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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. E^(ln r) e^(i?) e^(2pin)
The Complex Numbers
Euler Formula
i^1
e^(ln z)

2. The modulus of the complex number z= a + ib now can be interpreted as
Every complex number has the 'Standard Form': a + bi for some real a and b.
the distance from z to the origin in the complex plane
integers
complex

3. Every complex number has the 'Standard Form':
Polar Coordinates - Multiplication
i^1
a + bi for some real a and b.
Complex Number Formula

4. Any number not rational
The Complex Numbers
irrational
z1 ^ (z2)
Polar Coordinates - z?¹

5. (e^(iz) - e^(-iz)) / 2i
Any polynomial O(xn) - (n > 0)
0 if and only if a = b = 0
rational
sin z

6. ½(e^(-y) +e^(y)) = cosh y
Any polynomial O(xn) - (n > 0)
cos iy
Polar Coordinates - r
We say that c+di and c-di are complex conjugates.

7. A + bi
Complex numbers are points in the plane
radicals
standard form of complex numbers
Real Numbers

8. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
We say that c+di and c-di are complex conjugates.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler Formula
z - z*

9. I
v(-1)
Complex numbers are points in the plane
real
Complex Conjugate

10. ½(e^(iz) + e^(-iz))
|z-w|
Euler Formula
cos z
Complex Number Formula

11. 1
z1 ^ (z2)
i^4
irrational
i^0

12. 1
multiplying complex numbers
Polar Coordinates - z
i^0
Irrational Number

13. Multiply moduli and add arguments
Polar Coordinates - Multiplication
point of inflection
z + z*
|z-w|

14. When two complex numbers are subtracted from one another.
Square Root
Complex Subtraction
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Liouville's Theorem -

15. z1z2* / |z2|²
conjugate pairs
-1
z1 / z2
i^2

16. The field of all rational and irrational numbers.
Any polynomial O(xn) - (n > 0)
radicals
Real Numbers
Euler Formula

17. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^2
z1 ^ (z2)
standard form of complex numbers

18. (e^(-y) - e^(y)) / 2i = i sinh y
Affix
sin iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division

19. (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
Imaginary Numbers
x-axis in the complex plane
multiplying complex numbers
How to multiply complex nubers(2+i)(2i-3)

20. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
point of inflection
i^2 = -1
the distance from z to the origin in the complex plane

21. 2ib
rational
non-integers
z - z*
cos iy

22. 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
Polar Coordinates - Arg(z*)
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Unit
Real and Imaginary Parts

23. Root negative - has letter i
point of inflection
Polar Coordinates - Arg(z*)
imaginary
-1

24. I
Liouville's Theorem -
a + bi for some real a and b.
-1
i^1

25. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - Arg(z*)
Imaginary number
Affix
Imaginary Numbers

26. (a + bi)(c + bi) =
'i'
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. 3rd. Rule of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the distance from z to the origin in the complex plane
Imaginary number
For real a and b - a + bi = 0 if and only if a = b = 0

28. 2nd. Rule of Complex Arithmetic

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29. V(x² + y²) = |z|
conjugate pairs
Polar Coordinates - r
Polar Coordinates - Division
Liouville's Theorem -

30. Like pi
can't get out of the complex numbers by adding (or subtracting) or multiplying two
subtracting complex numbers
transcendental
Affix

31. A complex number and its conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + c) + ( b + d)i
the vector (a -b)
conjugate pairs

32. 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
rational
Complex Conjugate
|z| = mod(z)

33. A number that cannot be expressed as a fraction for any integer.
transcendental
Irrational Number
conjugate pairs
conjugate

34. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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35. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
cos iy
complex numbers
-1
Field

36. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z?¹
subtracting complex numbers
Absolute Value of a Complex Number

37. Cos n? + i sin n? (for all n integers)
i^3
|z-w|
Imaginary Unit
(cos? +isin?)n

38. R^2 = x
Polar Coordinates - cos?
real
Square Root
Polar Coordinates - z?¹

39. For real a and b - a + bi =
standard form of complex numbers
0 if and only if a = b = 0
x-axis in the complex plane
z - z*

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

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41. A² + b² - real and non negative
real
x-axis in the complex plane
zz*
Complex Number Formula

42. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Subtraction
How to add and subtract complex numbers (2-3i)-(4+6i)
-1
Real Numbers

43. When two complex numbers are multipiled together.
conjugate
Polar Coordinates - Multiplication
Euler's Formula
Complex Multiplication

44. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
-1
ln z
How to find any Power
Complex Division

45. 5th. Rule of Complex Arithmetic
e^(ln z)
i^2 = -1
i^1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

46. When two complex numbers are divided.
Complex Number Formula
cos iy
Complex Division
a + bi for some real a and b.

47. The reals are just the
i^2 = -1
Any polynomial O(xn) - (n > 0)
x-axis in the complex plane
How to solve (2i+3)/(9-i)

48. 3
Polar Coordinates - Multiplication by i
subtracting complex numbers
(cos? +isin?)n
i^3

49. Where the curvature of the graph changes
(a + c) + ( b + d)i
point of inflection
the complex numbers
irrational

50. 4th. Rule of Complex Arithmetic
standard form of complex numbers
subtracting complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
adding complex numbers