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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. A number that cannot be expressed as a fraction for any integer.
Irrational Number
interchangeable
How to find any Power
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

2. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - sin?
complex
the distance from z to the origin in the complex plane
complex numbers

3. No i
(a + c) + ( b + d)i
real
natural
How to find any Power

4. (a + bi)(c + bi) =
Subfield
complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
radicals

5. 1
Polar Coordinates - cos?
i^2
cosh²y - sinh²y
standard form of complex numbers

6. x / r
How to find any Power
Polar Coordinates - cos?
cos iy
transcendental

7. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
0 if and only if a = b = 0
Polar Coordinates - z
radicals
ln z

8. 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
sin iy
Rules of Complex Arithmetic
irrational
Complex Number Formula

9. Imaginary number

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10. We can also think of the point z= a+ ib as
the vector (a -b)
Complex Numbers: Add & subtract
Field
Imaginary Numbers

11. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
a + bi for some real a and b.
(cos? +isin?)n
Complex Number
multiplying complex numbers

12. Where the curvature of the graph changes
Subfield
point of inflection
z - z*
integers

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

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14. ½(e^(-y) +e^(y)) = cosh y
cos iy
Rational Number
cosh²y - sinh²y
Euler Formula

15. (e^(-y) - e^(y)) / 2i = i sinh y
Real Numbers
Imaginary number
sin iy
zz*

16. E ^ (z2 ln z1)
Every complex number has the 'Standard Form': a + bi for some real a and b.
standard form of complex numbers
Complex Number
z1 ^ (z2)

17. Not on the numberline
non-integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to multiply complex nubers(2+i)(2i-3)
Integers

18. 1st. Rule of Complex Arithmetic
non-integers
i²
i^2 = -1
i^0

19. 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.
Euler's Formula
We say that c+di and c-di are complex conjugates.
z1 ^ (z2)
Complex numbers are points in the plane

20. Numbers on a numberline
integers
Rational Number
Polar Coordinates - z
zz*

21. (e^(iz) - e^(-iz)) / 2i
sin z
De Moivre's Theorem
natural
complex

22. When two complex numbers are added together.
Complex Addition
0 if and only if a = b = 0
Absolute Value of a Complex Number
i^3

23. 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
Irrational Number
i^4
We say that c+di and c-di are complex conjugates.
'i'

24. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
z1 / z2
Euler's Formula
Imaginary Unit

25. V(x² + y²) = |z|
z + z*
Polar Coordinates - r
a + bi for some real a and b.
Irrational Number

26. xpressions such as ``the complex number z'' - and ``the point z'' are now
complex numbers
subtracting complex numbers
interchangeable
Absolute Value of a Complex Number

27. The reals are just the
radicals
i^2 = -1
z1 ^ (z2)
x-axis in the complex plane

28. Derives z = a+bi
0 if and only if a = b = 0
integers
Euler Formula
natural

29. Have radical
standard form of complex numbers
|z-w|
Imaginary Numbers
radicals

30. 2ib
z - z*
Absolute Value of a Complex Number
x-axis in the complex plane
Imaginary number

31. Written as fractions - terminating + repeating decimals
Polar Coordinates - z
How to find any Power
i²
rational

32. The square root of -1.
Imaginary Unit
Polar Coordinates - cos?
Square Root
Real and Imaginary Parts

33. 1
Rules of Complex Arithmetic
z + z*
cosh²y - sinh²y
0 if and only if a = b = 0

34. (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
i²
Polar Coordinates - z?¹
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to solve (2i+3)/(9-i)

35. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
i^1
(a + c) + ( b + d)i
Square Root
conjugate

36. 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.
De Moivre's Theorem
Polar Coordinates - Arg(z*)
We say that c+di and c-di are complex conjugates.
Complex Numbers: Multiply

37. 1
Complex Conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Multiplication
i²

38. A+bi
adding complex numbers
Complex Number Formula
i^0
e^(ln z)

39. Cos n? + i sin n? (for all n integers)
cosh²y - sinh²y
x-axis in the complex plane
(cos? +isin?)n
(a + c) + ( b + d)i

40. For real a and b - a + bi =
(a + c) + ( b + d)i
Complex Multiplication
0 if and only if a = b = 0
Polar Coordinates - cos?

41. x + iy = r(cos? + isin?) = re^(i?)
irrational
Euler Formula
v(-1)
Polar Coordinates - z

42. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex numbers are points in the plane
Integers
Real Numbers
Complex Exponentiation

43. 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
Polar Coordinates - Arg(z*)
cos z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers

44. z1z2* / |z2|²
four different numbers: i - -i - 1 - and -1.
0 if and only if a = b = 0
For real a and b - a + bi = 0 if and only if a = b = 0
z1 / z2

45. V(zz*) = v(a² + b²)
|z| = mod(z)
Complex Number Formula
Roots of Unity
multiply the numerator and the denominator by the complex conjugate of the denominator.

46. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
cos z
Field
interchangeable
z - z*

47. 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
Irrational Number
Complex Numbers: Add & subtract
non-integers
Polar Coordinates - z

48. A complex number may be taken to the power of another complex number.
De Moivre's Theorem
Complex Exponentiation
Complex Number
has a solution.

49. (a + bi) = (c + bi) =
Complex Numbers: Add & subtract
integers
(a + c) + ( b + d)i
Rules of Complex Arithmetic

50. A subset within a field.
Imaginary number
How to find any Power
We say that c+di and c-di are complex conjugates.
Subfield