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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. x / r
i²
De Moivre's Theorem
point of inflection
Polar Coordinates - cos?

2. Root negative - has letter i
Polar Coordinates - Multiplication by i
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Exponentiation
imaginary

3. y / r
x-axis in the complex plane
Polar Coordinates - sin?
i^2 = -1
Argand diagram

4. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
sin iy
|z-w|
i^2

5. R?¹(cos? - isin?)
|z-w|
Polar Coordinates - z?¹
Complex Number Formula
Polar Coordinates - cos?

6. x + iy = r(cos? + isin?) = re^(i?)
For real a and b - a + bi = 0 if and only if a = b = 0
Rules of Complex Arithmetic
Polar Coordinates - z
Field

7. 2a
conjugate
i^0
real
z + z*

8. 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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9. Cos n? + i sin n? (for all n integers)
the distance from z to the origin in the complex plane
conjugate
(cos? +isin?)n
x-axis in the complex plane

10. Equivalent to an Imaginary Unit.
natural
Real and Imaginary Parts
z1 ^ (z2)
Imaginary number

11. I^2 =
Complex Number Formula
standard form of complex numbers
-1
i^3

12. A + bi
x-axis in the complex plane
Polar Coordinates - sin?
complex numbers
standard form of complex numbers

13. (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)
How to find any Power
Complex Division
complex numbers

14. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Integers
integers
i²
How to add and subtract complex numbers (2-3i)-(4+6i)

15. When two complex numbers are subtracted from one another.
radicals
x-axis in the complex plane
Complex Subtraction
Real and Imaginary Parts

16. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
point of inflection
Complex Conjugate
z1 ^ (z2)
the complex numbers

17. A complex number and its conjugate
Polar Coordinates - z
Imaginary number
conjugate pairs
Complex Subtraction

18. When two complex numbers are divided.
z1 ^ (z2)
Complex Division
Imaginary Unit
conjugate pairs

19. 2nd. Rule of Complex Arithmetic

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20. We can also think of the point z= a+ ib as
Real Numbers
Field
the vector (a -b)
Affix

21. (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
Complex Number Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to solve (2i+3)/(9-i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

22. The field of all rational and irrational numbers.
four different numbers: i - -i - 1 - and -1.
i²
Real Numbers
Complex Numbers: Add & subtract

23. The product of an imaginary number and its conjugate is
e^(ln z)
Polar Coordinates - r
a real number: (a + bi)(a - bi) = a² + b²
The Complex Numbers

24. A+bi
radicals
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Number Formula
z1 / z2

25. R^2 = x
Complex Conjugate
Irrational Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Square Root

26. Every complex number has the 'Standard Form':
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Number Formula
z1 / z2
a + bi for some real a and b.

27. Has exactly n roots by the fundamental theorem of algebra
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)
Complex Conjugate
Real Numbers

28. ? = -tan?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Arg(z*)
(cos? +isin?)n
How to add and subtract complex numbers (2-3i)-(4+6i)

29. When two complex numbers are added together.
Polar Coordinates - z?¹
Polar Coordinates - sin?
Complex Addition
Complex Multiplication

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
Liouville's Theorem -
four different numbers: i - -i - 1 - and -1.
Rules of Complex Arithmetic
real

31. All numbers
has a solution.
complex
Complex Subtraction
|z| = mod(z)

32. Real and imaginary numbers
real
complex numbers
complex
imaginary

33. Imaginary number

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34. No i
i^3
How to add and subtract complex numbers (2-3i)-(4+6i)
real
Argand diagram

35. Numbers on a numberline
Argand diagram
Irrational Number
integers
z + z*

36. Any number not rational
(cos? +isin?)n
We say that c+di and c-di are complex conjugates.
Irrational Number
irrational

37. A number that can be expressed as a fraction p/q where q is not equal to 0.
subtracting complex numbers
Complex Subtraction
Polar Coordinates - Division
Rational Number

38. 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
Polar Coordinates - Multiplication
has a solution.
We say that c+di and c-di are complex conjugates.
standard form of complex numbers

39. I
Real Numbers
e^(ln z)
i^1
Liouville's Theorem -

40. The square root of -1.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Imaginary Unit
Imaginary Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

41. 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
i^2 = -1
Imaginary Unit
subtracting complex numbers
Complex Division

42. Rotates anticlockwise by p/2
Affix
v(-1)
z + z*
Polar Coordinates - Multiplication by i

43. 1
i^0
Irrational Number
z - z*
transcendental

44. 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
integers
multiplying complex numbers
natural

45. (a + bi) = (c + bi) =
Polar Coordinates - cos?
interchangeable
(a + c) + ( b + d)i
ln z

46. Divide moduli and subtract arguments
Polar Coordinates - Division
Complex Multiplication
Rational Number
point of inflection

47. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - Multiplication
Absolute Value of a Complex Number
i^0
Subfield

48. Derives z = a+bi
a real number: (a + bi)(a - bi) = a² + b²
irrational
Euler Formula
Complex Addition

49. Written as fractions - terminating + repeating decimals
multiply the numerator and the denominator by the complex conjugate of the denominator.
Irrational Number
Complex Conjugate
rational

50. ½(e^(-y) +e^(y)) = cosh y
Polar Coordinates - sin?
cos iy
non-integers
We say that c+di and c-di are complex conjugates.