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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.
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. Equivalent to an Imaginary Unit.
Absolute Value of a Complex Number
standard form of complex numbers
Imaginary number
Polar Coordinates - z

2. ½(e^(-y) +e^(y)) = cosh y
the vector (a -b)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Division
cos iy

3. The reals are just the
has a solution.
Imaginary Unit
x-axis in the complex plane
z + z*

4. 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
v(-1)
Complex Multiplication
the vector (a -b)
Rules of Complex Arithmetic

5. A number that cannot be expressed as a fraction for any integer.
Euler's Formula
Rational Number
cosh²y - sinh²y
Irrational Number

6. x + iy = r(cos? + isin?) = re^(i?)
complex
Polar Coordinates - z
sin iy
integers

7. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Liouville's Theorem -
Real Numbers
The Complex Numbers

8. Where the curvature of the graph changes
Affix
point of inflection
sin iy
Polar Coordinates - Multiplication

9. When two complex numbers are divided.
the vector (a -b)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Division
cos z

10. Real and imaginary numbers
-1
complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Arg(z*)

11. Has exactly n roots by the fundamental theorem of algebra
Imaginary Unit
Any polynomial O(xn) - (n > 0)
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 ^ (z2)

12. I
How to solve (2i+3)/(9-i)
z - z*
the complex numbers
v(-1)

13. We see in this way that the distance between two points z and w in the complex plane is
interchangeable
Square Root
|z-w|
real

14. I = imaginary unit - i² = -1 or i = v-1
For real a and b - a + bi = 0 if and only if a = b = 0
transcendental
Imaginary Numbers
point of inflection

15. Imaginary number

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16. R^2 = x
sin iy
Square Root
Complex Number Formula
Affix

17. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
multiplying complex numbers
x-axis in the complex plane
We say that c+di and c-di are complex conjugates.
integers

18. For real a and b - a + bi =
Imaginary Numbers
conjugate
0 if and only if a = b = 0
Polar Coordinates - Arg(z*)

19. (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
Complex Conjugate
The Complex Numbers
i^2 = -1
How to solve (2i+3)/(9-i)

20. 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
Roots of Unity
transcendental
(a + c) + ( b + d)i
Real and Imaginary Parts

21. No i
real
Polar Coordinates - r
|z-w|
Complex numbers are points in the plane

22. All the powers of i can be written as
cos iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
real
four different numbers: i - -i - 1 - and -1.

23. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Affix
i^0
Irrational Number
How to add and subtract complex numbers (2-3i)-(4+6i)

24. 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
the complex numbers
How to solve (2i+3)/(9-i)
Imaginary Numbers
Complex numbers are points in the plane

25. Divide moduli and subtract arguments
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Division
a real number: (a + bi)(a - bi) = a² + b²
Complex Multiplication

26. 2a
interchangeable
multiplying complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z + z*

27. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
For real a and b - a + bi = 0 if and only if a = b = 0
radicals
multiply the numerator and the denominator by the complex conjugate of the denominator.
Roots of Unity

28. The square root of -1.
(cos? +isin?)n
Imaginary Numbers
Imaginary Unit
We say that c+di and c-di are complex conjugates.

29. V(zz*) = v(a² + b²)
Any polynomial O(xn) - (n > 0)
|z| = mod(z)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix

30. In this amazing number field every algebraic equation in z with complex coefficients
e^(ln z)
'i'
has a solution.
|z-w|

31. 1
0 if and only if a = b = 0
zz*
rational
i²

32. V(x² + y²) = |z|
Polar Coordinates - r
Square Root
conjugate
Any polynomial O(xn) - (n > 0)

33. Have radical
Absolute Value of a Complex Number
ln z
radicals
Complex Multiplication

34. The modulus of the complex number z= a + ib now can be interpreted as
Field
the distance from z to the origin in the complex plane
Integers
rational

35. (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
four different numbers: i - -i - 1 - and -1.
i^1
How to multiply complex nubers(2+i)(2i-3)
the distance from z to the origin in the complex plane

36. Not on the numberline
Complex Division
Polar Coordinates - r
non-integers
cos z

37. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Complex Addition
Complex Numbers: Add & subtract
|z| = mod(z)

38. 1
e^(ln z)
Field
cosh²y - sinh²y
multiply the numerator and the denominator by the complex conjugate of the denominator.

39. A complex number and its conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Liouville's Theorem -
conjugate pairs
Complex Exponentiation

40. Any number not rational
irrational
We say that c+di and c-di are complex conjugates.
Polar Coordinates - cos?
Argand diagram

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

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42. 1
Rules of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
i^2
zz*

43. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - z?¹
the distance from z to the origin in the complex plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rational Number

44. (a + bi) = (c + bi) =
Argand diagram
rational
(a + c) + ( b + d)i
z1 ^ (z2)

45. Root negative - has letter i
interchangeable
We say that c+di and c-di are complex conjugates.
conjugate pairs
imaginary

46. To simplify a complex fraction
Argand diagram
The Complex Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Liouville's Theorem -

47. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
the vector (a -b)
Complex Number
Euler Formula

48. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
point of inflection
z - z*
radicals

49. 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.
conjugate pairs
i^1
v(-1)
Complex numbers are points in the plane

50. 2ib
z - z*
v(-1)
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