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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. I
Complex Multiplication
cos iy
four different numbers: i - -i - 1 - and -1.
v(-1)

2. 1
Imaginary Unit
i^2
zz*
multiplying complex numbers

3. 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
zz*
the complex numbers
i^0
i^2 = -1

4. 2a
Roots of Unity
the complex numbers
z1 ^ (z2)
z + z*

5. Have radical
complex numbers
a + bi for some real a and b.
Subfield
radicals

6. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Argand diagram
Complex numbers are points in the plane
Complex Conjugate

7. A+bi
conjugate pairs
Complex Number Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
Every complex number has the 'Standard Form': a + bi for some real a and b.

8. Where the curvature of the graph changes
subtracting complex numbers
conjugate
transcendental
point of inflection

9. A number that cannot be expressed as a fraction for any integer.
transcendental
ln z
Irrational Number
Polar Coordinates - z

10. 5th. Rule of Complex Arithmetic
irrational
the vector (a -b)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

11. Cos n? + i sin n? (for all n integers)
complex numbers
Affix
How to solve (2i+3)/(9-i)
(cos? +isin?)n

12. A complex number may be taken to the power of another complex number.
Complex Exponentiation
interchangeable
complex
the complex numbers

13. When two complex numbers are divided.
|z-w|
Complex Division
standard form of complex numbers
We say that c+di and c-di are complex conjugates.

14. Any number not rational
Complex Division
irrational
i^3
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

15. In this amazing number field every algebraic equation in z with complex coefficients
Liouville's Theorem -
sin z
Argand diagram
has a solution.

16. (e^(iz) - e^(-iz)) / 2i
Affix
Subfield
real
sin z

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

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18. A plot of complex numbers as points.
The Complex Numbers
Argand diagram
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

19. Numbers on a numberline
radicals
Complex Division
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
integers

20. Root negative - has letter i
i^1
imaginary
'i'
cos z

21. Has exactly n roots by the fundamental theorem of algebra
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
cos z
Any polynomial O(xn) - (n > 0)

22. 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 Subtraction
Imaginary Numbers
i^1
Complex Number

23. Derives z = a+bi
Polar Coordinates - z
Euler Formula
radicals
Complex Multiplication

24. Rotates anticlockwise by p/2
Polar Coordinates - cos?
Complex Addition
Polar Coordinates - Multiplication by i
Argand diagram

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

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26. ½(e^(-y) +e^(y)) = cosh y
For real a and b - a + bi = 0 if and only if a = b = 0
the vector (a -b)
-1
cos iy

27. The modulus of the complex number z= a + ib now can be interpreted as
Subfield
radicals
the distance from z to the origin in the complex plane
Euler's Formula

28. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
(cos? +isin?)n
(a + bi) = (c + bi) = (a + c) + ( b + d)i
De Moivre's Theorem

29. A number that can be expressed as a fraction p/q where q is not equal to 0.
-1
Rational Number
cosh²y - sinh²y
v(-1)

30. 2nd. Rule of Complex Arithmetic

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31. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z1 ^ (z2)
x-axis in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)

32. 2ib
'i'
radicals
z - z*
Any polynomial O(xn) - (n > 0)

33. z1z2* / |z2|²
subtracting complex numbers
How to solve (2i+3)/(9-i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 / z2

34. Not on the numberline
'i'
conjugate pairs
non-integers
standard form of complex numbers

35. A complex number and its conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiplying complex numbers
conjugate pairs
Subfield

36. Like pi
(a + c) + ( b + d)i
transcendental
interchangeable
can't get out of the complex numbers by adding (or subtracting) or multiplying two

37. Divide moduli and subtract arguments
Polar Coordinates - Division
i^0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition

38. (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
Affix
How to multiply complex nubers(2+i)(2i-3)
i^0
real

39. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Any polynomial O(xn) - (n > 0)
Affix
Complex Conjugate

40. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
(cos? +isin?)n
Complex Subtraction
Absolute Value of a Complex Number
has a solution.

41. ? = -tan?
Polar Coordinates - Arg(z*)
conjugate
sin iy
Polar Coordinates - Division

42. (a + bi)(c + bi) =
sin iy
We say that c+di and c-di are complex conjugates.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Affix

43. I = imaginary unit - i² = -1 or i = v-1
Irrational Number
Imaginary Numbers
i^2 = -1
a + bi for some real a and b.

44. 3
-1
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Division
i^3

45. 1
Complex Multiplication
cos z
Polar Coordinates - r
i²

46. Starts at 1 - does not include 0
Polar Coordinates - z
non-integers
has a solution.
natural

47. When two complex numbers are subtracted from one another.
Complex Subtraction
cos z
z1 / z2
0 if and only if a = b = 0

48. The square root of -1.
Euler's Formula
Imaginary Unit
Imaginary Numbers
Rules of Complex Arithmetic

49. 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
(a + c) + ( b + d)i
-1
i^3
adding complex numbers

50. 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
z + z*
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Division
Rules of Complex Arithmetic