exponentiation
cos iy
How to multiply complex nubers(2+i)(2i-3)
i²

47. When two complex numbers are subtracted from one another.
Real and Imaginary Parts
How to multiply complex nubers(2+i)(2i-3)
irrational
Complex Subtraction

48. I
i^2 = -1
-1
Polar Coordinates - sin?
i^1

49. When two complex numbers are divided.
e^(ln z)
Roots of Unity
Complex Division
The Complex Numbers

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

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