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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. 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.'
rational
integers
radicals
Complex Number

2. To simplify the square root of a negative number
Real and Imaginary Parts
natural
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Every complex number has the 'Standard Form': a + bi for some real a and b.

3. Rotates anticlockwise by p/2
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication by i
Complex numbers are points in the plane
How to find any Power

4. 1
complex
i²
Complex numbers are points in the plane
Complex Number

5. All the powers of i can be written as
Complex numbers are points in the plane
|z| = mod(z)
four different numbers: i - -i - 1 - and -1.
Imaginary number

6. 2ib
i^1
zz*
Subfield
z - z*

7. I
Imaginary Numbers
Rules of Complex Arithmetic
v(-1)
Complex Number Formula

8. Starts at 1 - does not include 0
natural
sin z
Complex Number Formula
Affix

9. A subset within a field.
We say that c+di and c-di are complex conjugates.
Subfield
the complex numbers
point of inflection

10. Real and imaginary numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^3
z1 / z2
complex numbers

11. A plot of complex numbers as points.
Complex Number
Argand diagram
i²
Polar Coordinates - z

12. R?¹(cos? - isin?)
Polar Coordinates - z?¹
ln z
z + z*
irrational

13. 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
Complex numbers are points in the plane
Real and Imaginary Parts
How to multiply complex nubers(2+i)(2i-3)
We say that c+di and c-di are complex conjugates.

14. When two complex numbers are added together.
Complex Exponentiation
sin z
Complex Addition
point of inflection

15. Where the curvature of the graph changes
point of inflection
has a solution.
Complex Numbers: Add & subtract
non-integers

16. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
z - z*
Integers
Complex Multiplication
complex numbers

17. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
real
-1
z1 ^ (z2)

18. Equivalent to an Imaginary Unit.
Imaginary Numbers
Imaginary number
De Moivre's Theorem
Complex Multiplication

19. All numbers
e^(ln z)
complex
x-axis in the complex plane
i^2 = -1

20. When two complex numbers are subtracted from one another.
cosh²y - sinh²y
cos iy
-1
Complex Subtraction

21. y / r
Affix
Field
Polar Coordinates - sin?
Complex numbers are points in the plane

22. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Square Root
Imaginary number
point of inflection

23. ½(e^(-y) +e^(y)) = cosh y
a + bi for some real a and b.
cos iy
Integers
rational

24. 1st. Rule of Complex Arithmetic
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Division
i^2 = -1
De Moivre's Theorem

25. A + bi
irrational
standard form of complex numbers
Complex Addition
Liouville's Theorem -

26. Have radical
We say that c+di and c-di are complex conjugates.
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Subtraction
radicals

27. The complex number z representing a+bi.
Affix
imaginary
cos z
cos iy

28. Root negative - has letter i
sin iy
How to find any Power
imaginary
complex

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 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
subtracting complex numbers
|z-w|
Polar Coordinates - sin?
i^2

31. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Euler Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
the vector (a -b)
imaginary

32. I^2 =
the vector (a -b)
irrational
-1
Polar Coordinates - Multiplication

33. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Rational Number
the vector (a -b)
Complex Exponentiation

34. To simplify a complex fraction
(cos? +isin?)n
cos iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers

35. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Integers
Subfield
(cos? +isin?)n

36. 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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37. ? = -tan?
Irrational Number
cos z
Polar Coordinates - Arg(z*)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

38. Any number not rational
Polar Coordinates - z?¹
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Subtraction
irrational

39. (e^(iz) - e^(-iz)) / 2i
(cos? +isin?)n
Complex Number Formula
sin z
We say that c+di and c-di are complex conjugates.

40. 1
Rules of Complex Arithmetic
Field
'i'
i^4

41. 1
Polar Coordinates - Multiplication
Complex Conjugate
rational
cosh²y - sinh²y

42. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
x-axis in the complex plane
Polar Coordinates - Division
0 if and only if a = b = 0

43. Written as fractions - terminating + repeating decimals
transcendental
rational
real
conjugate

44. I
Polar Coordinates - Multiplication by i
conjugate pairs
i^1
Polar Coordinates - z

45. 1
'i'
i^4
i^2
Complex Multiplication

46. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Euler Formula
zz*
Real Numbers
Roots of Unity

47. The modulus of the complex number z= a + ib now can be interpreted as
|z-w|
radicals
the distance from z to the origin in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.

48. Derives z = a+bi
|z-w|
(a + c) + ( b + d)i
Euler Formula
sin iy

49. Cos n? + i sin n? (for all n integers)
Complex Numbers: Multiply
v(-1)
(cos? +isin?)n
point of inflection

50. (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
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Add & subtract
Real Numbers
complex numbers