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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. I
complex numbers
|z-w|
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^1

2. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Affix
conjugate
Field
Roots of Unity

3. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0
The Complex Numbers

4. Real and imaginary numbers
z1 / z2
complex numbers
i^2 = -1
Complex Exponentiation

5. A subset within a field.
We say that c+di and c-di are complex conjugates.
Subfield
Polar Coordinates - r
Real Numbers

6. A complex number may be taken to the power of another complex number.
Imaginary Unit
Complex Exponentiation
Subfield
Absolute Value of a Complex Number

7. Divide moduli and subtract arguments
v(-1)
How to solve (2i+3)/(9-i)
Polar Coordinates - Division
Euler's Formula

8. 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
rational
Polar Coordinates - Multiplication
Polar Coordinates - sin?

9. A + bi
Polar Coordinates - Multiplication by i
has a solution.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
standard form of complex numbers

10. Imaginary number

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11. When two complex numbers are added together.
Complex Addition
complex numbers
(a + c) + ( b + d)i
z1 / z2

12. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
the complex numbers
standard form of complex numbers
i²
ln z

13. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
i^2
the distance from z to the origin in the complex plane
conjugate
Polar Coordinates - r

14. (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
has a solution.
How to solve (2i+3)/(9-i)
i^2
Polar Coordinates - r

15. A+bi
standard form of complex numbers
Complex Addition
a + bi for some real a and b.
Complex Number Formula

16. 2nd. Rule of Complex Arithmetic

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17. (e^(iz) - e^(-iz)) / 2i
sin z
cosh²y - sinh²y
Affix
Complex Numbers: Add & subtract

18. R^2 = x
Complex Numbers: Multiply
Square Root
Imaginary number
cosh²y - sinh²y

19. Have radical
radicals
i^1
cosh²y - sinh²y
Complex numbers are points in the plane

20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Imaginary Unit
integers
Absolute Value of a Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i

21. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
i^1
How to find any Power
How to multiply complex nubers(2+i)(2i-3)

22. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Roots of Unity
i^3
the complex numbers

23. Starts at 1 - does not include 0
multiply the numerator and the denominator by the complex conjugate of the denominator.
'i'
Euler's Formula
natural

24. (a + bi) = (c + bi) =
How to multiply complex nubers(2+i)(2i-3)
(a + c) + ( b + d)i
ln z
Subfield

25. Root negative - has letter i
Polar Coordinates - Division
imaginary
i^1
standard form of complex numbers

26. V(x² + y²) = |z|
the distance from z to the origin in the complex plane
z + z*
Polar Coordinates - r
cos iy

27. 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.
irrational
How to find any Power
conjugate pairs
transcendental

28. 1
i^2 = -1
The Complex Numbers
Complex numbers are points in the plane
i^4

29. Where the curvature of the graph changes
(cos? +isin?)n
point of inflection
Polar Coordinates - Multiplication
Imaginary Numbers

30. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
0 if and only if a = b = 0
The Complex Numbers
four different numbers: i - -i - 1 - and -1.
point of inflection

31. xpressions such as ``the complex number z'' - and ``the point z'' are now
z - z*
|z-w|
interchangeable
The Complex Numbers

32. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Affix
natural
integers

33. 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.
Integers
Complex Numbers: Multiply
the vector (a -b)
Euler's Formula

34. E ^ (z2 ln z1)
z1 ^ (z2)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
Polar Coordinates - Arg(z*)

35. Multiply moduli and add arguments
non-integers
Rules of Complex Arithmetic
Argand diagram
Polar Coordinates - Multiplication

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
Polar Coordinates - Arg(z*)
Rational Number
Polar Coordinates - r

37. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
How to multiply complex nubers(2+i)(2i-3)
|z| = mod(z)
multiplying complex numbers
De Moivre's Theorem

38. Every complex number has the 'Standard Form':
has a solution.
complex numbers
a + bi for some real a and b.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

39. 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
Affix
Euler's Formula
Polar Coordinates - z

40. I
x-axis in the complex plane
irrational
v(-1)
complex numbers

41. A plot of complex numbers as points.
z + z*
Real Numbers
Complex Multiplication
Argand diagram

42. 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
Complex numbers are points in the plane
adding complex numbers
Real and Imaginary Parts
standard form of complex numbers

43. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Imaginary Numbers
standard form of complex numbers
How to solve (2i+3)/(9-i)

44. Written as fractions - terminating + repeating decimals
i^2 = -1
interchangeable
Complex Conjugate
rational

45. We can also think of the point z= a+ ib as
|z| = mod(z)
(a + c) + ( b + d)i
the vector (a -b)
Polar Coordinates - Arg(z*)

46. I^2 =
Polar Coordinates - Division
Polar Coordinates - Arg(z*)
-1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

47. 1
Polar Coordinates - Multiplication
The Complex Numbers
Absolute Value of a Complex Number
i²

48. The reals are just the
x-axis in the complex plane
Complex Division
the distance from z to the origin in the complex plane
imaginary

49. ½(e^(iz) + e^(-iz))
cos z
z1 / z2
sin iy
Complex Addition

50. For real a and b - a + bi =
0 if and only if a = b = 0
Argand diagram
-1
i²