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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. R?¹(cos? - isin?)
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z?¹
De Moivre's Theorem
Rational Number

2. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
natural
i^4
Absolute Value of a Complex Number

3. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
cosh²y - sinh²y
z1 / z2
Polar Coordinates - Arg(z*)

4. Given (4-2i) the complex conjugate would be (4+2i)
Complex Subtraction
Complex Conjugate
point of inflection
z + z*

5. (a + bi) = (c + bi) =
Any polynomial O(xn) - (n > 0)
Complex Multiplication
(a + c) + ( b + d)i
Polar Coordinates - sin?

6. Where the curvature of the graph changes
i^2 = -1
De Moivre's Theorem
four different numbers: i - -i - 1 - and -1.
point of inflection

7. 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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8. When two complex numbers are multipiled together.
Euler Formula
Complex Multiplication
We say that c+di and c-di are complex conjugates.
Complex Number

9. (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)
cos z
Polar Coordinates - Multiplication
Field

10. 3
Complex Multiplication
(cos? +isin?)n
i^3
Square Root

11. V(zz*) = v(a² + b²)
|z| = mod(z)
radicals
sin iy
point of inflection

12. I
Euler's Formula
v(-1)
Polar Coordinates - Multiplication by i
zz*

13. When two complex numbers are divided.
Complex Division
i^1
The Complex Numbers
Integers

14. Not on the numberline
Euler Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
non-integers
ln z

15. x + iy = r(cos? + isin?) = re^(i?)
irrational
|z-w|
Polar Coordinates - z
has a solution.

16. ½(e^(iz) + e^(-iz))
i^1
cos z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex

17. The complex number z representing a+bi.
Affix
De Moivre's Theorem
integers
Liouville's Theorem -

18. A plot of complex numbers as points.
The Complex Numbers
z1 / z2
Argand diagram
Real Numbers

19. R^2 = x
Square Root
Polar Coordinates - sin?
natural
-1

20. 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.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate pairs

21. Starts at 1 - does not include 0
standard form of complex numbers
zz*
For real a and b - a + bi = 0 if and only if a = b = 0
natural

22. E ^ (z2 ln z1)
For real a and b - a + bi = 0 if and only if a = b = 0
Liouville's Theorem -
z1 ^ (z2)
Imaginary Unit

23. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex Number Formula
v(-1)
the complex numbers

24. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
For real a and b - a + bi = 0 if and only if a = b = 0
the distance from z to the origin in the complex plane
0 if and only if a = b = 0

25. 1
Euler Formula
cosh²y - sinh²y
z - z*
Argand diagram

26. 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.
Complex Numbers: Multiply
cos z
i^4
i^3

27. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
cosh²y - sinh²y
Field
integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

28. 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 Multiplication
Complex Numbers: Add & subtract
i^2
i^0

29. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
transcendental
standard form of complex numbers

30. 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
Complex Conjugate
Rules of Complex Arithmetic
a real number: (a + bi)(a - bi) = a² + b²
Subfield

31. A number that can be expressed as a fraction p/q where q is not equal to 0.
cosh²y - sinh²y
|z| = mod(z)
Polar Coordinates - Multiplication
Rational Number

32. ½(e^(-y) +e^(y)) = cosh y
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos iy
radicals
irrational

33. We can also think of the point z= a+ ib as
interchangeable
the vector (a -b)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex

34. Equivalent to an Imaginary Unit.
Square Root
Complex Conjugate
zz*
Imaginary number

35. 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
-1
How to multiply complex nubers(2+i)(2i-3)
Euler Formula
subtracting complex numbers

36. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
z1 / z2
cos iy
Absolute Value of a Complex Number
Polar Coordinates - Division

37. 1
The Complex Numbers
i^2
can't get out of the complex numbers by adding (or subtracting) or multiplying two
multiplying complex numbers

38. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Argand diagram
Complex Multiplication
Complex Subtraction
Roots of Unity

39. (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
How to solve (2i+3)/(9-i)
Polar Coordinates - Arg(z*)
the distance from z to the origin in the complex plane
v(-1)

40. 1
Irrational Number
i^2
i^0
For real a and b - a + bi = 0 if and only if a = b = 0

41. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
i^2 = -1
z + z*
e^(ln z)

42. When two complex numbers are added together.
irrational
Imaginary number
Complex Addition
i²

43. A complex number may be taken to the power of another complex number.
Polar Coordinates - cos?
De Moivre's Theorem
Complex Exponentiation
has a solution.

44. (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)
Complex Number Formula
integers
Field

45. Any number not rational
zz*
irrational
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate pairs

46. Divide moduli and subtract arguments
has a solution.
Polar Coordinates - r
Polar Coordinates - Division
Field

47. 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.
Imaginary number
Complex numbers are points in the plane
(cos? +isin?)n
How to add and subtract complex numbers (2-3i)-(4+6i)

48. Cos n? + i sin n? (for all n integers)
Euler Formula
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(cos? +isin?)n

49. When two complex numbers are subtracted from one another.
ln z
Complex Subtraction
adding complex numbers
the complex numbers

50. A+bi
Absolute Value of a Complex Number
rational
Complex Number Formula
Complex Numbers: Multiply