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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.
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. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Complex Number
has a solution.
adding complex numbers
the distance from z to the origin in the complex plane

2. E ^ (z2 ln z1)
z1 ^ (z2)
Polar Coordinates - z?¹
z - z*
-1

3. Have radical
conjugate pairs
subtracting complex numbers
x-axis in the complex plane
radicals

4. A+bi
z - z*
How to solve (2i+3)/(9-i)
Complex numbers are points in the plane
Complex Number Formula

5. When two complex numbers are divided.
Complex Numbers: Add & subtract
Complex Division
the distance from z to the origin in the complex plane
Polar Coordinates - r

6. Equivalent to an Imaginary Unit.
|z| = mod(z)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number
z - z*

7. I
i^1
(cos? +isin?)n
Field
Euler's Formula

8. When two complex numbers are added together.
i^0
subtracting complex numbers
Euler Formula
Complex Addition

9. 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.
real
i^4
Argand diagram
Complex Numbers: Multiply

10. x / r
the vector (a -b)
Polar Coordinates - cos?
Complex Addition
Complex Numbers: Add & subtract

11. For real a and b - a + bi =
0 if and only if a = b = 0
Polar Coordinates - Multiplication by i
Argand diagram
has a solution.

12. 1st. Rule of Complex Arithmetic
i^2 = -1
cosh²y - sinh²y
Rational Number
natural

13. Starts at 1 - does not include 0
Polar Coordinates - r
cos z
transcendental
natural

14. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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15. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
How to add and subtract complex numbers (2-3i)-(4+6i)
subtracting complex numbers
point of inflection

16. 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
multiply the numerator and the denominator by the complex conjugate of the denominator.
cos iy
Polar Coordinates - Division
Rules of Complex Arithmetic

17. 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*)
subtracting complex numbers
Polar Coordinates - Multiplication
Complex Division

18. 1
multiplying complex numbers
Complex Division
i^0
Polar Coordinates - z?¹

19. No i
complex
transcendental
i^4
real

20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
How to find any Power
Complex Conjugate
imaginary

21. Cos n? + i sin n? (for all n integers)
Polar Coordinates - cos?
|z-w|
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
(cos? +isin?)n

22. 1
has a solution.
i²
Real Numbers
Polar Coordinates - z

23. A number that can be expressed as a fraction p/q where q is not equal to 0.
i^0
Rational Number
zz*
Polar Coordinates - cos?

24. Like pi
i^0
the distance from z to the origin in the complex plane
Complex Division
transcendental

25. 2nd. Rule of Complex Arithmetic

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26. Divide moduli and subtract arguments
ln z
Polar Coordinates - Division
sin z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. 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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28. All numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex
z + z*
Liouville's Theorem -

29. A subset within a field.
Subfield
complex
Polar Coordinates - z?¹
rational

30. I = imaginary unit - i² = -1 or i = v-1
Complex Exponentiation
Complex Numbers: Multiply
Imaginary Numbers
i^3

31. 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
multiply the numerator and the denominator by the complex conjugate of the denominator.
Euler Formula
non-integers
We say that c+di and c-di are complex conjugates.

32. 2a
real
z + z*
multiplying complex numbers
complex numbers

33. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - Multiplication by i
Polar Coordinates - r
standard form of complex numbers
Polar Coordinates - z

34. The field of all rational and irrational numbers.
Polar Coordinates - sin?
Complex Division
Real Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

35. 2ib
z - z*
complex numbers
e^(ln z)
De Moivre's Theorem

36. ½(e^(-y) +e^(y)) = cosh y
i^4
cos iy
i^3
Complex Number Formula

37. E^(ln r) e^(i?) e^(2pin)
ln z
multiplying complex numbers
i^3
e^(ln z)

38. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Polar Coordinates - r
cos iy
Complex Multiplication

39. Where the curvature of the graph changes
rational
Polar Coordinates - z?¹
Complex Conjugate
point of inflection

40. 1
Affix
Rules of Complex Arithmetic
i^4
conjugate

41. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
'i'
integers
complex numbers

42. R^2 = x
the distance from z to the origin in the complex plane
Polar Coordinates - sin?
conjugate pairs
Square Root

43. Numbers on a numberline
'i'
De Moivre's Theorem
natural
integers

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
v(-1)
Polar Coordinates - Division
cos iy
Complex Numbers: Add & subtract

45. (a + bi) = (c + bi) =
For real a and b - a + bi = 0 if and only if a = b = 0
imaginary
Polar Coordinates - r
(a + c) + ( b + d)i

46. Derives z = a+bi
sin z
Euler Formula
Liouville's Theorem -
Complex Number Formula

47. Any number not rational
irrational
(a + c) + ( b + d)i
(cos? +isin?)n
Real Numbers

48. R?¹(cos? - isin?)
i^4
Polar Coordinates - z?¹
point of inflection
Subfield

49. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Rules of Complex Arithmetic
multiplying complex numbers
Roots of Unity
Field

50. Imaginary number

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