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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. 2ib
cosh²y - sinh²y
z - z*
Real and Imaginary Parts
interchangeable

2. 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
Complex Numbers: Add & subtract
imaginary
Polar Coordinates - Multiplication by i
conjugate pairs

3. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
complex
natural
ln z

4. V(x² + y²) = |z|
Any polynomial O(xn) - (n > 0)
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - r
For real a and b - a + bi = 0 if and only if a = b = 0

5. 2nd. Rule of Complex Arithmetic

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6. Written as fractions - terminating + repeating decimals
point of inflection
x-axis in the complex plane
rational
Imaginary Numbers

7. Numbers on a numberline
i^1
radicals
i^2
integers

8. 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.
e^(ln z)
subtracting complex numbers
Field
Complex Numbers: Multiply

9. (e^(-y) - e^(y)) / 2i = i sinh y
Liouville's Theorem -
e^(ln z)
sin iy
Irrational Number

10. All numbers
Integers
Complex Division
Euler's Formula
complex

11. When two complex numbers are added together.
Imaginary number
Polar Coordinates - Multiplication by i
Complex Addition
Polar Coordinates - Division

12. A plot of complex numbers as points.
imaginary
Argand diagram
z + z*
Imaginary Unit

13. (a + bi) = (c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + c) + ( b + d)i
the vector (a -b)
Euler Formula

14. When two complex numbers are multipiled together.
Complex Number
Complex Multiplication
the distance from z to the origin in the complex plane
a + bi for some real a and b.

15. Given (4-2i) the complex conjugate would be (4+2i)
Affix
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - z
Complex Conjugate

16. Equivalent to an Imaginary Unit.
real
z1 / z2
How to find any Power
Imaginary number

17. 2a
the distance from z to the origin in the complex plane
x-axis in the complex plane
z + z*
the complex numbers

18. Every complex number has the 'Standard Form':
a + bi for some real a and b.
conjugate pairs
Any polynomial O(xn) - (n > 0)
sin z

19. A subset within a field.
(cos? +isin?)n
Subfield
zz*
point of inflection

20. No i
Argand diagram
|z| = mod(z)
0 if and only if a = b = 0
real

21. 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
the vector (a -b)
sin z
v(-1)

22. The modulus of the complex number z= a + ib now can be interpreted as
i^4
Rules of Complex Arithmetic
the distance from z to the origin in the complex plane
Polar Coordinates - z

23. y / r
a + bi for some real a and b.
ln z
Polar Coordinates - sin?
interchangeable

24. Have radical
radicals
Euler Formula
Polar Coordinates - Multiplication
(cos? +isin?)n

25. (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
z - z*
i^3
i^1
How to multiply complex nubers(2+i)(2i-3)

26. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Rational Number
Polar Coordinates - z?¹
How to add and subtract complex numbers (2-3i)-(4+6i)
Field

27. 1
i^0
|z-w|
i^2
For real a and b - a + bi = 0 if and only if a = b = 0

28. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin z
a + bi for some real a and b.
Roots of Unity

29. 3
i^3
i^0
transcendental
sin iy

30. 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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31. 4th. Rule of Complex Arithmetic
real
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Irrational Number
interchangeable

32. V(zz*) = v(a² + b²)
i²
complex numbers
conjugate
|z| = mod(z)

33. The product of an imaginary number and its conjugate is
interchangeable
sin z
imaginary
a real number: (a + bi)(a - bi) = a² + b²

34. 1
Polar Coordinates - z?¹
Complex Exponentiation
i²
the distance from z to the origin in the complex plane

35. 1st. Rule of Complex Arithmetic
e^(ln z)
Euler Formula
subtracting complex numbers
i^2 = -1

36. A + bi
subtracting complex numbers
interchangeable
standard form of complex numbers
complex numbers

37. xpressions such as ``the complex number z'' - and ``the point z'' are now
Rational Number
Irrational Number
interchangeable
Integers

38. To simplify the square root of a negative number
Real and Imaginary Parts
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication by i
z - z*

39. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
The Complex Numbers
Polar Coordinates - Division
integers

40. When two complex numbers are subtracted from one another.
the complex numbers
adding complex numbers
Complex Subtraction
|z| = mod(z)

41. Root negative - has letter i
Irrational Number
imaginary
i^0
z + z*

42. I^2 =
Affix
four different numbers: i - -i - 1 - and -1.
-1
z - z*

43. The field of all rational and irrational numbers.
complex
Real Numbers
Square Root
a real number: (a + bi)(a - bi) = a² + b²

44. The square root of -1.
sin z
Imaginary Unit
Polar Coordinates - Arg(z*)
cosh²y - sinh²y

45. 1
Irrational Number
cosh²y - sinh²y
-1
standard form of complex numbers

46. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Division
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rational Number
Roots of Unity

47. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
How to multiply complex nubers(2+i)(2i-3)
Imaginary Unit
conjugate
Polar Coordinates - Division

48. Where the curvature of the graph changes
Complex Number Formula
De Moivre's Theorem
i^1
point of inflection

49. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
four different numbers: i - -i - 1 - and -1.
How to add and subtract complex numbers (2-3i)-(4+6i)
standard form of complex numbers
Absolute Value of a Complex Number

50. 1
non-integers
multiplying complex numbers
standard form of complex numbers
i^2