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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. (a + bi)(c + bi) =
the distance from z to the origin in the complex plane
Complex Division
imaginary
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

2. Rotates anticlockwise by p/2
i^3
De Moivre's Theorem
point of inflection
Polar Coordinates - Multiplication by i

3. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r
Complex numbers are points in the plane

4. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
How to add and subtract complex numbers (2-3i)-(4+6i)
Field
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Argand diagram

5. 3rd. Rule of Complex Arithmetic
z1 ^ (z2)
For real a and b - a + bi = 0 if and only if a = b = 0
a real number: (a + bi)(a - bi) = a² + b²
radicals

6. Not on the numberline
non-integers
irrational
Any polynomial O(xn) - (n > 0)
Euler Formula

7. 1
i^3
i²
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Division

8. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Complex Multiplication
sin z
0 if and only if a = b = 0

9. ½(e^(iz) + e^(-iz))
i^3
Integers
subtracting complex numbers
cos z

10. To simplify the square root of a negative number
Euler Formula
radicals
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
sin z

11. The modulus of the complex number z= a + ib now can be interpreted as
Euler's Formula
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the distance from z to the origin in the complex plane
|z-w|

12. (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
sin z
How to solve (2i+3)/(9-i)
Any polynomial O(xn) - (n > 0)
Complex Addition

13. A subset within a field.
Imaginary number
Subfield
Complex Numbers: Multiply
0 if and only if a = b = 0

14. 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
i^3
has a solution.
|z-w|

15. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Square Root
Roots of Unity
ln z
radicals

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

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17. A number that can be expressed as a fraction p/q where q is not equal to 0.
Roots of Unity
Rational Number
Field
i^2

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

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19. E ^ (z2 ln z1)
Complex Division
Polar Coordinates - z
z1 ^ (z2)
For real a and b - a + bi = 0 if and only if a = b = 0

20. Where the curvature of the graph changes
Rules of Complex Arithmetic
point of inflection
i^1
Subfield

21. 5th. Rule of Complex Arithmetic
Affix
Real and Imaginary Parts
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z

22. To simplify a complex fraction
Complex Subtraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Division
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. Given (4-2i) the complex conjugate would be (4+2i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^0
Complex Conjugate
cos iy

24. 1st. Rule of Complex Arithmetic
Field
i^2 = -1
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Multiplication

25. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
irrational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)
multiplying complex numbers

26. 1
Complex Number Formula
i^0
Euler's Formula
sin z

27. E^(ln r) e^(i?) e^(2pin)
Euler's Formula
z - z*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)

28. For real a and b - a + bi =
Field
0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic

29. 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
e^(ln z)
Roots of Unity
Complex Numbers: Add & subtract
standard form of complex numbers

30. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
transcendental
Integers
i^3
cos z

31. Real and imaginary numbers
transcendental
complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0

32. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
has a solution.
How to add and subtract complex numbers (2-3i)-(4+6i)
z - z*
the complex numbers

33. Derives z = a+bi
i^2 = -1
Complex Multiplication
Euler Formula
i^4

34. 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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35. In this amazing number field every algebraic equation in z with complex coefficients
multiply the numerator and the denominator by the complex conjugate of the denominator.
has a solution.
zz*
Any polynomial O(xn) - (n > 0)

36. A complex number may be taken to the power of another complex number.
Field
cosh²y - sinh²y
Complex Exponentiation
adding complex numbers

37. The complex number z representing a+bi.
i^0
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Arg(z*)
Affix

38. All numbers
Polar Coordinates - Arg(z*)
complex
Polar Coordinates - Division
adding complex numbers

39. 4th. Rule of Complex Arithmetic
transcendental
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number Formula
z1 ^ (z2)

40. 3
i^3
Polar Coordinates - Division
Imaginary number
Roots of Unity

41. Equivalent to an Imaginary Unit.
Imaginary number
point of inflection
0 if and only if a = b = 0
Irrational Number

42. A complex number and its conjugate
Euler Formula
conjugate pairs
Polar Coordinates - Multiplication by i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

43. 2a
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Roots of Unity
z + z*
|z| = mod(z)

44. Imaginary number

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45. We see in this way that the distance between two points z and w in the complex plane is
Imaginary number
|z-w|
z1 / z2
Euler's Formula

46. Written as fractions - terminating + repeating decimals
'i'
z1 / z2
rational
0 if and only if a = b = 0

47. 1
Absolute Value of a Complex Number
interchangeable
complex
i^4

48. V(x² + y²) = |z|
Subfield
adding complex numbers
The Complex Numbers
Polar Coordinates - r

49. xpressions such as ``the complex number z'' - and ``the point z'' are now
Imaginary Numbers
has a solution.
radicals
interchangeable

50. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
radicals
Complex Numbers: Multiply
How to find any Power
z1 / z2