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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. Every complex number has the 'Standard Form':
a + bi for some real a and b.
i^4
point of inflection
Real Numbers

2. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Imaginary Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Division

3. x / r
Polar Coordinates - cos?
Complex numbers are points in the plane
imaginary
Roots of Unity

4. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Multiplication by i
ln z
|z-w|
Polar Coordinates - z

5. The complex number z representing a+bi.
Affix
i^2
'i'
point of inflection

6. Divide moduli and subtract arguments
Roots of Unity
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Division
integers

7. For real a and b - a + bi =
How to solve (2i+3)/(9-i)
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
the complex numbers

8. When two complex numbers are multipiled together.
Polar Coordinates - Arg(z*)
Complex Multiplication
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)

9. The square root of -1.
Imaginary Unit
integers
Complex Number
Complex Conjugate

10. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
z1 / z2
i^2 = -1
cos z

11. 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
conjugate
z1 / z2
i^4

12. 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.
non-integers
Polar Coordinates - r
i^2 = -1
Complex Numbers: Multiply

13. y / r
Imaginary Numbers
Polar Coordinates - sin?
How to find any Power
Complex numbers are points in the plane

14. 1
cos iy
i^0
(cos? +isin?)n
Polar Coordinates - z?¹

15. Where the curvature of the graph changes
sin iy
i^3
point of inflection
How to multiply complex nubers(2+i)(2i-3)

16. ½(e^(iz) + e^(-iz))
cos z
Polar Coordinates - Multiplication
Irrational Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

17. E ^ (z2 ln z1)
Polar Coordinates - sin?
z1 ^ (z2)
i^4
interchangeable

18. The field of all rational and irrational numbers.
sin z
Real Numbers
the complex numbers
z1 ^ (z2)

19. 5th. Rule of Complex Arithmetic
Rules of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)

20. A complex number and its conjugate
Imaginary number
conjugate pairs
ln z
Rules of Complex Arithmetic

21. I
v(-1)
four different numbers: i - -i - 1 - and -1.
Complex Number Formula
radicals

22. Multiply moduli and add arguments
Complex Numbers: Multiply
x-axis in the complex plane
v(-1)
Polar Coordinates - Multiplication

23. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
|z| = mod(z)
natural
Polar Coordinates - z?¹
How to find any Power

24. ? = -tan?
e^(ln z)
cosh²y - sinh²y
Polar Coordinates - Arg(z*)
imaginary

25. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^1
Polar Coordinates - Multiplication
x-axis in the complex plane

26. Imaginary number

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27. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Polar Coordinates - sin?
How to solve (2i+3)/(9-i)
non-integers

28. 1st. Rule of Complex Arithmetic
How to find any Power
can't get out of the complex numbers by adding (or subtracting) or multiplying two
rational
i^2 = -1

29. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Argand diagram
point of inflection
Absolute Value of a Complex Number
the vector (a -b)

30. The reals are just the
has a solution.
x-axis in the complex plane
Polar Coordinates - Multiplication
a + bi for some real a and b.

31. Real and imaginary numbers
i^3
complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate pairs

32. (e^(iz) - e^(-iz)) / 2i
i^4
Complex Division
sin z
complex

33. I = imaginary unit - i² = -1 or i = v-1
For real a and b - a + bi = 0 if and only if a = b = 0
i^0
Complex Number
Imaginary Numbers

34. z1z2* / |z2|²
z1 / z2
Polar Coordinates - Division
Complex Exponentiation
Field

35. 3
Any polynomial O(xn) - (n > 0)
imaginary
i^3
'i'

36. Numbers on a numberline
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Conjugate
Field
integers

37. Written as fractions - terminating + repeating decimals
interchangeable
|z-w|
transcendental
rational

38. A number that cannot be expressed as a fraction for any integer.
Any polynomial O(xn) - (n > 0)
z1 / z2
adding complex numbers
Irrational Number

39. I^2 =
x-axis in the complex plane
cosh²y - sinh²y
-1
Every complex number has the 'Standard Form': a + bi for some real a and b.

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

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41. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Polar Coordinates - Division
ln z
can't get out of the complex numbers by adding (or subtracting) or multiplying two

42. A + bi
Complex Multiplication
standard form of complex numbers
zz*
Euler's Formula

43. When two complex numbers are subtracted from one another.
Polar Coordinates - Multiplication
Complex Subtraction
integers
Polar Coordinates - Arg(z*)

44. V(x² + y²) = |z|
Polar Coordinates - r
Polar Coordinates - cos?
Polar Coordinates - sin?
transcendental

45. When two complex numbers are divided.
Complex Division
subtracting complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
real

46. 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
Irrational Number
Real and Imaginary Parts
Absolute Value of a Complex Number
conjugate

47. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
x-axis in the complex plane
sin iy
ln z
point of inflection

48. A plot of complex numbers as points.
Argand diagram
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Subtraction
Absolute Value of a Complex Number

49. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary number
Imaginary Numbers

50. 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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