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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. ? = -tan?
Polar Coordinates - Arg(z*)
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field
i^2 = -1

2. V(zz*) = v(a² + b²)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)
Polar Coordinates - Multiplication by i
Complex Subtraction

3. The complex number z representing a+bi.
ln z
(a + c) + ( b + d)i
Complex Numbers: Add & subtract
Affix

4. A+bi
z1 / z2
Complex Number Formula
Polar Coordinates - cos?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

5. (a + bi)(c + bi) =
Polar Coordinates - Multiplication by i
Argand diagram
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
zz*

6. In this amazing number field every algebraic equation in z with complex coefficients
non-integers
z + z*
has a solution.
Subfield

7. Has exactly n roots by the fundamental theorem of algebra
-1
i^0
Integers
Any polynomial O(xn) - (n > 0)

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

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9. Multiply moduli and add arguments
Complex Number
Polar Coordinates - Multiplication
0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)

10. Imaginary number

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11. R^2 = x
Square Root
the vector (a -b)
adding complex numbers
a real number: (a + bi)(a - bi) = a² + b²

12. ½(e^(-y) +e^(y)) = cosh y
Imaginary Unit
Every complex number has the 'Standard Form': a + bi for some real a and b.
cos iy
point of inflection

13. (e^(-y) - e^(y)) / 2i = i sinh y
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - z?¹
(a + c) + ( b + d)i
sin iy

14. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
point of inflection
How to add and subtract complex numbers (2-3i)-(4+6i)

15. (e^(iz) - e^(-iz)) / 2i
complex
Imaginary Unit
Polar Coordinates - z
sin z

16. We see in this way that the distance between two points z and w in the complex plane is
Complex Numbers: Multiply
z1 ^ (z2)
Complex Conjugate
|z-w|

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

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19. E ^ (z2 ln z1)
complex numbers
z1 ^ (z2)
cos iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

20. 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.'
Subfield
Rules of Complex Arithmetic
|z-w|
Complex Number

21. ½(e^(iz) + e^(-iz))
real
cos z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

22. 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
Rules of Complex Arithmetic
Complex Numbers: Multiply
Real Numbers
Polar Coordinates - sin?

23. A plot of complex numbers as points.
complex
standard form of complex numbers
Real Numbers
Argand diagram

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
the distance from z to the origin in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
-1

25. y / r
conjugate pairs
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler Formula
Polar Coordinates - sin?

26. All numbers
complex
Liouville's Theorem -
standard form of complex numbers
Complex numbers are points in the plane

27. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
a real number: (a + bi)(a - bi) = a² + b²
Argand diagram
Absolute Value of a Complex Number
rational

28. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division
Square Root
can't get out of the complex numbers by adding (or subtracting) or multiplying two

29. Cos n? + i sin n? (for all n integers)
complex
z + z*
(cos? +isin?)n
Square Root

30. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Addition
has a solution.
Integers
non-integers

31. Have radical
z + z*
multiplying complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
radicals

32. 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
z1 ^ (z2)
conjugate
Square Root
the complex numbers

33. When two complex numbers are divided.
Complex Division
point of inflection
i²
How to solve (2i+3)/(9-i)

34. 1
z1 ^ (z2)
x-axis in the complex plane
cosh²y - sinh²y
a + bi for some real a and b.

35. A complex number and its conjugate
conjugate pairs
The Complex Numbers
Rational Number
Imaginary number

36. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Affix
Polar Coordinates - Division
The Complex Numbers

37. When two complex numbers are multipiled together.
Complex Multiplication
Absolute Value of a Complex Number
Polar Coordinates - Division
Real Numbers

38. No i
interchangeable
i^2
-1
real

39. 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
standard form of complex numbers
'i'
i^4

40. Divide moduli and subtract arguments
z + z*
natural
Polar Coordinates - Division
Irrational Number

41. 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.
Polar Coordinates - Multiplication
Complex Numbers: Multiply
Polar Coordinates - sin?
Polar Coordinates - Arg(z*)

42. Root negative - has letter i
i^0
Polar Coordinates - z
imaginary
|z| = mod(z)

43. Given (4-2i) the complex conjugate would be (4+2i)
cos z
radicals
point of inflection
Complex Conjugate

44. A² + b² - real and non negative
Complex Addition
e^(ln z)
zz*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

45. x / r
Polar Coordinates - cos?
conjugate pairs
Irrational Number
Rules of Complex Arithmetic

46. When two complex numbers are subtracted from one another.
|z| = mod(z)
-1
i^0
Complex Subtraction

47. I
Square Root
v(-1)
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Multiply

48. 1
Euler's Formula
i²
Polar Coordinates - z
Roots of Unity

49. Equivalent to an Imaginary Unit.
adding complex numbers
Imaginary number
Polar Coordinates - z
a real number: (a + bi)(a - bi) = a² + b²

50. The product of an imaginary number and its conjugate is
Rules of Complex Arithmetic
adding complex numbers
Irrational Number
a real number: (a + bi)(a - bi) = a² + b²