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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. Any number not rational
x-axis in the complex plane
Polar Coordinates - Division
irrational
Complex numbers are points in the plane

2. 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
subtracting complex numbers
sin z
i^4
Imaginary Numbers

3. Real and imaginary numbers
complex numbers
Polar Coordinates - r
|z| = mod(z)
i^3

4. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
sin z
(a + c) + ( b + d)i
Integers
Euler Formula

5. 1
Complex Multiplication
Polar Coordinates - z?¹
i^2
i^1

6. (e^(iz) - e^(-iz)) / 2i
|z-w|
transcendental
sin z
the complex numbers

7. Divide moduli and subtract arguments
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
-1
Polar Coordinates - Division

8. I
How to multiply complex nubers(2+i)(2i-3)
v(-1)
natural
standard form of complex numbers

9. 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
z - z*
Rational Number
the vector (a -b)
the complex numbers

10. x + iy = r(cos? + isin?) = re^(i?)
Liouville's Theorem -
Polar Coordinates - z
Complex Division
e^(ln z)

11. 1
i^4
Roots of Unity
Real and Imaginary Parts
the complex numbers

12. 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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13. For real a and b - a + bi =
0 if and only if a = b = 0
conjugate pairs
the vector (a -b)
De Moivre's Theorem

14. Every complex number has the 'Standard Form':
subtracting complex numbers
a + bi for some real a and b.
complex
Complex Number Formula

15. The square root of -1.
Integers
|z-w|
Imaginary Unit
Polar Coordinates - Division

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

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17. All the powers of i can be written as
Complex Addition
Polar Coordinates - cos?
four different numbers: i - -i - 1 - and -1.
z + z*

18. A complex number may be taken to the power of another complex number.
Polar Coordinates - cos?
Complex Exponentiation
Complex Numbers: Multiply
Imaginary number

19. Root negative - has letter i
e^(ln z)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
radicals
imaginary

20. V(x² + y²) = |z|
Polar Coordinates - r
|z-w|
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
We say that c+di and c-di are complex conjugates.

21. A number that can be expressed as a fraction p/q where q is not equal to 0.
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?
Rational Number

22. y / r
Complex Exponentiation
z - z*
Polar Coordinates - r
Polar Coordinates - sin?

23. Where the curvature of the graph changes
imaginary
point of inflection
|z| = mod(z)
non-integers

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
Complex numbers are points in the plane
adding complex numbers
complex
Subfield

25. 2a
Affix
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4
z + z*

26. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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27. I
Real and Imaginary Parts
i^1
Polar Coordinates - z
Field

28. (a + bi) = (c + bi) =
Polar Coordinates - cos?
(a + c) + ( b + d)i
e^(ln z)
Every complex number has the 'Standard Form': a + bi for some real a and b.

29. Multiply moduli and add arguments
Any polynomial O(xn) - (n > 0)
Argand diagram
Polar Coordinates - Multiplication
z1 ^ (z2)

30. (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)
cos z
Imaginary Unit
(a + bi) = (c + bi) = (a + c) + ( b + d)i

31. 1
i²
Roots of Unity
Complex Addition
Polar Coordinates - Multiplication by i

32. Cos n? + i sin n? (for all n integers)
ln z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(cos? +isin?)n
'i'

33. Starts at 1 - does not include 0
Square Root
Polar Coordinates - sin?
natural
cos iy

34. E^(ln r) e^(i?) e^(2pin)
(a + c) + ( b + d)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
non-integers
e^(ln z)

35. 1
i^0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)
'i'

36. When two complex numbers are subtracted from one another.
i^2 = -1
the complex numbers
Complex Subtraction
Roots of Unity

37. Have radical
Complex Addition
Complex Subtraction
radicals
We say that c+di and c-di are complex conjugates.

38. Numbers on a numberline
integers
irrational
sin iy
Complex Number Formula

39. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - z
Polar Coordinates - cos?
Any polynomial O(xn) - (n > 0)
Imaginary Unit

40. We can also think of the point z= a+ ib as
Any polynomial O(xn) - (n > 0)
-1
the vector (a -b)
Complex Exponentiation

41. 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
four different numbers: i - -i - 1 - and -1.
Complex Conjugate
Complex Multiplication
We say that c+di and c-di are complex conjugates.

42. 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
Any polynomial O(xn) - (n > 0)
point of inflection
interchangeable

43. V(zz*) = v(a² + b²)
De Moivre's Theorem
|z| = mod(z)
Complex numbers are points in the plane
Complex Addition

44. A+bi
conjugate pairs
cosh²y - sinh²y
Complex Number Formula
|z-w|

45. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
Real and Imaginary Parts
conjugate pairs
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

46. Rotates anticlockwise by p/2
Rational Number
Euler Formula
Polar Coordinates - Multiplication by i
i^4

47. A complex number and its conjugate
Polar Coordinates - sin?
i^4
How to find any Power
conjugate pairs

48. R^2 = x
Real and Imaginary Parts
Square Root
Euler Formula
Integers

49. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
standard form of complex numbers
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Roots of Unity

50. Equivalent to an Imaginary Unit.
Imaginary number
-1
natural
rational