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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. V(zz*) = v(a² + b²)
four different numbers: i - -i - 1 - and -1.
Euler's Formula
|z| = mod(z)
transcendental

2. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Exponentiation
non-integers
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to add and subtract complex numbers (2-3i)-(4+6i)

3. When two complex numbers are subtracted from one another.
Polar Coordinates - r
Roots of Unity
Complex Subtraction
sin iy

4. A complex number and its conjugate
Rules of Complex Arithmetic
i^1
Polar Coordinates - Division
conjugate pairs

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

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6. 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 + bi for some real a and b.
subtracting complex numbers
Complex Numbers: Multiply
adding complex numbers

7. A complex number may be taken to the power of another complex number.
point of inflection
Complex Exponentiation
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

8. For real a and b - a + bi =
How to add and subtract complex numbers (2-3i)-(4+6i)
0 if and only if a = b = 0
De Moivre's Theorem
-1

9. 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
x-axis in the complex plane
Imaginary number
Real and Imaginary Parts
How to solve (2i+3)/(9-i)

10. I^2 =
a real number: (a + bi)(a - bi) = a² + b²
e^(ln z)
v(-1)
-1

11. 2a
Complex Number Formula
Affix
z + z*
z1 ^ (z2)

12. Not on the numberline
transcendental
non-integers
0 if and only if a = b = 0
e^(ln z)

13. The square root of -1.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary Unit
z - z*
subtracting complex numbers

14. 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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15. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
cosh²y - sinh²y
Complex Subtraction
rational

16. 3
Argand diagram
Any polynomial O(xn) - (n > 0)
z1 ^ (z2)
i^3

17. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
the vector (a -b)
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate
Complex Numbers: Add & subtract

18. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Imaginary Unit
Field
Complex Numbers: Add & subtract
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

19. x + iy = r(cos? + isin?) = re^(i?)
|z-w|
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z
complex numbers

20. In this amazing number field every algebraic equation in z with complex coefficients
zz*
has a solution.
De Moivre's Theorem
How to solve (2i+3)/(9-i)

21. The product of an imaginary number and its conjugate is
Rules of Complex Arithmetic
z - z*
Complex Number Formula
a real number: (a + bi)(a - bi) = a² + b²

22. No i
Polar Coordinates - cos?
Subfield
sin iy
real

23. A+bi
How to multiply complex nubers(2+i)(2i-3)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number Formula
sin z

24. 1
i^0
Imaginary Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
'i'

25. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
imaginary
Polar Coordinates - sin?
Imaginary Unit
Roots of Unity

26. 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.
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to find any Power
i²
adding complex numbers

27. 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
How to solve (2i+3)/(9-i)
How to add and subtract complex numbers (2-3i)-(4+6i)
The Complex Numbers

28. ? = -tan?
Complex Division
Polar Coordinates - Arg(z*)
conjugate
i^3

29. 1st. Rule of Complex Arithmetic
i^2 = -1
complex
Polar Coordinates - cos?
adding complex numbers

30. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
zz*
Integers
How to solve (2i+3)/(9-i)

31. Divide moduli and subtract arguments
Polar Coordinates - Division
Any polynomial O(xn) - (n > 0)
ln z
Complex Number

32. Every complex number has the 'Standard Form':
a + bi for some real a and b.
complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
subtracting complex numbers

33. Written as fractions - terminating + repeating decimals
z - z*
Real and Imaginary Parts
rational
radicals

34. (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 find any Power
standard form of complex numbers
Polar Coordinates - cos?
How to solve (2i+3)/(9-i)

35. 1
Square Root
i^2
has a solution.
i^4

36. 1
multiply the numerator and the denominator by the complex conjugate of the denominator.
i²
Complex Exponentiation
Real and Imaginary Parts

37. Derives z = a+bi
z + z*
cosh²y - sinh²y
z - z*
Euler Formula

38. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
Complex Multiplication
standard form of complex numbers

39. Numbers on a numberline
integers
non-integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Division

40. Equivalent to an Imaginary Unit.
complex numbers
Imaginary number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Unit

41. A plot of complex numbers as points.
Argand diagram
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
zz*
The Complex Numbers

42. 4th. Rule of Complex Arithmetic
Field
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^4

43. V(x² + y²) = |z|
a real number: (a + bi)(a - bi) = a² + b²
(cos? +isin?)n
Polar Coordinates - r
Polar Coordinates - z?¹

44. Starts at 1 - does not include 0
has a solution.
ln z
natural
How to multiply complex nubers(2+i)(2i-3)

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

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46. 5th. Rule of Complex Arithmetic
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
(cos? +isin?)n
conjugate

47. We see in this way that the distance between two points z and w in the complex plane is
the vector (a -b)
Integers
|z-w|
interchangeable

48. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
i^4

49. Imaginary number

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50. 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.
i^2 = -1
Complex numbers are points in the plane
Imaginary Unit
Imaginary Numbers