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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. Rotates anticlockwise by p/2
cos z
z1 ^ (z2)
Polar Coordinates - Multiplication by i
non-integers

2. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
cos z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication

3. x + iy = r(cos? + isin?) = re^(i?)
'i'
Polar Coordinates - z
z + z*
Rational Number

4. (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
x-axis in the complex plane
Polar Coordinates - z
How to solve (2i+3)/(9-i)
Complex Conjugate

5. 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.
Complex numbers are points in the plane
Polar Coordinates - Division
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Conjugate

6. A complex number and its conjugate
conjugate pairs
point of inflection
(cos? +isin?)n
|z| = mod(z)

7. z1z2* / |z2|²
Polar Coordinates - r
cos iy
Integers
z1 / z2

8. E ^ (z2 ln z1)
ln z
z1 ^ (z2)
interchangeable
Polar Coordinates - Division

9. 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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10. 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
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex numbers are points in the plane
interchangeable

11. 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.
Roots of Unity
Complex Exponentiation
multiplying complex numbers
Complex Numbers: Multiply

12. A number that can be expressed as a fraction p/q where q is not equal to 0.
four different numbers: i - -i - 1 - and -1.
the distance from z to the origin in the complex plane
Euler's Formula
Rational Number

13. 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
Real Numbers
Complex Numbers: Add & subtract
radicals

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

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15. I
natural
i^1
Complex Division
Rules of Complex Arithmetic

16. Written as fractions - terminating + repeating decimals
-1
Polar Coordinates - Multiplication
rational
How to find any Power

17. When two complex numbers are subtracted from one another.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Subtraction
For real a and b - a + bi = 0 if and only if a = b = 0
ln z

18. The field of all rational and irrational numbers.
sin iy
Imaginary Numbers
Real Numbers
conjugate

19. 1
Polar Coordinates - Division
i^4
irrational
Euler Formula

20. 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
De Moivre's Theorem
non-integers
Complex Numbers: Add & subtract
rational

21. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - z
|z-w|
Rational Number
We say that c+di and c-di are complex conjugates.

22. A+bi
complex numbers
point of inflection
Polar Coordinates - Division
Complex Number Formula

23. Multiply moduli and add arguments
z1 / z2
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Multiplication
Field

24. 4th. Rule of Complex Arithmetic
sin iy
Polar Coordinates - z
i^1
(a + bi) = (c + bi) = (a + c) + ( b + d)i

25. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - r
z1 ^ (z2)
Affix

26. 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
Polar Coordinates - Division
the complex numbers
|z-w|
adding complex numbers

27. The square root of -1.
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
Imaginary Unit

28. A² + b² - real and non negative
Complex Division
zz*
point of inflection
Complex Exponentiation

29. 3rd. Rule of Complex Arithmetic
the vector (a -b)
cos iy
Real Numbers
For real a and b - a + bi = 0 if and only if a = b = 0

30. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Irrational Number
point of inflection
For real a and b - a + bi = 0 if and only if a = b = 0

31. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
adding complex numbers
has a solution.
For real a and b - a + bi = 0 if and only if a = b = 0

32. The complex number z representing a+bi.
Imaginary Unit
Affix
z1 ^ (z2)
Polar Coordinates - r

33. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Multiplication by i
Complex Number
Complex Multiplication

34. (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
i^0
Euler Formula
Subfield
How to multiply complex nubers(2+i)(2i-3)

35. V(x² + y²) = |z|
Polar Coordinates - r
cosh²y - sinh²y
Polar Coordinates - Division
Real and Imaginary Parts

36. 1
Irrational Number
cosh²y - sinh²y
complex
'i'

37. Where the curvature of the graph changes
0 if and only if a = b = 0
Complex Conjugate
point of inflection
Complex Division

38. 1
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Division
i^2
i²

39. Divide moduli and subtract arguments
complex numbers
a + bi for some real a and b.
Field
Polar Coordinates - Division

40. R?¹(cos? - isin?)
point of inflection
Polar Coordinates - z?¹
interchangeable
cos iy

41. A + bi
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
non-integers
standard form of complex numbers

42. A complex number may be taken to the power of another complex number.
Euler Formula
Complex Exponentiation
Euler's Formula
Polar Coordinates - Arg(z*)

43. 1
Complex Numbers: Add & subtract
Complex Division
i^2 = -1
i²

44. Any number not rational
Complex Conjugate
Complex Division
irrational
a real number: (a + bi)(a - bi) = a² + b²

45. Have radical
i^4
radicals
We say that c+di and c-di are complex conjugates.
'i'

46. 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.'
Polar Coordinates - Multiplication by i
Complex Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - sin?

47. 3
We say that c+di and c-di are complex conjugates.
ln z
How to find any Power
i^3

48. 1st. Rule of Complex Arithmetic
ln z
radicals
Imaginary Unit
i^2 = -1

49. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
|z-w|
ln z
Absolute Value of a Complex Number
complex

50. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
x-axis in the complex plane
ln z
Euler's Formula