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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. ? = -tan?
integers
imaginary
Polar Coordinates - Arg(z*)
adding complex numbers

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
ln z
How to solve (2i+3)/(9-i)
imaginary
Complex Numbers: Add & subtract

3. 2ib
z - z*
x-axis in the complex plane
standard form of complex numbers
multiplying complex numbers

4. All numbers
complex
ln z
Polar Coordinates - z?¹
Subfield

5. 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
How to solve (2i+3)/(9-i)
Rules of Complex Arithmetic
Imaginary Numbers
Affix

6. (a + bi) = (c + bi) =
Argand diagram
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a real number: (a + bi)(a - bi) = a² + b²
(a + c) + ( b + d)i

7. When two complex numbers are divided.
Polar Coordinates - Division
Roots of Unity
Complex Division
0 if and only if a = b = 0

8. Imaginary number

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9. I = imaginary unit - i² = -1 or i = v-1
Complex Exponentiation
Imaginary Numbers
the vector (a -b)
multiply the numerator and the denominator by the complex conjugate of the denominator.

10. x + iy = r(cos? + isin?) = re^(i?)
Complex Number
i^2 = -1
Polar Coordinates - z
Irrational Number

11. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Polar Coordinates - cos?
'i'
cos z
ln z

12. In this amazing number field every algebraic equation in z with complex coefficients
Complex Exponentiation
Complex numbers are points in the plane
has a solution.
i^3

13. E^(ln r) e^(i?) e^(2pin)
z - z*
Complex Number Formula
conjugate
e^(ln z)

14. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
ln z
Irrational Number
|z| = mod(z)

15. A complex number may be taken to the power of another complex number.
point of inflection
Complex Exponentiation
multiply the numerator and the denominator by the complex conjugate of the denominator.
z - z*

16. x / r
Polar Coordinates - cos?
i^2 = -1
point of inflection
Every complex number has the 'Standard Form': a + bi for some real a and b.

17. 2nd. Rule of Complex Arithmetic

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18. 1
Imaginary number
Polar Coordinates - r
i^0
'i'

19. (e^(iz) - e^(-iz)) / 2i
e^(ln z)
radicals
x-axis in the complex plane
sin z

20. All the powers of i can be written as
Rules of Complex Arithmetic
Polar Coordinates - sin?
cosh²y - sinh²y
four different numbers: i - -i - 1 - and -1.

21. (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)
rational
'i'
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

22. Any number not rational
irrational
v(-1)
Polar Coordinates - r
Absolute Value of a Complex Number

23. Root negative - has letter i
imaginary
Imaginary Unit
four different numbers: i - -i - 1 - and -1.
i^2 = -1

24. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
adding complex numbers
i^2 = -1
Roots of Unity
Polar Coordinates - z

25. (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)
Argand diagram
Euler's Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

26. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Polar Coordinates - Multiplication
-1
Imaginary number

27. 1
cosh²y - sinh²y
Complex Exponentiation
z + z*
i^3

28. (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)
We say that c+di and c-di are complex conjugates.
conjugate
the vector (a -b)

29. 3
i^3
i²
radicals
0 if and only if a = b = 0

30. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
zz*
Imaginary Unit
The Complex Numbers
multiplying complex numbers

31. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Affix
i^1
Complex Exponentiation

32. Starts at 1 - does not include 0
natural
non-integers
Imaginary Numbers
Complex Number

33. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
i^0
Roots of Unity
x-axis in the complex plane
De Moivre's Theorem

34. z1z2* / |z2|²
i^3
z1 / z2
natural
Imaginary Unit

35. 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.
Rational Number
Euler Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.

36. A number that cannot be expressed as a fraction for any integer.
We say that c+di and c-di are complex conjugates.
Irrational Number
Polar Coordinates - Multiplication
irrational

37. When two complex numbers are subtracted from one another.
i^1
Complex Addition
Complex Subtraction
Polar Coordinates - Multiplication

38. E ^ (z2 ln z1)
Polar Coordinates - z?¹
z1 ^ (z2)
How to multiply complex nubers(2+i)(2i-3)
For real a and b - a + bi = 0 if and only if a = b = 0

39. A plot of complex numbers as points.
Polar Coordinates - z
Argand diagram
i^3
i^1

40. A complex number and its conjugate
complex
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate pairs
a + bi for some real a and b.

41. R^2 = x
the distance from z to the origin in the complex plane
Argand diagram
Square Root
the complex numbers

42. We can also think of the point z= a+ ib as
Complex Number
the vector (a -b)
a real number: (a + bi)(a - bi) = a² + b²
'i'

43. Divide moduli and subtract arguments
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Division
adding complex numbers
Subfield

44. Cos n? + i sin n? (for all n integers)
We say that c+di and c-di are complex conjugates.
(cos? +isin?)n
Affix
Any polynomial O(xn) - (n > 0)

45. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate
the complex numbers
Polar Coordinates - Multiplication

46. (a + bi)(c + bi) =
How to multiply complex nubers(2+i)(2i-3)
the vector (a -b)
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

47. A + bi
irrational
standard form of complex numbers
Polar Coordinates - z
Absolute Value of a Complex Number

48. Given (4-2i) the complex conjugate would be (4+2i)
i^1
x-axis in the complex plane
cosh²y - sinh²y
Complex Conjugate

49. ½(e^(iz) + e^(-iz))
Complex Division
Any polynomial O(xn) - (n > 0)
rational
cos z

50. When two complex numbers are multipiled together.
Complex Multiplication
Field
multiplying complex numbers
cos iy