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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. ½(e^(-y) +e^(y)) = cosh y
cosh²y - sinh²y
cos iy
Subfield
Euler's Formula

2. Every complex number has the 'Standard Form':
irrational
i^2
the distance from z to the origin in the complex plane
a + bi for some real a and b.

3. Numbers on a numberline
integers
Euler Formula
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division

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

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5. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Real and Imaginary Parts
Square Root
Euler's Formula

6. x + iy = r(cos? + isin?) = re^(i?)
0 if and only if a = b = 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - cos?
Polar Coordinates - z

7. Multiply moduli and add arguments
Complex Conjugate
Polar Coordinates - Multiplication
rational
standard form of complex numbers

8. Real and imaginary numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
complex numbers
real
Polar Coordinates - z?¹

9. 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.
imaginary
z1 / z2
Complex numbers are points in the plane
integers

10. ? = -tan?
Polar Coordinates - Arg(z*)
complex
How to add and subtract complex numbers (2-3i)-(4+6i)
For real a and b - a + bi = 0 if and only if a = b = 0

11. 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
irrational
natural
Complex Numbers: Add & subtract
Euler's Formula

12. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Complex Exponentiation
standard form of complex numbers
Complex Number

13. Has exactly n roots by the fundamental theorem of algebra
Imaginary number
sin iy
Any polynomial O(xn) - (n > 0)
i^0

14. The field of all rational and irrational numbers.
Roots of Unity
Real Numbers
cos z
Square Root

15. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
i^0
'i'
Polar Coordinates - z

16. When two complex numbers are added together.
i^2 = -1
subtracting complex numbers
the complex numbers
Complex Addition

17. 1
Polar Coordinates - z?¹
i^0
Subfield
Absolute Value of a Complex Number

18. Written as fractions - terminating + repeating decimals
rational
real
the distance from z to the origin in the complex plane
Imaginary Unit

19. E^(ln r) e^(i?) e^(2pin)
i^4
e^(ln z)
Real Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)

20. A number that can be expressed as a fraction p/q where q is not equal to 0.
the complex numbers
Imaginary number
Rational Number
i^1

21. I
The Complex Numbers
Euler Formula
i^1
How to add and subtract complex numbers (2-3i)-(4+6i)

22. 1
i²
complex
Rational Number
0 if and only if a = b = 0

23. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Euler's Formula
adding complex numbers
(a + c) + ( b + d)i
The Complex Numbers

24. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
i^2
For real a and b - a + bi = 0 if and only if a = b = 0
interchangeable

25. E ^ (z2 ln z1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
'i'
Roots of Unity
z1 ^ (z2)

26. Starts at 1 - does not include 0
i^2 = -1
Roots of Unity
natural
How to find any Power

27. R?¹(cos? - isin?)
Integers
Polar Coordinates - z?¹
Irrational Number
complex

28. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - r
(cos? +isin?)n
Imaginary Numbers

29. Derives z = a+bi
Polar Coordinates - cos?
'i'
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler Formula

30. Cos n? + i sin n? (for all n integers)
The Complex Numbers
(cos? +isin?)n
multiply the numerator and the denominator by the complex conjugate of the denominator.
Rational Number

31. x / r
Real Numbers
Polar Coordinates - cos?
0 if and only if a = b = 0
Rational Number

32. A+bi
Complex Number Formula
-1
standard form of complex numbers
Complex Conjugate

33. 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.
the vector (a -b)
a + bi for some real a and b.
How to find any Power
i^0

34. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Polar Coordinates - z?¹
interchangeable
i^4

35. (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
How to solve (2i+3)/(9-i)
v(-1)
the vector (a -b)
Imaginary number

36. Where the curvature of the graph changes
point of inflection
How to find any Power
cosh²y - sinh²y
Polar Coordinates - Multiplication

37. y / r
four different numbers: i - -i - 1 - and -1.
How to solve (2i+3)/(9-i)
Polar Coordinates - sin?
standard form of complex numbers

38. 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
natural
Complex Number
Complex Addition
adding complex numbers

39. 1
non-integers
Integers
i^2
Irrational Number

40. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix
four different numbers: i - -i - 1 - and -1.
z1 ^ (z2)

41. z1z2* / |z2|²
multiplying complex numbers
z1 / z2
zz*
z + z*

42. A complex number and its conjugate
Real Numbers
radicals
standard form of complex numbers
conjugate pairs

43. A number that cannot be expressed as a fraction for any integer.
Real Numbers
Polar Coordinates - Division
Irrational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

44. I^2 =
-1
multiplying complex numbers
standard form of complex numbers
zz*

45. 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
point of inflection
Polar Coordinates - Multiplication
Argand diagram

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.'
Complex Number
Polar Coordinates - z?¹
Every complex number has the 'Standard Form': a + bi for some real a and b.
a + bi for some real a and b.

47. Root negative - has letter i
z - z*
imaginary
z + z*
i^1

48. 3rd. Rule of Complex Arithmetic
transcendental
For real a and b - a + bi = 0 if and only if a = b = 0
|z| = mod(z)
Complex Number

49. A plot of complex numbers as points.
the complex numbers
Argand diagram
Complex Multiplication
multiplying complex numbers

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

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