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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. The field of all rational and irrational numbers.
Any polynomial O(xn) - (n > 0)
Real Numbers
Field
De Moivre's Theorem

2. The reals are just the
v(-1)
x-axis in the complex plane
integers
Complex numbers are points in the plane

3. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
|z| = mod(z)
Polar Coordinates - sin?

4. Given (4-2i) the complex conjugate would be (4+2i)
0 if and only if a = b = 0
Square Root
Complex Conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.

5. Derives z = a+bi
Irrational Number
Polar Coordinates - z?¹
Polar Coordinates - cos?
Euler Formula

6. Multiply moduli and add arguments
How to multiply complex nubers(2+i)(2i-3)
Imaginary number
Polar Coordinates - Multiplication
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

7. (a + bi) = (c + bi) =
Euler's Formula
|z-w|
a + bi for some real a and b.
(a + c) + ( b + d)i

8. 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^0
Integers
conjugate

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

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10. Where the curvature of the graph changes
transcendental
point of inflection
We say that c+di and c-di are complex conjugates.
Polar Coordinates - cos?

11. Starts at 1 - does not include 0
x-axis in the complex plane
the distance from z to the origin in the complex plane
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i

12. x + iy = r(cos? + isin?) = re^(i?)
Real and Imaginary Parts
Polar Coordinates - z
point of inflection
Field

13. 4th. Rule of Complex Arithmetic
How to add and subtract complex numbers (2-3i)-(4+6i)
Rules of Complex Arithmetic
z - z*
(a + bi) = (c + bi) = (a + c) + ( b + d)i

14. I
v(-1)
z1 ^ (z2)
Polar Coordinates - z
non-integers

15. R?¹(cos? - isin?)
Affix
Polar Coordinates - z?¹
Imaginary number
zz*

16. 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
Complex Subtraction
Imaginary Unit
the complex numbers
(cos? +isin?)n

17. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
i^2 = -1
(a + c) + ( b + d)i
Complex Numbers: Add & subtract

18. 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
Complex Subtraction
subtracting complex numbers
cos iy
We say that c+di and c-di are complex conjugates.

19. R^2 = x
zz*
Polar Coordinates - Division
Integers
Square Root

20. V(x² + y²) = |z|
zz*
Euler's Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r

21. A² + b² - real and non negative
four different numbers: i - -i - 1 - and -1.
zz*
has a solution.
z1 ^ (z2)

22. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Irrational Number
conjugate
zz*
Integers

23. Equivalent to an Imaginary Unit.
a + bi for some real a and b.
|z-w|
Imaginary number
Polar Coordinates - Multiplication by i

24. A + bi
How to solve (2i+3)/(9-i)
transcendental
four different numbers: i - -i - 1 - and -1.
standard form of complex numbers

25. Imaginary number

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26. x / r
z + z*
De Moivre's Theorem
i^0
Polar Coordinates - cos?

27. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Multiplication
ln z
Polar Coordinates - Multiplication by i
Rational Number

28. 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
Polar Coordinates - sin?
Real and Imaginary Parts
multiply the numerator and the denominator by the complex conjugate of the denominator.
Any polynomial O(xn) - (n > 0)

29. 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
adding complex numbers
the complex numbers
Real Numbers
transcendental

30. A complex number may be taken to the power of another complex number.
i^2 = -1
Complex Exponentiation
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
-1

31. (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
Absolute Value of a Complex Number
How to solve (2i+3)/(9-i)
z1 ^ (z2)
Complex Exponentiation

32. ? = -tan?
Complex Conjugate
Polar Coordinates - Arg(z*)
Complex Numbers: Add & subtract
Polar Coordinates - z

33. I
sin z
z - z*
i^1
sin iy

34. xpressions such as ``the complex number z'' - and ``the point z'' are now
Argand diagram
Complex Number
How to find any Power
interchangeable

35. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Absolute Value of a Complex Number
Polar Coordinates - r
point of inflection

36. Like pi
transcendental
How to find any Power
Integers
ln z

37. All the powers of i can be written as
Roots of Unity
a real number: (a + bi)(a - bi) = a² + b²
sin z
four different numbers: i - -i - 1 - and -1.

38. 3
Complex Division
Square Root
i^3
z + z*

39. 1
i²
Square Root
the complex numbers
(a + c) + ( b + d)i

40. Root negative - has letter i
De Moivre's Theorem
Euler Formula
imaginary
Polar Coordinates - cos?

41. The modulus of the complex number z= a + ib now can be interpreted as
Integers
the distance from z to the origin in the complex plane
Euler Formula
Imaginary Unit

42. A+bi
rational
transcendental
i^4
Complex Number Formula

43. E ^ (z2 ln z1)
imaginary
Complex Number
z1 ^ (z2)
Complex Number Formula

44. 1
integers
i^4
sin iy
Absolute Value of a Complex Number

45. When two complex numbers are subtracted from one another.
cosh²y - sinh²y
subtracting complex numbers
i^3
Complex Subtraction

46. All numbers
z1 ^ (z2)
Argand diagram
complex
real

47. A plot of complex numbers as points.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers
Argand diagram
Imaginary number

48. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
cosh²y - sinh²y
Absolute Value of a Complex Number
Complex Exponentiation
Affix

49. Not on the numberline
Imaginary Numbers
Imaginary number
point of inflection
non-integers

50. E^(ln r) e^(i?) e^(2pin)
Absolute Value of a Complex Number
real
Euler's Formula
e^(ln z)