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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
interchangeable
ln z
point of inflection
Polar Coordinates - cos?

2. Multiply moduli and add arguments
Square Root
Polar Coordinates - Multiplication
radicals
How to solve (2i+3)/(9-i)

3. Written as fractions - terminating + repeating decimals
Imaginary Unit
-1
Irrational Number
rational

4. All numbers
complex
Polar Coordinates - sin?
i^0
Rules of Complex Arithmetic

5. We see in this way that the distance between two points z and w in the complex plane is
the complex numbers
Euler Formula
|z-w|
real

6. A number that can be expressed as a fraction p/q where q is not equal to 0.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Rational Number
Complex Addition
(a + bi) = (c + bi) = (a + c) + ( b + d)i

7. R?¹(cos? - isin?)
i^4
Polar Coordinates - Multiplication by i
Polar Coordinates - z?¹
Imaginary Unit

8. 3
z - z*
i^3
four different numbers: i - -i - 1 - and -1.
Argand diagram

9. 1
Polar Coordinates - Multiplication by i
i^0
We say that c+di and c-di are complex conjugates.
Imaginary Numbers

10. 2nd. Rule of Complex Arithmetic

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11. y / r
Polar Coordinates - z?¹
radicals
i^1
Polar Coordinates - sin?

12. 2ib
z - z*
cos z
radicals
z1 / z2

13. Not on the numberline
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Numbers: Add & subtract
i^2
non-integers

14. 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
Affix
standard form of complex numbers
i^1

15. ? = -tan?
Polar Coordinates - Arg(z*)
multiplying complex numbers
complex
the complex numbers

16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
How to add and subtract complex numbers (2-3i)-(4+6i)
Argand diagram
-1
Roots of Unity

17. When two complex numbers are divided.
Irrational Number
Complex Division
adding complex numbers
the complex numbers

18. A complex number may be taken to the power of another complex number.
Complex Numbers: Add & subtract
Integers
Complex Exponentiation
Polar Coordinates - cos?

19. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
ln z
Complex Conjugate
i^3

20. 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
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Number Formula
integers

21. Any number not rational
complex
cos z
For real a and b - a + bi = 0 if and only if a = b = 0
irrational

22. Has exactly n roots by the fundamental theorem of algebra
four different numbers: i - -i - 1 - and -1.
ln z
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0

23. To simplify a complex fraction
0 if and only if a = b = 0
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^3
zz*

24. The product of an imaginary number and its conjugate is
z1 / z2
x-axis in the complex plane
Complex Conjugate
a real number: (a + bi)(a - bi) = a² + b²

25. A + bi
Polar Coordinates - cos?
standard form of complex numbers
i²
zz*

26. Given (4-2i) the complex conjugate would be (4+2i)
non-integers
Complex Conjugate
Complex Numbers: Multiply
Complex numbers are points in the plane

27. No i
real
radicals
imaginary
the complex numbers

28. 4th. Rule of Complex Arithmetic
the vector (a -b)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Subtraction
How to find any Power

29. I
v(-1)
irrational
natural
We say that c+di and c-di are complex conjugates.

30. (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
subtracting complex numbers
How to multiply complex nubers(2+i)(2i-3)
Complex Exponentiation
Complex Addition

31. Every complex number has the 'Standard Form':
De Moivre's Theorem
a + bi for some real a and b.
Complex Multiplication
cos iy

32. Imaginary number

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33. R^2 = x
Complex Subtraction
Square Root
cos z
a real number: (a + bi)(a - bi) = a² + b²

34. z1z2* / |z2|²
z1 / z2
i²
non-integers
-1

35. Derives z = a+bi
z1 ^ (z2)
Euler Formula
zz*
Subfield

36. I
Polar Coordinates - Arg(z*)
Real Numbers
i^1
ln z

37. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
Rational Number
Rules of Complex Arithmetic
How to solve (2i+3)/(9-i)

38. (a + bi)(c + bi) =
-1
cos z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
point of inflection

39. xpressions such as ``the complex number z'' - and ``the point z'' are now
For real a and b - a + bi = 0 if and only if a = b = 0
interchangeable
Polar Coordinates - Division
z1 / z2

40. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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41. To simplify the square root of a negative number
sin iy
How to find any Power
Polar Coordinates - Arg(z*)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

42. (e^(-y) - e^(y)) / 2i = i sinh y
zz*
sin iy
the vector (a -b)
z1 / z2

43. 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
Any polynomial O(xn) - (n > 0)
sin z
Complex Numbers: Add & subtract
For real a and b - a + bi = 0 if and only if a = b = 0

44. The field of all rational and irrational numbers.
Real Numbers
zz*
cos z
'i'

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

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46. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
transcendental
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Absolute Value of a Complex Number

47. Root negative - has letter i
transcendental
We say that c+di and c-di are complex conjugates.
imaginary
Liouville's Theorem -

48. A² + b² - real and non negative
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
zz*
the vector (a -b)
Imaginary Numbers

49. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
standard form of complex numbers
Any polynomial O(xn) - (n > 0)
multiplying complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.

50. 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.
Polar Coordinates - Division
Complex Number Formula
irrational